![home healthcare services, many service providers are struggling to find
cost-effective and efficient schedules to meet the expectations of the customers, the
management goals, as well as the desires of the healthcare workers. Home
healthcare service provides continue to expand. Consequently, healthcare worker
scheduling has become a large-scale combinatorial problem, inundated with a
myriad of constraints that have to be taken into account, including patients’ pref-
erences, visiting time windows, or travel times depending on the mode of transport.
Figure 1.5 presents a schematic of the home healthcare worker scheduling
problem. In general, the problem is defined thus a set of m available care workers
are given the responsibility to visit n patients for home-based medical care. Each
caregiver k (k = 1, 2, …, m) is supposed to serve a group of patients, where each
patient j (j = 1,2…, n) is to be visited within a given time window defined by
earliest start and latest start times, ej and lj, respectively. The aim is to minimize the
overall costs of visiting clients (Mutingi and Mbohwa 2013a, b). In this vein, a
penalty cost is incurred whenever a caregiver reaches the client earlier than ej or
later than lj. If aj denotes the caregiver’s arrival time at patient j, and pe and pl
denote the unit penalty costs incurred when the caregiver arrives too early or too
late, respectively, then max[0, ej − aj] and max[0, aj − lj] have to be minimized, to
maximize patient satisfaction. Furthermore, worker preferences should be taken
into account, if schedule quality is to be maximized (Mutingi and Mbohwa 2013a,
b). However, since planning for home healthcare schedules is especially compu-
tationally expensive, effective and efficient algorithms are very important.
A diagraph for a vehicle routing problem
Operations 1,2 3,4,5 6,7
Machines v1 v2 v3
Group structure
0
40 20
13
21
70
20
18
21
4
2
3
1
5
6
7
18
12
(a)
(b)
Fig. 1.4 A typical vehicle
routing schedule
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Grouping Genetic Algorithms: Advances and Applications 1st Edition Michael Mutingi
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- 4. Studies in Computational Intelligence 666 Michael Mutingi Charles Mbohwa Grouping Genetic Algorithms Advances and Applications
- 5. Studies in Computational Intelligence Volume 666 Series editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland e-mail: kacprzyk@ibspan.waw.pl
- 6. About this Series The series “Studies in Computational Intelligence” (SCI) publishes new develop- ments and advances in the various areas of computational intelligence—quickly and with a high quality. The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life sciences, as well as the methodologies behind them. The series contains monographs, lecture notes and edited volumes in computational intelligence spanning the areas of neural networks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, self-organizing systems, soft computing, fuzzy systems, and hybrid intelligent systems. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution, which enable both wide and rapid dissemination of research output. More information about this series at http://www.springer.com/series/7092
- 7. Michael Mutingi • Charles Mbohwa Grouping Genetic Algorithms Advances and Applications 123
- 8. Michael Mutingi Faculty of Engineering Namibia University of Science and Technology Windhoek Namibia and Faculty of Engineering and the Built Environment University of Johannesburg Johannesburg South Africa Charles Mbohwa Faculty of Engineering and the Built Environment University of Johannesburg Johannesburg South Africa ISSN 1860-949X ISSN 1860-9503 (electronic) Studies in Computational Intelligence ISBN 978-3-319-44393-5 ISBN 978-3-319-44394-2 (eBook) DOI 10.1007/978-3-319-44394-2 Library of Congress Control Number: 2016950866 © Springer International Publishing Switzerland 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 9. This book is dedicated to operations analysts, computational scientists, decision analysts, and industrial engineers
- 10. Preface Recent research trends have shown that industry is inundated with grouping problems that require efficient computational algorithms for grouping system entities based on specific guiding criteria. Grouping problems commonplace in industry include vehicle routing, container loading, equal piles problem, machine-part cell formation, cutting stock problem, job shop scheduling, assembly line balancing, and task assignment. These problems have a group structure with identifiable characteristic features, that is, the need to form efficient groups of entities according to guiding criteria, and the need to allocate those groups to specific assignees in order to satisfy the desired objectives. It is interesting to note that, across all the spectrum of these problems, grouping and allocation criteria are inherently very similar in nature. The wide spectrum of real-world grouping problems, the striking similarities between their features, and the multi-criteria decisions involved are three major motivating factors behind the research momentum in this area. However, more challenging issues in this field have appeared in recent researches. First, there is an ever-growing need to address uncertainties in various grouping problem situations. Second, decision analysts in the field often call for multi-criteria decision approa- ches by which multiple criteria can be handled simultaneously. Third, researchers and decision analysts have realized the need for interactive, population-based algorithms that can provide alternative solutions rather than prescribe a single solution to the decision maker. Examples of such approaches are tabu search, particle swarm optimization, ant colony optimization, simulated evolution algo- rithm, simulated metamorphosis algorithm, genetic algorithms, and grouping genetic algorithms. Thus, in sum, recent research has emphasized the need for development of interactive multi-criteria computational algorithms that can address grouping problems, even in uncertain or fuzzy environments. Evidently, notable research has focused on advances in genetic algorithms and related hybrid approaches, with application in various problem areas. Current research trends tend to show that there is a high potential for remarkable advances in genetic algorithms and its variants, specifically in grouping genetic algorithms. Genetic algorithm-based approaches offer a more user-friendly, flexible, and vii
- 11. adaptable population-based approach than related algorithms. Given these advan- tages, further developments and advances in grouping genetic algorithms are quite promising. The purpose of this book is to provide an account of recent research advances and, above all, applications of grouping genetic algorithm and its variants. The prospective audience of the book “Grouping Genetic Algorithms: Advances and Applications” includes research students, academicians, researchers, decision ana- lysts, software developers, and scientists. It is hoped that, by going through this book, readers will obtain an in-depth understanding of the novel unique features of the algorithm and apply it to specific areas of concern. The book comprises three parts. Part I presents an in-depth reader-friendly exposition of a wide range of practical grouping problems, and the emerging challenges often experienced in the decision process. Part II presents recent novel developments in grouping genetic algorithms, demonstrating new techniques and unique grouping genetic operators that can handle complex multi-criteria decision problems. Part III focuses on computational applications of grouping genetic algorithms across a wide range of real-world grouping problems, including fleet size and mix vehicle routing, heterogeneous vehicle routing, container loading, machine-part cell formation, cutting stock problem, job shop scheduling, assembly line balancing, task assignment, and other group technology applications. Finally, Part IV provides concluding remarks and suggests further research extensions. Johannesburg, South Africa Michael Mutingi Charles Mbohwa viii Preface
- 12. Contents Part I Introduction 1 Exploring Grouping Problems in Industry . . . . . . . . . . . . . . . . . . . . 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Identifying Grouping Problems in Industry. . . . . . . . . . . . . . . . . 5 1.2.1 Cell Formation in Manufacturing Systems. . . . . . . . . . . 5 1.2.2 Assembly Line Balancing . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Job Shop Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.4 Vehicle Routing Problem . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.5 Home Healthcare Worker Scheduling . . . . . . . . . . . . . . 10 1.2.6 Bin Packing Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.7 Task Assignment Problem. . . . . . . . . . . . . . . . . . . . . . . 13 1.2.8 Modular Product Design . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.9 Group Maintenance Planning. . . . . . . . . . . . . . . . . . . . . 15 1.2.10 Order Batching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.11 Team Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.12 Earnings Management . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.13 Economies of Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.14 Timetabling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.15 Student Grouping for Cooperative Learning . . . . . . . . . 22 1.2.16 Other Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3 Extant Modeling Approaches to Grouping Problems . . . . . . . . . 23 1.4 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Complicating Features in Industrial Grouping Problems. . . . . . . . . 31 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Research Findings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Complicating Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.1 Model Conceptualization . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.2 Myriad of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ix
- 13. 2.4.3 Fuzzy Management Goals . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.4 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Suggested Solution Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Part II Grouping Genetic Algorithms 3 Grouping Genetic Algorithms: Advances for Real-World Grouping Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Grouping Genetic Algorithm: An Overview . . . . . . . . . . . . . . . . 46 3.2.1 Group Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.1 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.2 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Grouping Genetic Algorithms: Advances and Innovations . . . . . 50 3.4.1 Group Encoding Strategies . . . . . . . . . . . . . . . . . . . . . . 50 3.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.3 Selection Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.4 Rank-Based Wheel Selection Strategy. . . . . . . . . . . . . . 54 3.4.5 Crossover Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4.6 Mutation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4.7 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.8 Replacement Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.9 Termination Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 Application Areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Fuzzy Grouping Genetic Algorithms: Advances for Real-World Grouping Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Preliminaries: Fuzzy Logic Control . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Fuzzy Grouping Genetic Algorithms: Advances and Innovations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.1 FGGA Coding Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.3 Fuzzy Fitness Evaluation. . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.4 Fuzzy Genetic Operators . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3.5 Fuzzy Dynamic Adaptive Operators . . . . . . . . . . . . . . . 80 4.3.6 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 Potential Application Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 x Contents
