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UBC Theses and Dissertations

Network planning and resource allocation for project control Arden, Nicholas Russel 1968

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NETWORK ND  P L A N N I N G  R E S O U R C E  FOR  A L L O C A T I O N  P R O J E C T  C O N T R O L  NICHOLAS RUSSEL ARDEN B . A . , Cambridge U n i v e r s i t y ,  1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF SCIENCE i n the Department of Computer Science  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA November,  1968  In p r e s e n t i n g an  advanced  the  this  degree  Library shall  I further for  agree  scholarly  by  his  of  this  written  fulfilment  of  at  University  of  Columbia,  the  make  that  it  for  freely  permission  purposes  thesis  may be It  financial  for  Computer Science  of  University 8,  of  British  Columbia  Canada  November 18th. 1968  by  the  understood  gain  for  extensive  granted  is  British  available  permission.  Vancouver  Date  in p a r t i a l  representatives.  Department The  thesis  shall  Head o f  be  requirements  reference copying  that  not  the  of  I agree and this  or  allowed  without  that  Study. thesis  my D e p a r t m e n t  copying  for  or  publication my  ii  ABSTRACT  The problems i n v o l v e d i n network p l a n n i n g f o r p r o j e c t a r e examined w i t h r e f e r e n c e t o t h e p u b l i s h e d work i n t h i s  control  field.  V a r i o u s s o l u t i o n s a r e d e s c r i b e d and compared. A d e t a i l e d i n v e s t i g a t i o n i s made o f t h e s t a n d a r d assumptions concerning expected a c t i v i t y d u r a t i o n s .  The d i f f e r e n t approaches t o  t h e e s t i m a t i o n o f t h e s e times a r e shown t o be i n c o n s i s t e n t w i t h t h e i r areas o f a p p l i c a t i o n . A g l o b a l h e u r i s t i c s o l u t i o n t o t h e problem o f f i n d i n g t h e minimum v a l u e o f the maximum r e s o u r c e requirement i s presented.  T h i s procedure  during a project  uses a m o d i f i e d r e s o u r c e p r o f i l e .  r e s u l t s o f a comparison between t h i s s o l u t i o n and a s t a n d a r d  The  local  s o l u t i o n i n d i c a t e a s l i g h t improvement w i t h a c o n s i d e r a b l e i n c r e a s e i n computing t i m e . combination  The new approach p e r m i t s e a s i e r s u b m i n i m i z a t i o n .  A  o f t h e s e methods i s proposed.  EXAMINERS  TABLE OF CONTENTS PART ' A ' CHAPTER I. II.  ' INTRODUCTION  p  . . . . . . .  2  HISTORICAL BACKGROUND  . . . . . .  h  Introduction  It  Bar Charts  •.' . •  h  Reporting Systems Networks  •  . . . . . . . . . . . . . . . . . . . . . .  CPM  . . . .  PERT  5 6 6 6  Subsequent Developments  . . . . . . . . .  Summary III.  AGS  7 8  THE NETWORK  9  Introduction  9  Activities  9  Split Activities  10  Repetitive A c t i v i t i e s Events  . . . . . . . . . . . . . . .  11  . . . . . . . . . . . . . . . . . . . . . . .  11  Types of Networks Subnetworks  . . . . . . •  13 1U  Numbering of Events and A c t i v i t i e s  15  Network Errors  15  . . . . . . . . . . . . . . . . . . .  iv  CHAPTER  PAGE Network Construction Summary  IV.  16  •  17  .  19  . . . . . . . .  CRITICALITT Introduction  . . . . . . . . . . . . . . . . . . .  The C r i t i c a l Path  V.  . . .  .  19  . .  19  Slack and Float  20  Types of Float  22  C r i t i c a l and N o n - c r i t i c a l A c t i v i t i e s  23  Summary  25"  TIME Introduction Time Estimates  .  26  • •  26  • •  27  Extensions of the P r o b a b i l i t y Concept Errors inherent i n the D i s t r i b u t i o n Assumptions  29 . •  Conclusions Summary VI.  31 3k  .  .  .  .  .  COST  35 .  Introduction  37 37  The Cost Problem  . • •  •  37  The A c t i v i t y Time/Cost Relationship  39  The Solution  39  . .  . .  