TEXTS IN OPERATIONAL RESEARCH

Transcription

1 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.

2 TEXTS IN OPERATIONAL RESEARCH Linear Programming Mik Wisniewski Critical Path Analysis Jonathan H. Klein

3 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 First published 2001 by PALGRAVE Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y 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 hardback ISBN ISBN (ebook) DOI / 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

4 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

5 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 Projects Characteristics of projects An introductory example: planning and monitoring a research project Structure of the text Critical Path Network analysis techniques Introduction The critical path method Work breakdown structure (WBS) Precedence relationships Activity-on-node (AON) diagram Activity-on-arrow (AOA) diagram Activity timings Critical path Dummy activities CPM: concluding comments The Gantt chart Introducing duration uncertainty into network schedules Trading off duration and cost Resource usage Critical Path Network software Introduction Commercial Critical Path network software Critical Path Network applications on spreadsheets 155

6 Contents vii 10 Practical application Introduction Critical Path Analysis and the life cycle of projects The size and detail of projects Critical Path Network approaches within organizations A survey of Critical Path Methods literature Introduction Introductory material Specialized texts Case studies and other material Current issues in Critical Path Network analysis Introduction Current issues in Critical Path Methods use Conclusion 173 References 175 Exercises 179

7 List of figures Linear Programming 2.1 Constraint Feasible area Objective function Feasible area and objective function Optimal solution Sensitivity analysis: objective function Minimization Infeasibility Simplex solutions Surplus variables Microsoft Excel Solver template Spreadsheet set-up Formulae Solver details Solution Sensitivity analysis XPRESS-MP model builder XPRESS-MP optimizer dialogue box Results dialogue box Model errors 75 Critical Path Analysis 7.1 Gantt chart (or schedule graph) for the consulting project A word breakdown structure for the EBSP Detector Project An activity-on-node (AON) diagram for the EBSP Detector Project An activity-on-arrow (AOA) diagram for the EBSP Detector Project 127 viii

8 List of figures ix 8.4 Activity-on-arrow diagram for the EBSP Detector Project A dummy activity is used to distinguish between activities B and C Activity D must be preceded by both of activities A and B, but activity C need only be preceded by activity A The EBSP Detector Project displayed as a Gantt chart Illustrations of the Beta probability distribution Activity-on-arrow diagram for the EBSP Detector Project with revised data The Normal probability distribution The Gantt chart of Figure 2.7, with activity C2 reduced by one day A minimum time schedule Gantt chart for the EBSP Detector Project Latest time schedule for the EBSP Detector Project 145

9 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 Dual solution Dual solution: minimization problem Tableau 4 and its Simplex solution reiterated Tableau 4 with RHS adjustments Simplex solution with constraints taking the form Optimal Simplex solution with additional product Basic variable B 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 Activities for the EBSP Detector Project Activity durations and precedence relationships for the EBSP Detector Project ES, LS, EF and LF times, total and free slack, and critical path status for the activities of the EBSP Detector Project Most likely, optimistic and pessimistic estimates for the activities of the EBSP Detector Project Probability of completion by various dates for the EBSP Detector Project 139 x

10 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 Activity reductions to derive a minimum time schedule for the EBSP Detector Project Cost profiles for earliest time (ET) and latest time (LT) schedules for the EBSP Detector Project 146