- 14. Part III Research Applications 5 Multi-Criterion Team Formation Using Fuzzy Grouping Genetic Algorithm Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Related Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3 The Multi-Criterion Team Formation Problem . . . . . . . . . . . . . . 91 5.3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.2 Fuzzy Multi-Criterion Modeling . . . . . . . . . . . . . . . . . . 92 5.4 A Fuzzy Grouping Genetic Algorithm Approach . . . . . . . . . . . . 94 5.4.1 Group Encoding Scheme. . . . . . . . . . . . . . . . . . . . . . . . 94 5.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4.3 Fuzzy Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4.4 Selection and Crossover . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4.5 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.4.6 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.4.7 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.5 Experimental Tests and Results . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.5.1 Experiment 1: Teaching Group Formation. . . . . . . . . . . 101 5.5.2 Experiment 2: Comparative FGGA Success Rates. . . . . 102 5.5.3 Experiment 3: Further Extensive Computations. . . . . . . 102 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6 Grouping Learners for Cooperative Learning: Grouping Genetic Algorithm Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3 Cooperative Learners’ Grouping Problem. . . . . . . . . . . . . . . . . . 109 6.4 A Grouping Genetic Algorithm Approach . . . . . . . . . . . . . . . . . 110 6.4.1 Group Encoding Scheme. . . . . . . . . . . . . . . . . . . . . . . . 111 6.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.4.3 Selection and Crossover . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4.4 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4.5 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.4.6 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.5 Computational Results and Discussions . . . . . . . . . . . . . . . . . . . 116 6.5.1 Preliminary Experiments . . . . . . . . . . . . . . . . . . . . . . . . 116 6.6 Comparative Results: GGA and Other Approaches. . . . . . . . . . . 117 6.6.1 Further Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Contents xi
- 15. 7 Optimizing Order Batching in Order Picking Systems: Hybrid Grouping Genetic Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.2 Order Batching Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2.1 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . 123 7.2.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.3 Related Solution Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.3.1 Routing Heuristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.3.2 Mathematical Programming Techniques . . . . . . . . . . . . 127 7.3.3 Constructive Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.3.4 Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.4 Hybrid Grouping Genetic Algorithm for Order Batching . . . . . . 128 7.4.1 Group Encoding Scheme. . . . . . . . . . . . . . . . . . . . . . . . 128 7.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.4.3 Selection and Crossover . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4.4 Mutation with Constructive Insertion. . . . . . . . . . . . . . . 131 7.4.5 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.4.6 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.5 Computation Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.6 Computational Results and Discussions . . . . . . . . . . . . . . . . . . . 135 7.6.1 Preliminary Experiments . . . . . . . . . . . . . . . . . . . . . . . . 135 7.6.2 Further Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8 Fleet Size and Mix Vehicle Routing: A Multi-Criterion Grouping Genetic Algorithm Approach. . . . . . . . . . . . . . . . . . . . . . . 141 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.2 Fleet Size and Mix Vehicle Routing Problem Description . . . . . 142 8.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.3.1 Vehicle Routing: A Background . . . . . . . . . . . . . . . . . . 143 8.3.2 Approaches to Fleet Size and Mix Vehicle Routing . . . 144 8.4 Multi-Criterion Grouping Genetic Algorithm Approach . . . . . . . 145 8.4.1 GGA Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.4.3 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.4.4 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.4.5 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.4.6 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.4.7 Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.4.8 GGA Computational Implementation. . . . . . . . . . . . . . . 153 xii Contents
- 16. 8.5 Computational Tests and Discussions . . . . . . . . . . . . . . . . . . . . . 154 8.5.1 Computational Experiments. . . . . . . . . . . . . . . . . . . . . . 154 8.5.2 Computational Results and Discussions. . . . . . . . . . . . . 154 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9 Multi-Criterion Examination Timetabling: A Fuzzy Grouping Genetic Algorithm Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 9.2 The Examination Timetabling Problem. . . . . . . . . . . . . . . . . . . . 162 9.3 Related Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.4 Fuzzy Grouping Genetic Algorithm for Multi-Criterion Timetabling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 9.4.1 Group Encoding Scheme. . . . . . . . . . . . . . . . . . . . . . . . 165 9.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.4.3 Fuzzy Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.4.4 Fuzzy Controlled Genetic Operators . . . . . . . . . . . . . . . 168 9.4.5 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.5 Numerical Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.6 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 10 Assembly Line Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.2 Assembly Line Balancing: Problem Description . . . . . . . . . . . . . 184 10.3 Approaches to Assembly Line Balancing . . . . . . . . . . . . . . . . . . 186 10.4 A Hybrid Grouping Genetic Algorithm Approach . . . . . . . . . . . 187 10.4.1 Encoding Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 10.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 10.4.3 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 10.4.4 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 10.4.5 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.4.6 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.4.7 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 10.5 Computational Tests and Results . . . . . . . . . . . . . . . . . . . . . . . . 191 10.5.1 Computational Results: Small-Scale Problems. . . . . . . . 192 10.5.2 Computational Results: Large-Scale Problems. . . . . . . . 193 10.5.3 Overall Computational Results . . . . . . . . . . . . . . . . . . . 195 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 11 Modeling Modular Design for Sustainable Manufacturing: A Fuzzy Grouping Genetic Algorithm Approach . . . . . . . . . . . . . . . 199 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 11.2 Sustainable Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Contents xiii
- 17. 11.3 Modular Product Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 11.4 Fuzzy Grouping Genetic Algorithm Approach . . . . . . . . . . . . . . 202 11.4.1 Group Encoding Scheme. . . . . . . . . . . . . . . . . . . . . . . . 202 11.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 11.4.3 Fitness Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 11.4.4 Fuzzy Dynamic Adaptive Operators . . . . . . . . . . . . . . . 205 11.4.5 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 12 Modeling Supplier Selection Using Multi-Criterion Fuzzy Grouping Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 12.3 A Subcontractor Selection Example . . . . . . . . . . . . . . . . . . . . . . 216 12.4 A Fuzzy Multi-Criterion Grouping Genetic Algorithm . . . . . . . . 218 12.4.1 FGGA Coding Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 219 12.4.2 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 12.4.3 Fuzzy Fitness Evaluation. . . . . . . . . . . . . . . . . . . . . . . . 220 12.4.4 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 12.4.5 Adaptive Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 12.4.6 Adaptive Mutation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 12.4.7 Adaptive Two-Point Inversion. . . . . . . . . . . . . . . . . . . . 224 12.4.8 Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 12.4.9 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 12.5 Summary and Further Research . . . . . . . . . . . . . . . . . . . . . . . . . 226 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Part IV Conclusions and Extensions 13 Further Research and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 13.2 Extension of the Application Domain. . . . . . . . . . . . . . . . . . . . . 232 13.3 Further Extensions to Grouping Genetic Algorithms . . . . . . . . . 234 13.3.1 Variants of Grouping Genetic Operators . . . . . . . . . . . . 234 13.3.2 Hybridizing GGA with Heuristic Algorithms . . . . . . . . 234 13.3.3 Further Use of Domain-Specific Heuristics . . . . . . . . . . 235 13.4 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 xiv Contents