Modifications  hi  Other Solutions  hZ  V  (CHAPTER  PAGED i f f e r e n t Problems Assumptions v Summary  VH.  h3  . . . . . . ...  . . .  . . . . . . .  .  U5  .  hi  MANPOWER AND OTHER RESOURCES Introduction  . . . .  The Problems  . . . . . . . . . . . .  Types o f S o l u t i o n  U7  . . . . . . . . . . . ... .  . . . . . . . . . . . . . . . .  The G e n e r a l S o l u t i o n t o t h e L e v e l l i n g Problem Criteria  hh  • •  h7 U8 h9 U9  . .  Other S o l u t i o n s  .  50  The General S o l u t i o n t o t h e R e s o u r c e - C o n s t r a i n e d Problem  50  . . . . . . . .  P r i o r i t y Systems Methods o f S o l u t i o n  . . . . . . . . . . . . . . . .  51  . . . . . . . . . . . . . . .  52 55  A n a l y t i c Approaches •  56  ... .  57  Other Approaches Summary VIII.  . . . . • • . .  .  60  MODELS AND SOLUTIONS Introduction  60  . . . . . . . . . . . . . . . . . .  M a t h e m a t i c a l Models S o l u t i o n s t o t h e Time/Cost Problems A n o t h e r Model  . . . . . . . . .  60 6k 65  vi  (CHAPTER  PAGE Analogues Summary  IX.  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  VARIATIONS ON A THEME Introduction  66 67  . . . . ........ . . . . . .  . . . . . . . . . . . . . . . . . .  69  PERT AND CPM  69  Cost Considerations  . . . . . . . . . . . . .  72  Resource Requirements  . . . . . . . . . . . . .  73 ?U  Other V a r i a t i o n s A G e n e r a l i z e d Network Approach  X.  69  . . . . . . . .  75  D i f f e r e n t Approaches  77  Summary  78  CONCLUSIONS  . . . . . . . . . . . . . . . . . . .  81  . . . . . . . . . . . . . . . . .  81  . . . . . . . . . . . . . . . . . . . .  81  . . . . . . . . . . . . . . . . .  82  Introduction PERT/CPM The Program Summary  . .  . . . . . . . .  . . .  83  PART »B«  XI.  AN INTRODUCTION TO THE PROBLEM  . . . . . . .  .  85  The Problem  85  The Approach  85  The S o l u t i o n  86  The P r e s e n t a t i o n  86  vii  CHAPTER XII.  PAGE  Introduction  XIII.  88  CRITICAL PATH ANALYSIS . . . . . . . . . . . . . . . .  •  Data P r e p a r a t i o n  88  Finding the C r i t i c a l Path  93  BOUNDS AND RESOURCE PROFILES  97  Introduction Bounds  . . . . . . . . .  . . . . . . . . .  .  Logical Implications SOLUTIONS  101 . . . . . . . . . . . . . .  . . . . .....  Introduction  107 110  . . . . . . . . .  110  The G l o b a l S o l u t i o n  110 . . . . . . .  llU  RESULTS, EXTENSIONS AND CONCLUSIONS  116  The L o c a l S o l u t i o n XV.  97 97  Bounds and P r o f i l e s  XIV.  88  Introduction  . . .. • . . . . . . . . • . ••  Comparative R e s u l t s  ..  D i f f e r e n t P r i o r i t y Systems  116 116  .  117  Percentage o f C r i t i c a l A c t i v i t i e s and E v e n t s . .  121  Developments  121  BIBLIOGRAPHY  . . . . .  . . . . .  126  viii  'APPENDIX  PAGE  •A  'THE BETA DISTRIBUTION AND PERT  •B'  THE ACTIVITY TIME/COST RELATIONSHIP  •C«  RESULTS  •D'  THE PROGRAM  1  . . . . . . .... . . . . .  .  138 lhh 1U7  ....................  15?  ix  LIST OF TABLES  TABLE I. II. III. IV. V. VI. VET. VT.U. IX.  PAGE MANAGEMENT PROJECT CONTROL SYSTEMS  70  COMPARISON OF TWO LOWER RESOURCE BOUND ESTIMATES A COMPARISON OF RESULTS  . .  .  118  A. COMPARISON OF TIMES  118  COMPARISON OF PRIORITY SYSTEMS  120  LIKELY PERCENTAGE OF CRITICAL ACTIVITIES  12U  LIKELY PERCENTAGE OF CRITICAL EVENTS  128  . . . . . . .  SAMPLE PARINGS OF X AND k  1^3  EVENTS/ACTIVITY RELATIONSHIP WITH VARIABLE CONTROL 'C»  Hi8  X-  SOLUTIONS TO THE RESOURCE MINIMAX PROBLEM  XT.  DETAILED COMPARISON OF TWO LOWER RESOURCE BOUND ESTIMATES  XU.  .  . . . .  •  152  153  THE RELATIONSHIP BETWEEN THE NUMBER OF ACTIVITIES AND THE NUMBER OF CRITICAL ACTIVITIES  XIV.  150  COMPARISON OF TWO PRIORITY SYSTEMS WITH THE CHOSEN SYSTEM P2  XILT.  101  . .  155  THE RELATIONSHIP BETWEEN THE NUMBER OF EVENTS AND THE NUMBER OF CRITICAL EVENTS  156  X  L I S T OF FIGURES  'FIGURE  PAGE  1.  