TEXTS IN OPERATIONAL RESEARCH Aimed at final year undergraduates and masters students, this is a new series of texts covering core subjects in Operational Research in an accessible student-friendly format. Each core subject will be paired with another core subject in order to provide maximum value for money for students. The Operational Research series aims to provide a new generation of European-originated texts focusing on the practical relevance of those topics to today s students. To guarantee accessibility, the texts are concise and take a non-mathematical orientation in favour of software applications and business relevance. These texts provide students with the grounding in Operational Research they need to become the practitioners, users and innovators of tomorrow.
TEXTS IN OPERATIONAL RESEARCH Linear Programming Mik Wisniewski Critical Path Analysis Jonathan H. Klein
Mik Wisniewski 2001 (Linear Programming) Jonathan Klein 2001 (Critical Path Analysis) All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1P 0LP. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2001 by PALGRAVE Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world PALGRAVE is the new global academic imprint of St. Martin s Press LLC Scholarly and Reference Division and Palgrave Publishers Ltd (formerly Macmillan Press Ltd). ISBN 978-0-333-76354-4 hardback ISBN 978-0-333-76355-1 ISBN 978-1-4039-3769-8 (ebook) DOI 10.1007/978-1-4039-3769-8 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. A catalogue record for this book is available from the British Library. Formatted by The Ascenders Partnership, Basingstoke 10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05 04 03 02 01
Contents Linear Programming 1 Introduction 3 The structure of the text 5 2 Linear Programming: an introduction 7 Business example 7 Formulating the problem 8 Solving an LP problem 10 Interpreting the solution 16 Simple sensitivity analysis 17 Other types of constraint and objective function 24 Solution to minimization problems 25 Infeasible and unbounded problems 26 Redundant constraints 28 Summary 29 3 The Simplex method 30 The Simplex formulation 30 The Simplex solution process 31 Summary of the Simplex method 40 Extensions to the Simplex 41 Minimization problems 48 The dual problem 49 Sensitivity analysis 56 Summary 66 4 Linear Programming and computer software 67 Microsoft Excel Solver 67 XPRESS-MP 71 Summary 75 5 LP in the real world 77 Santos seaport, Brazil 77 Brunswick Smelting, Canada 79 Aluminium recycling, Saudi Arabia 82 Summary 85 v
vi Contents 6 LP: where next? 87 Specialist LP applications 87 Development of other MP models 89 Algorithm development 91 Modelling and reporting developments 92 Conclusion 92 Bibliography 93 Exercises 96 Critical Path Analysis 7 Introduction 107 7.1 Projects 107 7.2 Characteristics of projects 110 7.3 An introductory example: planning and monitoring a research project 112 7.4 Structure of the text 117 8 Critical Path Network analysis techniques 119 8.1 Introduction 119 8.2 The critical path method 120 8.2.1 Work breakdown structure (WBS) 121 8.2.2 Precedence relationships 123 8.2.3 Activity-on-node (AON) diagram 124 8.2.4 Activity-on-arrow (AOA) diagram 126 8.2.5 Activity timings 127 8.2.6 Critical path 130 8.2.7 Dummy activities 130 8.2.8 CPM: concluding comments 132 8.3 The Gantt chart 132 8.4 Introducing duration uncertainty into network schedules 134 8.5 Trading off duration and cost 140 8.6 Resource usage 143 9 Critical Path Network software 148 9.1 Introduction 148 9.2 Commercial Critical Path network software 149 9.3 Critical Path Network applications on spreadsheets 155
Contents vii 10 Practical application 157 10.1 Introduction 157 10.2 Critical Path Analysis and the life cycle of projects 158 10.3 The size and detail of projects 160 10.4 Critical Path Network approaches within organizations 163 11 A survey of Critical Path Methods literature 165 11.1 Introduction 165 11.2 Introductory material 165 11.3 Specialized texts 166 11.4 Case studies and other material 167 12 Current issues in Critical Path Network analysis 168 12.1 Introduction 168 12.2 Current issues in Critical Path Methods use 168 12.3 Conclusion 173 References 175 Exercises 179
List of figures Linear Programming 2.1 Constraint 1 11 2.2 Feasible area 12 2.3 Objective function 14 2.4 Feasible area and objective function 15 2.5 Optimal solution 15 2.6 Sensitivity analysis: objective function 22 2.7 Minimization 26 2.8 Infeasibility 27 3.1 Simplex solutions 34 3.2 Surplus variables 43 4.1 Microsoft Excel Solver template 68 4.2 Spreadsheet set-up 68 4.3 Formulae 69 4.4 Solver details 69 4.5 Solution 70 4.6 Sensitivity analysis 71 4.7 XPRESS-MP model builder 72 4.8 XPRESS-MP optimizer dialogue box 73 4.9 Results dialogue box 74 4.10 Model errors 75 Critical Path Analysis 7.1 Gantt chart (or schedule graph) for the consulting project 114 8.1 A word breakdown structure for the EBSP Detector Project 122 8.2 An activity-on-node (AON) diagram for the EBSP Detector Project 125 8.3 An activity-on-arrow (AOA) diagram for the EBSP Detector Project 127 viii
List of figures ix 8.4 Activity-on-arrow diagram for the EBSP Detector Project 129 8.5 A dummy activity is used to distinguish between activities B and C 131 8.6 Activity D must be preceded by both of activities A and B, but activity C need only be preceded by activity A 131 8.7 The EBSP Detector Project displayed as a Gantt chart 133 8.8 Illustrations of the Beta probability distribution 135 8.9 Activity-on-arrow diagram for the EBSP Detector Project with revised data 138 8.10 The Normal probability distribution 139 8.11 The Gantt chart of Figure 2.7, with activity C2 reduced by one day 141 8.12 A minimum time schedule Gantt chart for the EBSP Detector Project 142 8.13 Latest time schedule for the EBSP Detector Project 145
List of tables Linear Programming Tableau 1 32 Tableau 2 33 Tableau 3 38 Tableau 4 39 Tableau 5 44 Tableau 6 45 Tableau 7 46 Tableau 8 47 Tables 3.1 Primal solution: Tableau 4 reiterated 51 3.2 Dual solution 51 3.3 Dual solution: minimization problem 55 3.4 Tableau 4 and its Simplex solution reiterated 57 3.5 Tableau 4 with RHS adjustments 58 3.6 Simplex solution with constraints taking the form 60 3.7 Optimal Simplex solution with additional product 64 3.8 Basic variable B 64 3.9 Basic variable D 65 E1 Raw materials for Exercise 5 97 E2 Exercise 7: 4-stage production process 98 Critical Path Analysis 7.1 Component activities for the consultancy project 113 8.1 Activities for the EBSP Detector Project 123 8.2 Activity durations and precedence relationships for the EBSP Detector Project 125 8.3 ES, LS, EF and LF times, total and free slack, and critical path status for the activities of the EBSP Detector Project 129 8.4 Most likely, optimistic and pessimistic estimates for the activities of the EBSP Detector Project 137 8.5 Probability of completion by various dates for the EBSP Detector Project 139 x
List of tables xi 8.7 Normal and crash durations, and costs of duration reductions, for activities C1, C2, C3 and C4 in the EBSP Detector Project 140 8.8 Activity reductions to derive a minimum time schedule for the EBSP Detector Project 142 8.9 Cost profiles for earliest time (ET) and latest time (LT) schedules for the EBSP Detector Project 146