- 19. Chapter 1 Exploring Grouping Problems in Industry 1.1 Introduction In many real-world industry settings, it is often desirable to improve the perfor- mance of systems, processes, and products by partitioning items into groups, based on suitable decision criteria. For instance, in logistics management, decision makers wish to minimize the overall transportation costs, the number of vehicles used, and the average waiting time experienced by the customer (Taillard 1999; Potvin and Bengio 1996). To achieve this, it is important to ensure that customers are portioned into efficient groups to be visited by a set of vehicles. As such, it is important to optimize grouping of customers to be visited by each vehicle, while considering the size, type, and capacity of the available vehicles. Similarly, when assigning a set of tasks to a team of workers, it is crucial to form cost-effective and efficient groups of tasks that can be assigned to workers in an optimal manner. Furthermore, manu- facturers may want to find the best way to group parts with similar characteristics so that similar parts can be produced using specific processes in specific departments. Some other well-known problems include bin packing problem, load balancing (or equal piles) problem, and machine cell formation problem in group technology. The major task in these problem situations is to group (or partition or cluster) a set of items into disjointed subsets or groups, in some optimal or near-optimal manner. Such problems are commonplace across a wide range of industries, from manu- facturing to service industry. In this book, these problem situations are called grouping problems. It is important to note at this point that grouping problems are known to be combinatorial, hard and computationally difficult to solve (Taillard 1999; Potvin and Bengio 1996; Moghadam and Seyedhosseini 2010). On the other hand, it is also important to realize that grouping problems have common grouping features and characteristics that can be potentially be exploited in order to develop more effective solution approaches. In a nutshell, further studies on the various types of grouping problems in the literature revealed interesting facts as outlined below: © Springer International Publishing Switzerland 2017 M. Mutingi and C. Mbohwa, Grouping Genetic Algorithms, Studies in Computational Intelligence 666, DOI 10.1007/978-3-319-44394-2_1 3
- 20. 1. The optimization of grouping problems is usually defined in terms of the composition of the groups of items and the overall array of all the groups; 2. Grouping problems possess a grouping structure that can be utilized for developing effective computational algorithms; 3. Grouping problems are highly combinatorial in nature, NP-hard, and compu- tationally expensive; 4. Grouping problems are highly constrained, which adds to their computational complexity; and 5. Some of the variables of grouping problems are not precise, so much so that fuzzy modeling is a useful option. Due to their computational complexity, the use of heuristics, expert systems, and metaheuristic approaches are a viable option for solving various grouping problems. Examples of these approaches are tabu search, particle swarm optimization (Mutingi and Mbohwa 2014a, b, c), genetic algorithms, grouping genetic algorithm (Mutingi and Mbohwa 2013a, b), and other evolutionary algorithms. Genetic algorithm (GA) is a potential solution approach for this class of prob- lems (Rochat and Taillard 1995; Badeau et al. 1997). GA is a metaheuristic approach based on the philosophy of genetics and natural selection. In its operation, GA encodes candidate solutions into chromosomes (or strings) and improves the strings by copying strings according to their objective function values and swap- ping partial chromosomes to generate successive solutions that improve over time. Its distinctive feature is the use of probabilistic genetic operators as tools to guide the search toward regions of the search space with likely improvement. Grouping genetic algorithm (GGA), originally developed by Falkenauer (1992), is a modi- fication of the conventional genetic algorithms for addressing grouping problems. Some recent remarkable improvements and applications of the GGA exist in the literature (Mutingi et al. 2012; Mutingi 2013; Mutingi and Mbohwa 2013a, b). Given the complexities of grouping problems, and their widespread occurrences in real-world industry, developing flexible, efficient, and effective solution methods is vital. The purpose of this chapter is to explore and identify grouping problems and explain their grouping characteristics. In this vein, the learning outcomes for the chapter are as follows: 1. To be able to identify various types of grouping problems and their common characteristics; 2. To develop an understanding of how to exploit grouping features of grouping problems for effective modeling; and 3. To understand the various modeling approaches for grouping problems and to open up research avenues for more efficient hybrid approaches. The rest of this chapter is organized as follows: The next section identifies typical grouping problems in industry, their group structures, and shows how the problems lend themselves to grouping algorithms. Section 1.3 outlines past mod- eling approaches for grouping problems. Finally, concluding remarks and further research prospects are presented in Sect. 1.4. 4 1 Exploring Grouping Problems in Industry
- 21. 1.2 Identifying Grouping Problems in Industry In this section, a number of grouping problems from a wide range of industry settings are explored and identified, briefly illustrating their group structure and how they lend themselves to grouping genetic algorithm approaches. It was observed in this study that many of these problems loosely fall into subcategories such as manufacturing systems, logistics and supply chain, healthcare services, design, and other services. Some of them, such as team formation, timetabling, and economics, frequently occur across several types of industries. Table 1.1 provides a summary of these problems, together with selected references for further reading. It is interesting to note that grouping problems are prevalent in a wide range of industry types. Several, if not all of these, problems possess similar characteristics and, therefore, lend themselves to a common group modeling and solution approach. By taking advantage of the knowledge of the common characteristics of the problems, a flexible computational algorithm can be developed for solving the problems. Such a computational algorithm is expected to be flexible and robust enough to be adapted to a wide range of problems with little or no fine-tuning. Apart from the ease of adaptation to problem situations, the algorithm is expected to be able to solve large-scale industrial problems within a reasonable time frame. In most industry settings, such as logistics, decision makers may need to make decisions on real time, or at least within a short space of time, so much so that efficient and flexible decision support is extremely crucial. In view of these and other related reasons, developing an efficient, flexible, and adaptable grouping algorithm is imperative and significant. 1.2.1 Cell Formation in Manufacturing Systems Cellular manufacturing is a lean system of making groups of products, each group with products that are similar in shape, size, and processing characteristics in a cell. A cell defines a group of team members, workstations, or equipment that are grouped together to facilitate operations by eliminating setup costs between oper- ations. Cell formation has a direct positive impact on the planning activities of a manufacturing system and is aimed at improving efficiency and productivity of the manufacturing system (Filho and Tiberti 2006; Mutingi et al. 2012; Mutingi and Onwubolu 2012). Machines are grouped together into efficient clusters, each operating on a product family with little or no inter-cell movement of the products. Figure 1.1 provides an example of a cellular manufacturing system and its group representation. Assume that the system is one of the solutions to cell formation problem. Part (a) indicates that the manufacturing system consists of 3 cells, that is, cell 1, cell 2, and cell 3, where each cell comprises machine groups (1,3,4), (2,6), and (5,7,8), respectively. Part (b) shows the group structure or group representation 1.2 Identifying Grouping Problems in Industry 5
- 22. Table 1.1 Identified grouping problems in industry No. Grouping problems Selected references 1 Assembly line balancing Rubinovitz and Levitin (1995), Sabuncuoglu et al. (2000a, b) 2 Bin packing Falkenauer (1996), Kaaouache and Bouamama (2015) 3 Job shop scheduling Chen et al. (2012), Phanden et al. (2012), Luh et al. (1998) 4 Cell formation De Lit et al. (2000), Onwubolu and Mutingi (2001), Filho and Tiberti (2006) 5 Container loading Althaus et al. (2007), Joung and Noh (2014) 6 Heterogeneous fixed fleet Vehicle Routing Gendreau et. (1999a, b), Tutuncu (2010), Tarantilis et al. (2003, 2004) 7 Cutting stock/material cutting Onwubolu and Mutingi (2003), Rostom et al. (2014), Hung et al. (2003) 8 Fleet size and Mix Vehicle Routing Liu et al. (2009), Brandao (2008), Renaud and Boctor (2002) 9 Equal piles problem Falkenauer (1995), Rekiek et al. (1999) 10 Group maintenance planning Li et al. (2011), Van Do et al. (2013), Gunn and Diallo (2015), De Jonge et al. (2016) 11 Handicapped person transportation Rekiek et al. (2006) 12 Task assignment Cheng et al. (2007), Mutingi and Mbohwa (2014a, b, c), Tarokh et al. (2011) 13 Home healthcare scheduling Mutingi and Mbohwa (2014a, b, c) 14 Multiple traveling salesperson Kivelevitch and Cohen (2013), Carter and Ragsdale (2006), Bektas (2006) 15 Modular product design Yu et al. (2011), Kreng and Lee (2004), Chen and Martinez (2012) 16 Order batching Henn and Wäscher (2012), Henn (2012) 17 Pickup and delivery Parragh et al. (2008), Chen (2013), Wang and Chen (2013) 18 Site/facility location Pitaksringkarn and Taylor (2005a, b), Yanik et al. (2016) 19 Wi-fi network deployment Landa-Toress et al. (2013), Agustın-Blas et al. (2011a, b) 20 Student/learners grouping Chen et al. (2012a, b), Wessner and Pfister (2001), Baker and Benn (2001) 21 Team formation Wi et al. (2009), Strnad and Guid (2010), Dereli et al. (2007) 22 Timetabling Dereli et al. (2007), Pillay and Banzhaf (2010), Rakesh et al. (2014) 23 Reviewer group construction Chen et al. (2011), Hettich and Pazzani (2006) 24 Estimating discretionary accruals Höglund (2013), Back et al. (1996), Bartov et al. (2000) 25 Economies of scale Falkenauer (1993, 1994) 26 Customer Grouping Sheu (2007), Chan (2008), Ho et al. (2012) 6 1 Exploring Grouping Problems in Industry
- 23. of the system. By considering process flows and the parts to be manufactured, multiple manufacturing system configurations can be generated and evaluated. In the presence of numerous possible configurations, this leads to a highly combinatorial optimization problem that is computationally expensive (Onwubolu and Mutingi 2001). 1.2.2 Assembly Line Balancing In assembly line balancing, individual work elements or tasks are assigned to workstations so that unit assembly cost is minimized as much as possible (Scholl 1999; Sabuncuoglu et al. 2000a, b; Scholl and Becker 2006). Line balancing decisions have a direct impact on the cost-effectiveness of a production process. As such, it is of utmost importance to develop optimal or near-optimal practical solution procedures that can assist decision makers in assembly line balancing decisions, yet with minimal computational requirements. Figure 1.2 shows a diagraph for a typical line balancing problem and its group representation. The problem consists of 6 different tasks to be allocated to 3 workstations. As illustrated, task groups (1,2), (3,4), and (4,6) are allocated to workstations 1, 2, and 3, respectively. Howbeit, there are several possible solutions that may be generated. The problem becomes highly combinatorial, demanding high computation requirements. (a) Cell manufacturing system Machines 1,3,4 2,6 5,7,8 Cells 1 2 3 (b) Group structure 1 2 3 4 6 cell 1 cell 2 8 cell 3 7 5 Fig. 1.1 A cellular manufacturing system and its group representation 1.2 Identifying Grouping Problems in Industry 7