THE ACTIVITY/EVENT RELATIONSHIP UNDER CONTROL « C  2.  POSSIBLE DEPENDENCIES AMONGST A C T I V I T I E S  3.  RELATIONSHIP BETWEEN NUMBER OF A C T I V I T I E S AND THE PERCENTAGE OF CRITICAL A C T I V I T I E S  IN  . .  .........  92 106  122  RELATIONSHIPMENT BETWEEN NUMBER OF EVENTS AND THE PERCENTAGE OF CRITICAL EVENTS  5.  .....  . .  POSSIBLE TIME/COST CURVES  . . . . . . . . . .  123 11x6  ACKNOWLEDGMENT  I t i s a s i n c e r e p l e a s u r e f o r me t o have t h i s o p p o r t u n i t y t o express my g r a t i t u d e t o t h e f o l l o w i n g f o r t h e i r a s s i s t a n c e i n t h e development and w r i t i n g o f t h i s  thesis.  To I n t e r n a t i o n a l B u s i n e s s Machines C o r p o r a t i o n , f o r t h e i r f i n a n c i a l a s s i s t a n c e by t h e award o f Summer F e l l o w s h i p s i n 1967 and  1968. To my S u p e r v i s o r , P r o f e s s o r J.M. Kennedy, f o r much u s e f u l advice i n the preparation o f the t h e s i s . To my f r i e n d , R o b e r t J . Epp, f o r h i s p a t i e n t h e l p i n t h e c o d i n g and debugging o f t h e programs. To my d e a r w i f e , G l o r i a , f o r h e r e x c e l l e n t t y p i n g and f o r h e r c o n s t a n t encouragement and u n f a i l i n g u n d e r s t a n d i n g phase o f t h e t h e s i s .  throughout  every  PART  'A'  CHAPTER I Introduction  P a r t 'A' comprises  a comprehensive survey o f t h e problems  and s o l u t i o n s i n v o l v e d i n p r o j e c t management c o n t r o l , and o f t h e p u b l i s h e d papers and books t h a t d i s c u s s them. A s k e t c h o f t h e h i s t o r i c a l development o f p r o j e c t management t e c h n i q u e s , from t h e b e g i n n i n g o f t h e t w e n t i e t h c e n t u r y up t o t h e i n t r o d u c t i o n o f network p l a n n i n g , i s p r e s e n t e d i n Chapter I I .  This i s  f o l l o w e d , i n t h e next s e c t i o n , by a d e s c r i p t i o n o f the v a r i o u s components and t y p e s o f p r o j e c t network.  The e s t a b l i s h m e n t o f the b a s i c  network l e a d s t o t h e c o n s i d e r a t i o n o f the concept in  Chapter  o f c r i t i c a l ! t y and,  V, t o a d i s c u s s i o n c o n c e r n i n g time e s t i m a t e s .  The s i x t h  and s e v e n t h c h a p t e r s examine problems r e l a t i n g t o c o s t , and t o manpower and o t h e r r e s o u r c e s .  Mathematical  models and a n a l y t i c o r analogue s o l u -  t i o n s a r e n o t d e s c r i b e d i n any d e t a i l u n t i l Chapter  VIII.  Preceding the  f i n a l c o n c l u s i o n s i n Chapter X i s a s e c t i o n w h i c h p r e s e n t s those  systems  t h a t implement t h e s o l u t i o n s g i v e n e a r l i e r . I n a d d i t i o n t o t h e b i b l i o g r a p h y , t h e f i r s t two appendices to  P a r t 'A'.  Appendix 'A' i s an a n a l y s i s o f t h e b e t a  assumption f o r a c t i v i t y d u r a t i o n e s t i m a t e s .  refer  distribution  Appendix"S5 examines t h e  i m p l i c a t i o n s o f t h e v a r i o u s assumptions made about t h e form o f t h e a c t i v i t y time/cost f u n c t i o n .  3 Although most a r t i c l e s quote references,  three works i n p a r t i -  c u l a r , by Davies (2h), by IBM ( l ) and by Sobczak (11), and useful b i b l i o g r a p h i e s .  give  extensive  The paper by Davies also gives excellent  coverage to the major work done i n resource a l l o c a t i o n and minimum cost scheduling.  A w e l l - w r i t t e n , comprehensive book, which includes a wide  variety of t h e o r e t i c a l applications, i s and Scheduling , by Battersby (9).  'Network Analysis for Planning  A t r i l o g y by Martino (80)  (81)  (10),  i n a simple presentation, adequately covers the three basic approaches, involving c r i t i c a l path analysis,  the time/cost tradeoff,  strained resource scheduling and l e v e l i n g .  