- 24. Due to the combinatorial nature of the assembly line balancing problem, metaheuristic algorithms, such as genetic algorithm, tabu search, simulated annealing, and particle swarm intelligence, are the most viable solution methods to the problem. Iterative metaheuristic approaches can offer reliable solutions within reasonable computational times. 1.2.3 Job Shop Scheduling The assignment of tasks in a flexible job shop problem environment is more challenging than the classical job shop problem. This requires proper selection of machines from a set of given machines to process each operation. The problem is best defined by three important features: the set of jobs, the set of machines, and the flexibility specification. These are defined as follows: 1. Jobs. J = {J1, J2, …, Jn} is a set of n independent jobs to be scheduled. Each job Ji consists of a sequence operation to be performed one after the other according to a given sequence. All jobs are assumed to be available at time 0. 2. Machines. M = {m1, m2, …, mm} is a set of m machines. Every machine pro- cesses only one operation at a time. All machines are assumed to be available at time 0. 3. Flexibility specification: The problem generally falls into two categories, namely: (a) total flexibility, where each operation can be processed on any of the machines; and (b) partial flexibility, where each operation can be performed only on a subset of the machines. Further to the above definition of the key features, the following simplifying assumptions are essential for formulation of the flexible job shop scheduling environment: (a) Line balancing problem Tasks 1,2 3,5 4,6 Workstations 1 2 3 (b) Grouping representation 6 3 1 2 4 5 Workstation 2 Workstation 1 Workstation 3 Fig. 1.2 A typical line balancing problem and its grouping representation 8 1 Exploring Grouping Problems in Industry
- 25. 1. Every job operation is performed on one and only one machine at any given point in time; 2. The processing times of the operations are machine-dependent, and the machines are independent from each other; 3. Once started, each operation must be performed to completion without interruption; 4. Setup times are of machines are included in the job processing time of the job; and 5. The transportation time of jobs between machines are negligible and are included in the processing times. Table 1.2 illustrates a job shop problem with 3 machines to process 3 jobs upon which 8 operations are to be performed. For instance, operation 11, 12, and 13 represent the first, second, and third operations of job 1, respectively. Machines capable of performing each operation are as shown. The objective is twofold: (i) to assign each operation to an appropriate machine, which is a routing problem, and (ii) to sequence the operations on specific machines, which is a sequencing prob- lem, so that make-span and the total working time of machines (total workload) are minimized. This becomes a complex multi-criteria optimization problem that demands significant computational resources. Figure 1.3 represents a typical solution to the problem, where operations 11, 13, and 31 are performed on machine m1, operations 21, 23, and 12 are performed on m2, while 22 and 32 are done on m3. A group representation scheme of the solution is shown in (b). Figure 1.3 represents a typical solution to the problem, where operations 11, 13, and 31 are performed on machine m1, operations 21, 23, and 12 are performed on m2, while 22 and 32 are done on m3. A group representation scheme of the solution is shown in (b). 1.2.4 Vehicle Routing Problem In transportation and distribution, planning for vehicle routing is a major challenge to decision makers in the logistics industry. In order to provide cost-effective and satisfactory delivery (and pickup) services to customers, it is important to optimize the routing of vehicles (Taillard 1999; Mutingi and Mbohwa 2012, b). The vehicle routing problem (VRP) is a hard combinatorial problem that seeks to assign a set of Table 1.2 A job shop problem with 3 machines and 3 jobs with 8 operations Jobs J1 J2 J3 Operations 11 12 13 21 22 23 31 32 Machines m1, m2, m3 m1, m2 m1, m2, m3 m1, m2, m3 m1, m3 m1, m2, m3 m1, m2, m3 m2, m3 1.2 Identifying Grouping Problems in Industry 9
- 26. groups of customers to a set of vehicles or drivers, so as to minimize the total costs incurred in visiting all the customers, subject to a number of constraints, such as follows: (i) use no more vehicles than those available; (ii) satisfy customer demand; (iii) visit each customer exactly once; (iv) vehicle routes start and finish at the depot; and (v) vehicle capacity is not violated. In most cases, the main objective is to minimize the total delivery costs incurred (Gendreau et al. 1999a, b; Mutingi 2013). Figure 1.4 illustrates a typical vehicle routing schedule, where part (a) shows a diagraph for the problem, and part (b) shows the group representation of the problem. The nodes represent 7 customer locations, and the arcs represent the distances between the customers, and from the depot (denoted by node 0). The schedule comprises 7 customers that are assigned to 3 available vehicles or drivers. Further, customer groups (1, 2), (3, 4, 5), and (6,7) are assigned to vehicles v1, v2, and v3, respectively. The sequence of customers in each group represents the sequence of customer visits or the direction of the route. This situation is similar to routing of healthcare workers in a homecare environment (Mutingi and Mbohwa 2013a, b) and handicapped person transportation problems (Rekiek et al. 2006). 1.2.5 Home Healthcare Worker Scheduling As aging populations continue to increase in most countries, healthcare authorities continue to face increasing demand for home-based medical care. Most families prefer their patients to be treated at their homes, rather than at retirement homes and hospitals (Mutingi and Mbohwa 2013a, b). With the ever-increasing demand for Operations: 11, 13, 31 Operations: 21, 23, 12 Operations: 22, 32 Job shop operations Operations 11,13,31 21,23,12 22,32 Machines m1 m2 m3 Group structure m1 m2 m3 (a) (b) Fig. 1.3 A typical job shop problem with 3 machines and 3 jobs 10 1 Exploring Grouping Problems in Industry
- 27. home healthcare services, many service providers are struggling to find cost-effective and efficient schedules to meet the expectations of the customers, the management goals, as well as the desires of the healthcare workers. Home healthcare service provides continue to expand. Consequently, healthcare worker scheduling has become a large-scale combinatorial problem, inundated with a myriad of constraints that have to be taken into account, including patients’ pref- erences, visiting time windows, or travel times depending on the mode of transport. Figure 1.5 presents a schematic of the home healthcare worker scheduling problem. In general, the problem is defined thus a set of m available care workers are given the responsibility to visit n patients for home-based medical care. Each caregiver k (k = 1, 2, …, m) is supposed to serve a group of patients, where each patient j (j = 1,2…, n) is to be visited within a given time window defined by earliest start and latest start times, ej and lj, respectively. The aim is to minimize the overall costs of visiting clients (Mutingi and Mbohwa 2013a, b). In this vein, a penalty cost is incurred whenever a caregiver reaches the client earlier than ej or later than lj. If aj denotes the caregiver’s arrival time at patient j, and pe and pl denote the unit penalty costs incurred when the caregiver arrives too early or too late, respectively, then max[0, ej − aj] and max[0, aj − lj] have to be minimized, to maximize patient satisfaction. Furthermore, worker preferences should be taken into account, if schedule quality is to be maximized (Mutingi and Mbohwa 2013a, b). However, since planning for home healthcare schedules is especially compu- tationally expensive, effective and efficient algorithms are very important. A diagraph for a vehicle routing problem Operations 1,2 3,4,5 6,7 Machines v1 v2 v3 Group structure 0 40 20 13 21 70 20 18 21 4 2 3 1 5 6 7 18 12 (a) (b) Fig. 1.4 A typical vehicle routing schedule 1.2 Identifying Grouping Problems in Industry 11
- 28. 1.2.6 Bin Packing Problem In bin packing, objects of different volumes and shapes must be packed into a finite number of bins, where each bin has a specific volume. Oftentimes, the goal is to ensure that the wasted space or number of bins used are minimized as much as possible (Allen et al. 2011; Pillay 2012). According to computational complexity theory, the bin packingproblem is a NP-complete problem that is computationally expensive and highly combinatorial, when formulated as a decision problem (Pillay 2012). Figure 1.6 presents an illustrative example of the bin packing problem and its group structure. In part (a), the problem shows three bins, that is, b1, b2, and b3, which are packed with groups of objects (1, 4, 8), (5, 6), and (2, 3, 7), respectively. This information can be presented conveniently in a group structure as indicated in part (b). (a) Home healthcare care worker scheduling patients 1,2,3 4,5 6 workers w1 w2 w3 (b) Group structure 0 4 2 3 1 5 6 w1 w2 w3 Fig. 1.5 Homecare worker schedule and its group structure Bin b1 Bin b2 Bin b3 1 5 4 2 8 3 7 6 Waste = 0 Waste = 1 Waste = 2 (a) A typical bin packing problem Objects 1,4,8 5,6 2,3,7 Bins b1 b2 b3 (b) Group structure Fig. 1.6 A bin packing problem and its representation 12 1 Exploring Grouping Problems in Industry
- 29. In real-world practice, the bin packing problem comes in different variations, including two-dimensional packing, linear packing, packing by weight, and packing by cost, with many applications. The concepts can be extended to other various situations, such as filling up containers, or loading trucks with weight capacity constraints, metal cutting, and other related problems (Ramesh 2001). In this view, it is important to develop a flexible and adaptive grouping algorithm which can solve this problem and its variants. 1.2.7 Task Assignment Problem The task assignment problem consists in assigning a set of tasks, T = {1, …, n} to an available set of available workers or processors, W = {1, …, w} in a manner that will minimize the overall assignment cost function (Tarokh et al. 2011; Mutingi and Mbohwa 2013a, b). The problem is also generally known as a task scheduling problem (Salcedo-Sanz et al. (2006). Basically, each task is defined by its duration and its time window defined by the task’s earliest start and latest start times (Mutingi and Mbohwa 2013a, b; Bachouch et al. 2010). Each worker or processor may have a specific scheduled time of day when it is available. In most practical cases, it is desired or required to minimize the workload variation to an acceptable degree, violation of time window constraints, and completion time of all the tasks, depending on the specific situation under consideration. Figure 1.7 presents an example of a task assignment problem in part (a) and its group structure in part (b). Seven tasks are to be assigned to three processors or workers (called assignees), while minimizing a specific assignment cost function, subject to hard and soft constraints. The general task assignment problem is a combinatorial problem that is known to be NP-hard due to its myriad of constraints and variables (Salcedo-Sanz et al. Task assignment problem Tasks 1,2 3,4,5 6,7 Assignees w1 w2 w3 Group structure 6 3 1 2 7 5 Worker w Worker w Worker w 4 3 2 1 (a) (b) Fig. 1.7 Task assignment and its group representation 1.2 Identifying Grouping Problems in Industry 13