and con-  A series of masters 1 theses  about project c o n t r o l edited by Mattel (12), includes a good description of the h i s t o r i c a l development and a detailed survey of commercial and i n d u s t r i a l applications.  The table of acronyms and meanings f o r project  management systems and t h e i r descriptions were compiled from a book by Baker and E r i s (8), another by Levin and K i r k p a t r i c k (79) and two a r t i c l e s by Sobczak (11)  (60).  CHAPTER I I  HISTORICAL BACKGROUND  Introduction The h i s t o r y o f management a i d s i n t h e p l a n n i n g and c o n t r o l o f p r o j e c t s i s t r a c e d from t h e i n t r o d u c t i o n o f b a r - c h a r t s , a t t h e b e g i n n i n g o f the t w e n t i e t h c e n t u r y , t o t h e i n t r o d u c t i o n o f t h e C r i t i c a l P a t h Method and o f t h e Program E v a l u a t i o n and Review Technique,  f i f t y years  later.  Some a p p l i c a t i o n s and t h e e n s u i n g development o f network t e c h n i q u e s a r e mentioned.  Bar  Charts The B a r - C h a r t i s a p l a n n i n g and p r o g r e s s e v a l u a t i o n t e c h n i q u e .  I t was i n t r o d u c e d a t t h e b e g i n n i n g o f t h e t w e n t i e t h c e n t u r y and r e p r e s e n ted  t h e f i r s t f o r m a l i z e d attempt  f i e l d of project control.  t o improve management e f f i c i e n c y i n t h e  A bar-chart i s a g r a p h i c a l i l l u s t r a t i o n o f the  s c h e d u l i n g o f t h e v a r i o u s a c t i v i t i e s t h a t comprise  a project.  The graph  c o n s i s t s o f a t i m e - s c a l e d h o r i z o n t a l a x i s and a s e r i e s o f h o r i z o n t a l l i n e s , e a c h r e p r e s e n t i n g an a c t i v i t y and drawn, from the planned time t o t h e expected f i n i s h i n g time o f t h e a c t i v i t y .  starting  The b a r - c h a r t f o r  a p r o j e c t was f i r s t drawn a t t h e p l a n n i n g stage when i t n o t o n l y i n d i c a t e d the e x p e c t e d d u r a t i o n o f t h e p r o j e c t b u t , b y t h e v e r y n a t u r e o f i t s cons t r u c t i o n , f o r c e d management i n t o a c a r e f u l c o n s i d e r a t i o n o f t h e components o f the p r o j e c t .  A t v a r i o u s s t a g e s b e f o r e t h e completion o f t h e p r o -  j e c t , t h e b a r - c h a r t was updated and t h e n a t u r e o f t h e n e c e s s a r y changes  s i n d i c a t e d to management the progress—or lack of progress—that made.  was being  The bar-chart was therefore useful both i n planning and i n progress  evaluation.  A comprehensive treatment o f the use of bar-charts i n the  planning and c o n t r o l of research and development projects i s given by Williams  (13).  In 1910, Henry Lawrence Gantt introduced the concept of a milestone. A milestone i s a p a r t i c u l a r moment of time usually i n d i c a t i n g the s t a r t or end of an a c t i v i t y . ment.  The designation of a milestone i s made by manage-  The GantVchart i s a v a r i a t i o n of the bar-chart.  It is basically  s i m i l a r but groups c e r t a i n a c t i v i t i e s i n t o a continuous horizontal l i n e , broken only by milestones, which are usually represented by c i r c l e s , each containing a reference to a p a r t i c u l a r milestone.  Reporting Systems Although there was no further change i n the basic technique for the next f o r t y years, considerably more sophisticated presentations of project status to management were developed.  Of p a r t i c u l a r note are the  Milestone and Line-of-Balance Reporting Systems, which are admirably described by Baar and Howard (?)• each milestone i s accomplished.  The Milestone System presents reports as The reports graphically compare the pro-  gress made with the o r i g i n a l p l a n .  