- 30. 2006; Chen 2007; Tarokh et al. 2011). It is desirable to develop heuristic algorithms that take advantage of the grouping structure of the problem. 1.2.8 Modular Product Design Modular design is a design approach that divides a system into modules that can be independently created for assembling a variety of different systems. There are three basic categories of modular design, namely function-based modular design, man- ufacturing design, and assembly-based modular design (Tseng et al. 2008). Modular design is important as manufacturers need to cope with multiple variations of product specifications and modules in a customized environment (Kreng and Lee 2003). Thus, well-developed modular designsystems will help with the production and control of mass customization. Modular design also focuses on environmental aspects based on grouping techniques. In the assembly-based modular design method, products are generally described by liaison graph (Tseng et al. 2008). Figure 1.8 shows an example of a Parker Pen assembly with 6 components to be grouped into 3 modules. There are three important stages in modular design: 1. Determination of liaison intensity (LI) of components; 2. Grouping or clustering of components using a grouping method; and 3. Evaluation of the clustering of grouping result. The goal is to maximize the liaison intensity within each module and to mini- mize the liaison intensity between modules. However, as the number of components and modules increase, the number of possible combinations increases (a) A liaison graph for a parker pen assembly Components: 1,3,4 5,6 2 Module : m1 m2 m3 (b) A Group structure for a typical modular design 2.Cap 3.Body 1.Button 4.Head 5.Tube Liaison intensities 6.ink Fig. 1.8 A Parker Pen assembly and its group structure 14 1 Exploring Grouping Problems in Industry
- 31. exponentially, and the complexity of the problem increases rapidly; therefore, developing efficient grouping techniques is imperative (Kamrani and Gonzalez 2003; Tseng et al. 2008). 1.2.9 Group Maintenance Planning Most industrial systems, such as production systems, pipe networks, mining equipment, aerospace industry, oil and gas, and military equipment, are made up of multi-component systems that require high reliability. Moreover, these systems are normally required to operate with minimal stoppages and breakdowns and, there- fore, are usually supported by preventive maintenance/replacement procedures at intervals defined by operating hours in terms of mean time to failure. In the pres- ence of multiple components and subsystems, multiple replacements and mainte- nance tasks are involved at high costs. Since the reliabilities of the components and subsystems contribute to the overall system reliability, it is essential to formulate the best maintenance strategies for the components and subsystems. Figure 1.9 shows a typical schedule for system components to be replaced in groups. Part (a) may represent, for example, a set P = {P1, …, P10} of component pipes in a pipe network that are to be replaced in optimal groups, and the grouping process is supposed to minimize the total costs in terms of distance traveled to repair, preparation, and setup costs. Part (b) is a group structure for the group maintenance schedule, where the 10 pipes are scheduled into 4 groups: {1, 5}, {2, 3, 4, 7}, {6, 10, 8}, and {9}. High fixed costs are often incurred in transporting repair equipment to repair facilities and to set them up for the required maintenance procedures. As a result, it is generally more economical to conduct the maintenance or replacements of related (a) (b) A schedule of pipeline maintenance jobs Pipes: 1,5 2,3,4,7 6,10,8 9 Groups: G1 G2 G3 G4 Group structurefor group maintenance schedule 6 3 1 5 7 10 Group G1 Group G3 4 9 2 8 Group G2 Group G4 Fig. 1.9 A pipeline maintenance schedule and its group structure 1.2 Identifying Grouping Problems in Industry 15
- 32. components at one goal. This means that specific groups of components or sub- systems have to be cautiously defined so that each group can undergo preventive maintenance within a defined time window. The overall aim is to minimize maintenance costs (setup, preparation costs) while maximizing the reliability of the systems. This grouping problem is twofold (Dekker et al. 1997): 1. Fixed group models, where all components are always jointly maintained as a group; and 2. Optimized-groups models, where several groups are optimally generated, either directly or indirectly. The major challenge in solving the group maintenance problem is its compu- tational complexity due to exponential growth of the number of variables as the number of components or subsystems increase. Efficient and robust metaheuristic methods are a potential option. 1.2.10 Order Batching Order batching is a decision problem that is commonplace in warehouse and dis- tribution systems. It is concerned with the search and retrieval of items from their respective storage areas in the warehouse in order to satisfy customer orders. In the real-world practice, customer orders come in small volumes of various types; this makes the retrieval process even more complex. As a result, manual order picking systems need to put in place effective methods to collect items in batches in a more efficient way. Customer orders should be grouped into picking orders of limited sizes, such that the total distance traversed by order pickers is minimized; the total length of all the tours traveled by order pickers should be minimized. Figure 1.10 shows a plan for three order pickers who are scheduled to pick eight customer orders shown by shaded squares in part (a). The three tours, shown by dotted lines, begin and end at the depot (that is the origin). Part (b) illustrates the group representation for the schedule, where three order pickers O1, O2, and O3 are assigned to pick groups of orders {5, 28, 19}, {45, 51, 85}, and {98, 99}, respectively. Though the order batching problem can conveniently be modeled based on the group structure, finding the optimal or near-optimal solution poses a computational challenge. Due to the multiplicity of possible combinations, the problem is NP-hard and computationally expensive. However, by taking advantage of the group structure of the problem, a grouping algorithm that iteratively explores improved solutions, while striving to preserve important information in the group structure, may be quite handy. 16 1 Exploring Grouping Problems in Industry
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- 34. These propositions the filibusters were already acquainted with, and had discussed their advantages; hence they did not take long to deliberate, for they had made up their mind beforehand, as their presence at Port Margot proved. "We accept your propositions, brother," Montbarts answered—"here is my hand, in the name of the filibusters I represent." "And here is mine," Lepoletais said, "in the name of the habitants and buccaneers." There was no other treaty but this honest shake of the hand between the adventurers; thus was concluded an alliance, which remained up to the dying day of buccaneering, as fresh and lively as when first made between the adventurers. "Now," Montbarts continued, "let us proceed orderly. How many brothers have you capable of fighting?" "Seventy," Lepoletais answered. "Very good; we will add to these one hundred and thirty more from the fleet, which will give us an effective strength of two hundred good fusils. And you, Chief, what can you do for us?" Up to this moment Omopoua had remained silent, listening to what was said with Indian gravity and decorum, and patiently waiting till his turn to speak arrived. "Omopoua will add two hundred Carib warriors, with long fusils, to the palefaces," he replied; "his sons are warned; they await the order of the Chief—L'Olonnais has seen them." "Good! These four hundred men will be commanded by myself; as this expedition is the most difficult and dangerous, I will undertake it. Michel le Basque will accompany me. I have aboard a guide, who will conduct us to Grand Fond. You, Drake, and you, David, will attack Leogane with your ships, while Bowline, with only fifteen men, will seize on Tortuga. Let us combine our movements, brothers, so that our three attacks may be simultaneous, and the Spaniards, surprised on three points at once, may not be able to
- 35. assist one another. Tomorrow you will sail, gentlemen, taking with you one hundred and eighty-five men, more than sufficient, I believe, to capture Leogane. As for you, Bowline, you will keep the lugger with the fifteen men left you, and remain here, while watching Tortuga closely. This is the fifth of the month, brothers; on the fifteenth we will attack, as ten days will be sufficient for all of us to reach our posts, and take all the necessary measures. Now, gentlemen, return aboard your vessels, and send ashore, under orders of their officers, the contingents I intend to take with me." The two Captains bowed to the Admiral, left the cabin, and returned to their ships. "As for you," Montbarts added, turning to Lepoletais, "this is what you will do, brother. You will go with Omopoua to the Grand Fond, as if hunting, but you will carefully watch the town of San Juan, and the hatto del Rincón; we must, if possible, make sure of the inhabitants of that hatto; they are rich and influential, and their capture may be of considerable importance to us. You will arrange with Omopoua on the subject of the allies he promises to bring us; perhaps it will be as well for the Chief to try and lead the Spaniards on to his track, and force them to quit their positions: by managing cleverly we might then be able to defeat them in detail. Have you understood me, brother?" "Zounds!" Lepoletais answered, "I should be an ass if I did not. All right! I will manoeuvre as you wish." Montbarts then turned to the engagé, and made him a sign. L'Olonnais drew nearer. "Go ashore with the Carib and Lepoletais," the Admiral whispered in his ear—"look at everything, hear everything, watch everything; in an hour you will receive through Bowline a letter, which you must deliver into the hands of Doña Clara de Bejar, who resides in the hatto on the Grand Fond." "That is easy," L'Olonnais answered, "if it must be, I will hand it to her in the midst of all her servants, in the hatto itself."