The Line-of—Balance Technique pre-  sents more detailed reports, usually at less frequent i n t e r v a l s .  These  reports p i c t o r i a l l y portray the comparison between the actual achievements i n such areas as production or d i s t r i b u t i o n and the planned objecti v e s , i n addition to the project progress chart of the Milestone System.  6 Networks For any project there are interdependencies amongst the a c t i v i t i e s caused by the need f o r c e r t a i n a c t i v i t i e s to be completed before other a c t i v i t i e s may s t a r t .  This p a r t i a l ordering cannot be shown on a  bar-chart and i s only indicated i n part on a Gantt-chart.  According to  Salveson (129), Boyan's lectures on Target Commitment Scheduling at the Massachusetts I n s t i t u t e of Technology i n 19U6 (89) were probably the f i r s t i n d i c a t i o n that networks could be used to f u l l y describe the i n t e r dependencies of the a c t i v i t i e s i n a p r o j e c t .  CPM However i t was not u n t i l Hayward and Robinson of E . I . Dupont de Nemours submitted a report (I4O) on December 30, 1956 which developed some basic concepts outlined by G . J . Fisher a year e a r l i e r , that the value of networks i n project control was recognized. Sperry-Rand Corporation, James E . K e l l e y J r . ,  In collaboration with the of Dr. John W. Mauchly's  Univac Applications Research Centre, developed the mathematical model (hh) and M.R. Walker developed the computer program for what was soon to be known as CPM—the C r i t i c a l Path Method.  In September 1957, the  first  application was successfully completed using a Univac I Computer.  PERT According to Fisher (12) i n reference to CPM, " i n about mid-1957, the o r i g i n a l program was shown to a group from the Navy Department who were v i s i t i n g various industries to become acquainted with new management  7  techniques." visit  The degree t o w h i c h t h e u n d e r s t a n d i n g garnered i n t h a t  a f f e c t e d the r e s u l t s o f the U n i t e d S t a t e s Navy Department's i n v e s t i -  g a t i o n i n t o i m p r o v i n g t h e i r methods o f management c o n t r o l i s n o t known. However the outcome o f t h e i r work (5) was n o t made p u b l i c u n t i l March 1958.  The Navy's team was  composed o f C.E. C l a r k , D.G. Malcolm and J .  Roseboom o f t h e Booz, A l l e n and Hamilton c o n s u l t i n g f i r m , R. Young and E.  Lennon from t h e M i s s i l e System D i v i s i o n o f Lockheed A i r c r a f t  t i o n , and W. F a z a r o f t h e Navy's S p e c i a l P r o j e c t s O f f i c e .  Corpora-  Their inten-  t i o n was t o s p e c i f i c a l l y c o n s i d e r r e s e a r c h and development p r o j e c t s and from t h e i r work e v o l v e d t h e Program E v a l u a t i o n and Review T e c h n i q u e — PERT.  The PERT system was programmed on t h e N a v a l Ordinance R e s e a r c h  C a l c u l a t o r and was f i r s t implemented on p a r t s o f t h e P o l a r i s Program. The immediate s u c c e s s a c h i e v e d l e d t o t h e c o n t r o l o f t h e e n t i r e  Polaris  Program by P E R T — t o which d e c i s i o n much o f t h e e n s u i n g s u c c e s s o f t h a t program has been c r e d i t e d .  Subsequent Developments The v a l u e o f b o t h CPM and PERT was r e c o g n i z e d by i n d u s t r y and government and much f u r t h e r development has o c c u r r e d s i n c e the f i r s t announcements.  A l t h o u g h i n t h e i r o r i g i n a l form the t e c h n i q u e s d e a l t  o n l y w i t h time as a v a r i a b l e , c o s t , manpower and o t h e r r e s o u r c e s were soon t o be c o n s i d e r e d .  Many s i m i l a r systems w i t h d i f f e r e n t  were developed and these a r e d i s c u s s e d i n C h a p t e r I X .  acronyms  The i d e a s o f  PERT/CPM—the g e n e r a l t i t l e by which t h e method i s r e f e r r e d — h a v e been i n t r o d u c e d i n t o many d i f f e r e n t f i e l d s , s u c h as Communications (121) o r  8 Assembly Line Theory (61*).  The o r i g i n a l systems have been applied with  varying degrees of success and PERT/CPM i s now a generally management t o o l .  accepted  The areas of application vary from planning a, National  Exposition (82) to the staging of a Broadway production (9)—the lack of appreciation 'Morgana' received from the theatre c r i t i c s i n no way r e f l e c t s on the C r i t i c a l Path Method.  Further descriptions of the introduc-  t i o n of PERT/CPM may be found i n (83),  (120) and (122).  Summary The f i r s t formalized attempt to improve management e f f i c i e n c y i n the f i e l d of project control occurred at the beginning of the twentieth century with the introduction of bar-charts  and Gantt-charts.  These  charts indicated the expected occurrence of the various a c t i v i t i e s comp r i s i n g the p r o j e c t .  They were used i n conjunction with Milestone and  Line-of-Balance Reporting Systems to compare actual progress with planned objectives.  The introduction of- networks solved the problem of i n d i c a t i n g  graphically the interdependencies of the a c t i v i t i e s .  The C r i t i c a l Path  Method—CPM—introduced i n 1957 by E . I . Dupont de Nemours and Sperry-Rand, and the Program Evaluation and Review Technique—PERT—developed a year l a t e r by the United States Navy Department i n collaboration with Booz, A l l e n and Hamilton and with Lockheed, were the forerunners of many s i m i l a r techniques which considered other resources besides time. have been applied i n a wide variety of areas.  These techniques  CHAPTER H I THE NETWORK  Introduction The  b a s i c components o f t h e p r o j e c t — t h e a c t i v i t y and t h e e v e n t —  a r e d e f i n e d and some o f t h e problems i n v o l v e d i n t h e i r s p e c i f i c a t i o n a r e described,  A d i s c u s s i o n o f t h e d i f f e r e n t t y p e s o f network, i n c l u d i n g  t h e concept o f a g e n e r a l i z e d network, l e a d s t o c o n s i d e r a t i o n o f s u b n e t works and m u l t i - p r o j e c t s .  The i - j numbering o f events t h r o u g h a t o p o l o -  g i c a l s o r t i s shown t o be u s e f u l b u t n o t e s s e n t i a l .  A f t e r considering  p o s s i b l e e r r o r s i n the network d e s c r i p t i o n , o p t i m i z a t i o n t e c h n i q u e s f o r t h i s d e s c r i p t i o n are introduced complicated  and, f i n a l l y , a method f o r drawing an un-  p r o j e c t network i s p r e s e n t e d .  Activities An a c t i v i t y i s an i n t e g r a l p a r t o f a p r o j e c t . and  Both i t s beginning  i t s end must be c l e a r l y d e f i n e d and, between t h e s e p o i n t s i n t i m e ,  some e x p e n d i t u r e  o f r e s o u r c e s must o c c u r .  The r e s o u r c e s  r e q u i r e d b y an  a c t i v i t y a r e u s u a l l y some combination o f t i m e , c o s t , manpower, machinery and m a t e r i a l s .  The a c t i v i t y i s r e p r e s e n t e d  a r c j o i n i n g two nodes. the  i n t h e network by a d i r e c t e d  A c t i v i t i e s a r e r e f e r r e d t o a s ' j o b s ' i n some o f  literature. There i s a p a r t i a l o r d e r i n g o f a c t i v i t i e s caused by t h e need f o r  some a c t i v i t i e s t o be completed b e f o r e  o t h e r a c t i v i t i e s can s t a r t .  This  i s i n d i c a t e d i n t h e network by e n s u r i n g t h a t a l l a c t i v i t i e s i n t h e former  10 group precede  their  c o r r e s p o n d i n g a c t i v i t i e s i n t h e l a t t e r group a l o n g  a l l paths i n t h e network. mal network.  