- 36. "Do nothing of the sort; arrange it so that she must come and fetch the letter." "Hang it! That is more difficult! Still, I will try to succeed." "You must succeed!" "Ah! In that case, on the word of a man, you may reckon on it— though, hang me if I know how I shall manage it!" Lepoletais had risen. "Farewell, brother," he said; "when you land tomorrow I shall be on my way to the Grand Fond; I shall, therefore, not see you again till we meet there; but do not be alarmed—you shall find everything in order when you arrive. Ah! By the way, shall I take my body of buccaneers with me?" "Certainly; they will be of the greatest use to you in watching the enemy; but hide them carefully." "All right," he said. At this moment Michael the Basque rushed suddenly into the cabin, with his features distorted by passion. "What is the matter, messmate? Come, recover yourself," Montbarts said coolly to him. "A great misfortune has happened to us," Michael exclaimed, as he passionately pulled out a handful of hair. "What is it? Come, speak like a man, messmate." "That villain, Antonio de la Ronda—" "Well?" Montbarts interrupted, with a nervous tremor. "He has escaped!" "Malediction!" "Ten men have set out in pursuit." "Stuff! It is all up now; they will not catch him. What is to be done?"
- 37. "What has happened?" Lepoletais asked. "Our guide has escaped." "Is it only that? I promise to find you another." "Yes, but this one is probably the cleverest spy the Spaniards possess; he knows enough of our secrets to make our expedition fail." "Heaven preserve us from it! Stuff!" the buccaneer added, carelessly —"Think no more about it, brother; what is done is done—let us go ahead all the same." And he left the cabin, apparently quite unaffected by the news. CHAPTER XXV. FRAY ARSENIO. Let us now tell the reader who these buccaneers were of whom we have several times spoken, and what was the origin of the name given them, and which they gave themselves. The red Caribs of the Antilles were accustomed, when they made prisoners in the obstinate contests they waged with each other, or which they carried on against the whites, to cut their prisoners into small pieces, and lay them upon a species of small hurdles, under which they lit a fire. These hurdles were called barbacoas, the spot where they were set up boucans, and the operation boucaning, to signify at the same time roasting and smoking. It was from this that the French boucaniers (anglicised into buccaneers) derived their name, with this difference, that they did to
- 38. animals what the others did to men. The first buccaneers were Spanish settlers on the Caribbean islands, who lived on intimate terms with the Indians; hence when they turned their attention to the chase, they accustomed themselves without reflection to employ these Indian terms, which were certainly characteristic, and for which it would have been difficult to substitute any others. The buccaneers carried on no other trade but hunting; they were divided into two classes, the first only hunting oxen to get their hides, the second killing boars, whose flesh they salted and sold to the planters. These two varieties of buccaneers were accoutred nearly in the same way, and had the same mode of life. The real buccaneers were those who pursued oxen, and they never called the others by any name but hunters. Their equipage consisted of a pack of twenty-four dogs, among which were two bloodhounds, whose duty it was to discover the animal; the price of these dogs, settled among themselves, was thirty livres. As we have said, their weapon was a long fusil, manufactured at Dieppe or Nantes; they always hunted together, two at the least, but sometimes more, and then everything was in common between them. As we advance in the history of these singular men, we shall enter into fuller details about their mode of life and strange habits. When Don Sancho and the Major-domo left them, Lepoletais and L'Olonnais had for a long time looked with a mocking glance after the two Spaniards, and then went on building their ajoupa and preparing their boucan, as if nothing had happened. So soon as the boucan was arranged, the fire lit, and the meat laid on the barbacoas, L'Olonnais set about curing the hide he had brought with him, while Lepoletais did the same to that of the bull which he had killed an hour previously.
- 39. He stretched the hide out on the ground, with the hairy side up, fastened it down by sixty-four pegs, driven into the earth, and then rubbed it vigorously with a mixture of ashes and salt, to make it dry more quickly. This duly accomplished, he turned his attention to supper, the preparations for which were neither long nor complicated. A piece of meat had been placed in a small cauldron, with water and salt, and soon boiled; L'Olonnais drew it out by means of a long pointed stick, and laid it on a palm leaf in lieu of a dish; then he collected the grease with a wooden spoon, and threw it into a calabash. Into this grease he squeezed the juice of a lemon, added a little pimento, stirred it all up, and the sauce, the famous pimentado, so liked by the buccaneers, was ready. Placing the meat in a pleasant spot in front of the ajoupa, with the calabash by its side, he called Lepoletais, and the men sitting down facing each other, armed themselves with their knife and a wooden spit instead of a fork, and began eating with a good appetite, carefully dipping each mouthful of meat in the pimentado, and surrounded by their dogs, which, though not daring to ask for anything, fixed greedy glances on the provisions spread out before them, and followed with eager eyes every morsel swallowed by the adventurers. They had been eating this in silence for some time, when the bloodhounds raised their heads, inhaling the air restlessly, and then gave several hoarse growls; almost immediately the whole pack began barking furiously. "Eh, eh!" Lepoletais said, after drinking a mouthful of brandy and water, and handing the gourd to the engagé, "What is the meaning of this?" "Some traveller, no doubt," L'Olonnais answered carelessly. "At this hour," the buccaneer went on, as he raised his eyes to the sky, and consulted the stars, "why hang it all, it is past eight o'clock at night."
- 40. "Zounds! I do not know what it is. But stay, I do not know whether I am mistaken, for I fancy I can hear a horse galloping." "It is really true, my son, you are not mistaken," the buccaneer continued, "it is indeed a horse; come, quiet, you devils," he shouted, addressing the dogs, which had redoubled their barking, and seemed ready to rush forward, "quiet, lie down, you ruffians." The dogs, doubtless accustomed for a long time to obey the imperious accents of this voice, immediately resumed their places, and ceased their deafening clamour, although they still continued to growl dully. In the meanwhile the galloping horses which the dogs had heard a great distance off, rapidly drew nearer; it soon became perfectly distinct, and at the end of a few minutes a horseman emerged from the forest, and became visible, although owing to the darkness it was not yet possible to see who this man might be. On turning into the savannah, he stopped his horse, seemed to look around him, with an air of indecision, for some minutes, then, loosening the rein again, he came up toward the boucan at a sharp trot. On reaching the two men, who continued their supper quietly, while keeping an eye on him, he bowed, and addressed them in Spanish— "Worthy friends," he said to them, "whoever you may be, I ask you, in the name of the Lord, to grant a traveller, who has lost his way, hospitality for this night." "Here is fire, and here is meat," the buccaneer replied, laconically, in the same language the traveller had employed; "rest yourself, and eat." "I thank you," he said. He dismounted: in the movement he made to leave the saddle, his cloak flew open, and the buccaneers perceived that the man was dressed in a religious garb. This discovery surprised them, though they did not allow it to be seen.