Consequently  t h e r e are no l o o p s i n t h e  nor-  There o f t e n o c c u r s a f u n c t i o n a l r e l a t i o n s h i p between c e r -  t a i n a c t i v i t i e s , which i s i n d i c a t e d i n t h e network by a 'dummy' a c t i v i t y . A dummy a c t i v i t y , t h e r e f o r e , r e q u i r e s no r e s o u r c e s and i s merely used t o i n d i c a t e an o r d e r i n g between a c t i v i t i e s .  Split  Activities The depth o f d e t a i l t o w h i c h t h e t o t a l workload  of a project i s  d i v i d e d i n t o a c t i v i t i e s i s n o r m a l l y l e f t to.management.  I f certain a c t i -  v i t i e s c a n s t a r t a f t e r an a c t i v i t y i s o n l y p a r t i a l l y completed,  then  t h a t a c t i v i t y s h o u l d be s p l i t i n t o two a c t i v i t i e s i n o r d e r t h a t t h e t r u e r e l a t i o n s h i p may be shown i n t h e network. t h a t i t i s p o s s i b l e f o r an a c t i v i t y ' s  However i t sometimes o c c u r s  p r o g r e s s t o be i n t e r r u p t e d a t  v a r i o u s s t a g e s w i t h o u t harming i t s s u c c e s s f u l c o m p l e t i o n .  Such ' s p l i t  a c t i v i t i e s ' a r e p e r m i t t e d i n some systems b u t are n o t c a t e r e d f o r i n / others.  K e l l e y ( i l l ) c l a i m s t h a t n o t t o p e r m i t a c t i v i t y s p l i t t i n g i s an  " u n d e s i r a b l e r e s t r i c t i o n on the performance o f work." practical  But there are  problems i n c u r r e d i n i n t e r r u p t i n g an a c t i v i t y , as i n d i c a t e d by  a s u p e r v i s i n g e n g i n e e r w i t h the S t e e l Company o f Wales (8I4), who c l a i m s t h a t doing so " i s v e r y unpopular w i t h the tradesmen, and i s n o t conducive t o h i g h q u a l i t y workmanship."  The s i m p l e s o l u t i o n t o t h i s  controversy  would appear t o be n o t t o a l l o w s p l i t a c t i v i t i e s , w h i l e s u b - d i v i d i n g t h o s e a c t i v i t i e s t h a t c o u l d be s p l i t i n t o an a p p r o p r i a t e number o f more d e tailed activities.  11 Repetitive A c t i v i t i e s As there i s no convenient f a c i l i t y for i n d i c a t i n g r e p e t i t i v e a c t i v i t i e s , which are frequently encountered i n a production environment, i n the normal networks, they are usually represented by a single activity.  Digman (9h) describes a system—PERT/LOB—in which r e p e t i t i v e  a c t i v i t i e s are permitted and are indicated as loops i n the network. An alternative procedure i s mentioned by Battersby ( 9 ) . A 'Ladder Diagram' permits t h i s type of production to be incorporated i n t o a network.  How-  ever t h i s method i s very l i m i t e d i n scope and can adversely affect the analysis of the network as a whole.  Events An event i s a p a r t i c u l a r moment of time usually representing the s t a r t or end o f one or more a c t i v i t i e s .  The only difference between an  event and a milestone i s that the milestone i s specified by management i n addition to the description of the project and that the event i s an inherent part of the project. event.  Every node of the network represents an  For s i m p l i c i t y , one node may represent the terminal event for a  few a c t i v i t i e s as w e l l as the i n i t i a l event for other a c t i v i t i e s . Battersby (9) suggests that the usual r a t i o of a c t i v i t i e s to events i s approximately three-to-two. Although i t i s perhaps natural to conceive of a single beginning and f i n a l event for a p r o j e c t , most systems permit more than one i n each category.  Thus a project s t a r t node i s defined as any node which i s not  terminal for any a c t i v i t y and, s i m i l a r l y , a project end node i s any  12 node which i s not i n i t i a l f o r any a c t i v i t y .  