- 41. On his side the stranger gave a start of terror, which was immediately suppressed, on perceiving that in his precipitation to seek a shelter for the night, he had come upon a boucan of French adventurers. The latter, however, had made him a place by their side, and while he was hobbling his horse, and removing its bridle, so that it might graze on the tall close grass of the savannah, they had placed for him, on a palm leaf, a lump of meat sufficient to still the appetite of a man who had been fasting for four and twenty hours. Somewhat reassured by the cordial manner of the adventurers, and, in his impossibility to do otherwise, bravely resolving to accept the awkward situation in which his awkwardness had placed him, the stranger sat down between his two hosts, and began to eat, while reflecting on the means of escaping from the difficult position in which he found himself. The adventurers, who had almost completed their meal before his arrival, left off eating long before him; they gave their dogs the food they had been expecting with so much impatience, then lit their pipes, and began smoking, paying no further attention to their guest beyond handing him the things he required. At length the stranger wiped his mouth, and, in order to prove to his hosts that he was quite as much at his ease as they, he produced a leaf of paper and tobacco, delicately rolled a cigarette, lit it, and smoked apparently as calmly as themselves. "I thank you for your generous hospitality, señores," he said, presently, understanding that along silence might be interpreted to his disadvantage, "I had a great necessity to recruit my strength, for I have been fasting since the morning." "That is very imprudent, señor," Lepoletais answered, "to embark thus without any biscuit, as we sailors say; the savannah is somewhat like the sea, you know when you start on it, but you never know when you will leave it again."
- 42. "What you say is perfectly true, señor; had it not been for you, I am afraid I should have passed a very bad night." "Pray say no more about that, señor; we have only done for you what we should wish to be done for us under similar circumstances. Hospitality is a sacred duty, which no one has a right to avoid: besides, you are a palpable proof of it." "How so?" "Why, you are a Spaniard, if I am not mistaken, while we, on the contrary, are French. Well, we forget for the moment our hatred of your nation, to welcome you at our fireside, as every guest sent by Heaven has the right to be received." "That is true, señor, and I thank you doubly, be assured." "Good Heavens!" the buccaneer replied, "I assure you that you act wrongly in dwelling so much on this subject. What we are doing at this moment is as much for you as in behalf of our honour, hence I beg you, señor, not to say any more about it, for it is really not worth the trouble." "Bless me, señor," L'Olonnais said with a laugh, "why, we are old acquaintances, though you little suspect it, I fancy." "Old acquaintances!" the stranger exclaimed, in surprise; "I do not understand you, señor." "And yet what I am saying is very clear." "If you would deign to explain," the stranger replied, completely thrown on his beam ends, as Lepoletais would have said, "perhaps I shall understand, which, I assure you, will cause me great pleasure." "I wish for nothing better than to explain myself, señor," L'Olonnais said, with a bantering air; "and in the first place, permit me to observe, that, though your cloak is so carefully buttoned, it is not sufficiently so to conceal the Franciscan garb you wear under it." "I am indeed a monk of that order," the stranger answered, rather disconcerted; "but that does not prove that you know me."
- 43. "Granted, but I am certain that I shall bring back your recollection by a single word." "I fancy you are mistaken, my dear señor, and that we never saw each other before." "Are you quite sure of that?" "Man, as you are aware, can never be sure of anything; still, it seems to me—" "And yet, it is so long since we met; it is true that you possibly did not pay any great attention to me." "On my honour, I know not what you mean," the monk remarked after attentively examining him for a minute or two. "Come," the engagé said with a laugh, "I will take pity on your embarrassment; and, as I promised you, dissipate all your doubts by a single word; we saw each other on the island of Nevis. Do you remember me?" At this revelation, the monk turned pale; he lost countenance, and for some minutes remained as if petrified; still the thought of denying the truth did not come to him for a second. "Where," L'Olonnais added, "you had a long conversation with Montbarts." "Still," the monk said with a hesitation that was not exempt from terror, "I do not understand—" "How I knew everything," L'Olonnais interrupted him laughingly, "then, you have not got to the end of your astonishment." "What, I am not at the end?" "Bah, Señor Padre, do you fancy that I should have taken the trouble to bother you about such a trifle? I know a good deal more." "What do you say?" the monk exclaimed, recoiling instinctively from this man whom he was not indisposed to regard as a sorcerer, the more so because he was a Frenchman, and a buccaneer to boot,
- 44. two peremptory reasons why Satan should nearly be master of his soul, if by chance he possessed one, which the worthy monk greatly doubted. "Zounds!" the engagé resumed, "You suppose, I think, that I do not know the motive of your journey, the spot where you have come from, where you are going, and more than that, the person you are about to see." "Oh, come, that is impossible," the monk said with a startled look. Lepoletais laughed inwardly at the ill-disguised terror of the Spaniard. "Take care, father," he whispered mysteriously in Fray Arsenio's ear, "that man knows everything; between ourselves, I believe him to be possessed by the demon." "Oh!" he exclaimed, rising hastily and crossing himself repeatedly, which caused the adventurers a still heartier laugh. "Come, resume your seat and listen to me," L'Olonnais continued as he seized him by the arm, and obliged him to sit down again, "my friend and I are only joking." "Excuse me, noble caballeros," the monk stammered, "I am in an extraordinary hurry, and must leave you at once, though most reluctantly." "Nonsense! Where could you go alone at this hour? Fall into a bog. Eh?" This far from pleasant prospect caused the monk to reflect; still, the terror he felt was the stronger. "No matter," he said, "I must be gone." "Nonsense, you will never find your road to the hatto del Rincón in this darkness." This time the monk was fairly conquered, this new revelation literally benumbed him, he fancied himself suffering from a terrible
- 45. nightmare, and did not attempt to continue an impossible struggle. "There," the engagé resumed, "now, you are reasonable; rest yourself, I will not torment you any more, and in order to prove to you that I am not so wicked as you suppose me, I undertake to find you a guide." "A guide," Fray Arsenio stammered, "Heaven guard me from accepting one at your hand." "Reassure yourself, señor Padre, it will not be a demon, though he may possibly have some moral and physical resemblance with the evil spirit; the guide I refer to is very simply a Carib." "Ah!" said the monk drawing a deep breath, as if a heavy weight had been removed from his chest, "If he is really a Carib." "Zounds! Who the deuce would you have it be?" Fray Arsenio crossed himself devoutly. "Excuse me," he said, "I did not wish to insult you." "Come, come, have patience, I will go myself and fetch the promised guide, for I see that you are really in a hurry to part company." L'Olonnais rose, took his fusil, whistled to a bloodhound, and went off at a rapid pace. "You will now be able," said Lepoletais, "to continue your journey without fear of going astray." "Has that worthy caballero really gone to fetch me a guide, as he promised?" Fray Arsenio asked, who did not dare to place full confidence in the engagé's word. "Hang it! I know no other reason why he should leave the boucan." "Then you are really a buccaneer, señor?" "At your service, padre." "Ah, ah! And do you often come to these parts?"
- 46. "Deuce take me if I do not believe you are questioning me, monk," Lepoletais said with a frown, and looking him in the face; "how does it concern you whether I come here or not?" "Me? Not at all." "That is true, but it may concern others, may it not? And you would not be sorry to know the truth." "Oh? can you suppose such a thing?" Fray Arsenio hastily said. "I do not suppose, by Heaven, I know exactly what I am saying, but, believe me, señor monk, you had better give up this habit of questioning, especially with buccaneers, people who through their character, do not like questions, or else you might some day run the risk of being played an ugly trick. It is only a simple piece of advice I venture to give you." "Thank you, señor, I will bear it in mind, though in saying what I did, I had not the intention you suppose." "All the better, but still profit by my hint." Thus rebuffed, the monk shut himself up in a timid silence; and in order to give a turn to his thoughts which, we are bound to say, were anything but rosy colored at this moment, he took up the rosary hanging from his girdle, and began muttering prayers in a low voice. Nearly an hour passed then without a word being exchanged between the two men; Lepoletais cut up tobacco, while humming a tune, and the monk prayed, or seemed to be doing so. At length a slight noise was heard a short distance off, and a few minutes later the engagé appeared, followed by an Indian, who was no other than Omopoua, the Carib chief. "Quick, quick, señor monk," L'Olonnais said gaily; "here is your guide, I answer for his fidelity; he will lead you in safety within two gun shots of the hatto."
- 47. The monk did not let the invitation be repeated, for anything seemed to him preferable to remaining any longer in the company of these two reprobates; besides, he thought that he had nothing to fear from an Indian. He rose at one bound, and bridled his horse again, which had made an excellent supper, and had had all the time necessary to rest. "Señores," he said, so soon as he was in the saddle, "I thank you for your generous hospitality, may the blessing of the Lord be upon you!" "Thanks," the engagé replied with a laugh, "but one last hint before parting; on arriving at the hatto, do not forget to tell Doña Clara from me, that I shall expect her here tomorrow; do you hear?" The monk uttered a cry of terror; without replying, he dug his spurs into his horse's flanks, and set off at a gallop, in the direction where the Carib was already going, with that quick, elastic step, with which a horse has a difficulty in keeping up. The two buccaneers watched his flight with a hearty laugh, then, stretching out their feet to the fire, and laying their weapons within reach, they prepared to sleep, guarded by their dogs, vigilant sentries that would not let them be surprised. CHAPTER XXVI. THE CONSEQUENCES OF A MEETING. Fray Arsenio followed his silent guide delightedly, although he was surrendered into the hands of an Indian, who must instinctively hate the Spaniards, those ferocious oppressors of his decimated and almost destroyed race. Still, the monk was glad at having escaped
- 48. safe and sound from the clutches of the adventurers, whom he feared not only as ladrones, that is to say, men without faith and steeped in vice, but also as demons, or at the least sorcerers in regular connection with Satan, for such were the erroneous ideas which the most enlightened of the Spaniards entertained about the filibusters and buccaneers. It had needed all the devotion which the monk professed for Doña Clara, and all the ascendancy that charming woman possessed over those who approached her, to make him consent to execute a plan so mad in his opinion, as that of entering into direct relation with one of the most renowned chiefs of the filibusters, and it was with a great tremor that he had accompanied his penitent to Nevis. When we met him, he was proceeding to the hatto, to inform Doña Clara, as had been arranged between them, of the arrival of the filibustering squadron at Port Margot, and consequently of Montbart's presence in the island of Saint Domingo. Unfortunately the monk, but little used to night journeys, across untrodden roads which he must guess at every step, lost himself on the savannah; overcome with terror, almost dead with hunger, and worn out by fatigue, the monk had seen the light of a fire flashing a short distance off; the sight of this had restored him hope, if not courage, and he had consequently ridden as fast as he could toward the fire, and tumbled headlong into a boucan of French adventurers. In doing this, he unconsciously followed the example of the silly moth, which feels itself irresistibly attracted to the candle in which it singes its wings. More fortunate than these insects, the monk had burned nothing at all; he had rested, eaten and drunk well, and, apart from a very honest terror at finding himself so unexpectedly in such company, he had escaped pretty well, or at least he supposed so, from this great danger, and had even succeeded in obtaining a guide. Everything, then, was for the best, the Lord had not ceased to watch over His servant, and the latter only needed to let himself be guarded by
- 49. Him. Moreover the monk's confidence was augmented by the taciturn carelessness of his guide who, without uttering a syllable, or even appearing to trouble himself about him the least in the world, walked in front of his horse, crossing the savannah obliquely, making a way through the tall grass, and seemed to direct himself as surely amid the darkness that surrounded him, as if he had been lit by the dazzling sunbeams. They went on thus for a long time following each other without the interchange of a word; like all the Spaniards, Fray Arsenio professed a profound contempt for the Indians, and it was much against his will that he ever entered into relations with them. For his part, the Carib was not at all anxious to carry on with this man, whom he regarded as a born foe of his race, a conversation which could only be an unimportant gossip. They had reached the top of a small hill, from which could be seen gleaming in the distance, like so many luminous dots, the watch fires of the soldiers encamped round the hatto, when all at once, instead of descending the hill and continuing his advance, Omopoua stopped, and looked round him anxiously, while strongly inhaling the air, and ordering the Spaniard by a wave of his hand to halt. The latter obeyed and remained motionless as an equestrian statue, while observing with a curiosity blended with a certain amount of discomfort, the manoeuvres of his guide. The Carib had laid himself down and was listening with his ear to the ground. At the end of a few minutes he rose again, though he did not cease listening. "What is the matter?" the monk, whom this conduct was beginning seriously to alarm, asked. "Horsemen are coming towards us at full speed." "Horsemen at this hour of night on the savannah?" Fray Arsenio remarked incredulously; "It is impossible."
- 50. "Why, you are here?" the Indian said with a jeering smile. "Hum! That is true," the monk muttered, struck by the logic of the answer; "who can they be!" "I do not know, but I will soon tell you," the Carib answered. And before the monk had the time to ask him what his scheme was, Omopoua glided through the tall grass and disappeared, leaving Fray Arsenio greatly disconcerted at this sudden flight, and extremely annoyed at finding himself thus left alone in the middle of the desert. A few minutes elapsed, during which the monk tried, though in vain, to hear the sound which the Indian's sharp sense of hearing had caused him to catch long before, amid the confused rumours of the savannah. The monk, believing himself decidedly deserted by his guide, was preparing to continue his journey, leaving to Providence the care of bringing him safely into port, when he heard a slight rustling in the bushes close to him, and the Indian reappeared. "I have seen them," he said. "Ah!" the monk replied; "And who are they?" "White men like you." "Spaniards in that case?" "Yes, Spaniards." "All the better," Fray Arsenio continued, whom the good news completely reassured; "are they numerous?" "Five or six at least; they are proceeding like yourself, towards the hatto, where, as far as I could understand, they are very eager to arrive." "That is famous; where are they at this moment?" "Two stones' throw at the most. According to the direction they are following, they will pass the spot where you are now standing."
- 51. "Better still. In that case we have only to wait." "You can do so, if you think proper; but I have no wish to meet them." "That is true, my friend," the monk remarked, with a paternal air. "And possibly such a meeting would not be agreeable to you; so pray accept my thanks for the manner in which you have guided me hitherto." "You are quite resolved on waiting for them, then? If you like, I can enable you to avoid them." "I have no motive for concealing myself from men of my own colour. Whoever they may be, I feel sure that I shall find friends in them." "Very good. Your affairs concern yourself, and I have nothing to do with them. But the sound is drawing nearer, and as they will speedily arrive, I will leave you, for it is unnecessary for them to find me here." "Farewell." "One last recommendation: if by chance they had a fancy to ask who served as your guide, do not tell them." "It is not at all probable they will ask this." "No matter. Promise me, if they do, to keep my secret." "Very good. I will be silent, since you wish it; although I do not understand the motive for such a recommendation." The monk had not finished the sentence, ere the Indian disappeared. The horsemen were rapidly approaching. The galloping of their steeds echoed on the ground like the rolling of thunder. Suddenly several shadows, scarcely distinguishable in the obscurity, rose as it were in the midst of the darkness, and a sharp voice shouted— "Who goes there?" "A friend!" the monk answered.
- 52. "Tell your name, ¡sangre de Dios!" the voice repeated, passionately, while the dry snap of a pistol being cocked, sounded disagreeably in the monk's ears. "At night there are friends in the desert!" "I am a poor Franciscan monk, proceeding to the hatto del Rincón; and my name is Fray Arsenio Mendoza." A hoarse cry replied to the monk's words—a cry whose meaning he had not the time to conjecture; that is to say, whether it was the result of pleasure or anger; for the horsemen came up with him like lightning, and surrounded him even before he could understand the reason of such a headlong speed to reach him. "Why, señores," he exclaimed, in a voice trembling with emotion, "what is the meaning of this? Have I to do with the ladrones?" "Good! Good! Calm yourself, Señor Padre," a rough voice answered, which he fancied he recognised. "We are not ladrones, but Spaniards like yourself; and nothing could cause us more pleasure than meeting you at this moment." "I am delighted at what you say to me, caballero. I confess that at first the suddenness of your movements alarmed me; but now I am completely reassured." "All the better," the stranger replied, ironically; "for I want to talk with you." "Talk with me, señor?" he said, with surprise. "The spot and the hour are badly chosen for an interview, I fancy. If you will wait till we reach the hatto, I will place myself at your disposal." "Enough talking. Get off your horse," the stranger observed, roughly; "unless you wish me to drag you off." The monk took a startled glance around him, but the horsemen looked at him savagely, and did not appear disposed to come to his help.
- 53. Fray Arsenio, through profession and temperament, was quite the opposite of a brave man. The way in which the adventure began was commencing seriously to alarm him. He did not yet know into what hands he had fallen, but everything led him to suppose that these individuals, whoever they might be, were not actuated by kindly feelings towards him. Still any resistance was impossible, and he resigned himself to obey; but it was not without a sigh of regret, intended for the Carib, whose judicious advice he had spurned, that he at length got off his horse, and placed himself in front of his stern questioner. "Light a torch!" the strange horseman said. "I wish this man to recognise me, so that, knowing who I am, he may be aware that he cannot employ any subterfuge with me, and that frankness alone will save him from the fate that menaces him." The monk understood less and less. He really believed himself suffering from an atrocious nightmare. By the horseman's orders, however, one of his suite had lighted a torch of ocote wood. So soon as the flame played over the stranger's feature, and illumined his face, the monk gave a start of surprise, and clasped his hands at the same time as his countenance suddenly reassumed its serenity. "Heaven be praised!" he said, with an accent of beatitude impossible to render. "Is it possible that it can be you, Don. Stenio de Bejar? I was so far from believing that I should have the felicity of meeting you this night, Señor Conde, that, on my faith, I did not recognise you, and felt almost frightened." The Count, for it was really he whom the monk had so unfortunately met, did not answer for the moment, but contented himself with smiling. Don Stenio de Bejar, who had left Saint Domingo at full speed, for the purpose of going to the hatto del Rincón, in order to convince himself of the truth of the information given him by Don Antonio de
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