The end of a p r o j e c t , then,  occurs at the l a t e s t of i t s project end nodes. Two types of events have been s p e c i f i c a l l y defined.  An event  which i s the f i n a l event f o r p r e c i s e l y one a c t i v i t y and also the i n i t i a l event f o r several a c t i v i t i e s i s c a l l e d a 'Burst E v e n t ' .  Conversely,  an event which i s i n i t i a l f o r only one and f i n a l for several a c t i v i t i e s i s known as a 'Merge E v e n t ' • A l l a c t i v i t i e s can be categorized i n terms of t h e i r i n i t i a l and terminal events.  In f a c t ,  the Minuteman PERT System (67) s p e c i f i c a l l y  requires that the starting and ending events f o r each a c t i v i t y be d i s crete.  Although t h i s r e s t r i c t i o n i s unusual, a common rule i s that no  two a c t i v i t i e s should s t a r t and end at the same event.  Consequently,  dummy a c t i v i t i e s are introduced i n conjunction with a d d i t i o n a l events to avoid t h i s . restriction.  The Burroughs PERT/Time System (69), f o r example, has t h i s The reason f o r the r u l e i s that networks may be described  by a binary connectivity matrix with both rows and columns representing events and an entry of '1' i n d i c a t i n g the presence of an a c t i v i t y between the appropriate events.  C l e a r l y , i n t h i s instance, each a c t i v i t y must  be uniquely described by i t s i n i t i a l and terminal event.  In p r a c t i c e ,  the binary entry i n the matrix i s often replaced by some or a l l of the resource requirements of the a c t i v i t y .  Even this rule i s not e s s e n t i a l  as i t i s always possible f o r a system to give each a c t i v i t y an i n t e r n a l number.  This fact i s i l l u s t r a t e d i n Chapter X I I .  13 Types o f Networks There are two fine a project.  The  d i f f e r e n t approaches by w h i c h management can events may  be s p e c i f i e d .  de-  In t h i s case, the ares  o f t h e network e s t a b l i s h the o r d e r i n g o f e v e n t s .  The nodes o f the n e t -  work a l l b e a r d e s c r i p t i o n s and t h i s i s known as an ' e v e n t - o r i e n t e d ' n e t work.  A l t e r n a t i v e l y , t h e a c t i v i t i e s may  i m p l i e s t h e nodes. is  called  be s p e c i f i e d and t h e i r o r d e r i n g  When the a c t i v i t i e s are d e s c r i b e d i n a network, i t  'activity-oriented'.  The  o r d e r i n g o f nodes o r a c t i v i t i e s i s  g i v e n by d e f i n i n g the s e t s o f e i t h e r the s u c c e s s o r o r the nodes o r a c t i v i t i e s . is  I n p r a c t i c e , a combination  predecessor  o f t h e s e two  approaches  used. The  d u a l network was  suggested by Roy  (59)  i n 1959.  The nodes o f  the network r e p r e s e n t a c t i v i t i e s and t h e i r s e q u e n t i a l r e l a t i o n s h i p i s shown by t h e a r c s .  The  ' d u r a t i o n ' between nodes i n d i c a t e s the time  differ-  e n t i a l between the s t a r t i n g - t i m e s o f t h e c o r r e s p o n d i n g a c t i v i t i e s , w h i c h i s more f l e x i b l e than the s t a n d a r d method.  Backward arrows are p e r m i t t e d  and i n d i c a t e t h a t an a c t i v i t y must end so many time u n i t s a f t e r t h e o t h e r activity.  Hence l o o p s can o c c u r .  The  s i m i l a r i t y o f the r e s u l t i n g theory  w i t h e l e c t r i c network t h e o r y - e x p l a i n s why 'Method o f P o t e n t i a l s ' . Diagramming.  i t i s o f t e n r e f e r r e d t o as the  I t i s a l s o c a l l e d Precedence o r A c t i v i t y - o n - N o d e  A l t h o u g h t h e d u a l network i s c o n s i d e r a b l y s i m p l i f i e d ,  l a c k o f events i s i n p r a c t i c e a severe  the  disadvantage.  E i s n e r (27) suggested a method o f g e n e r a l i z i n g the p r o j e c t n e t work by the i n t r o d u c t i o n o f ' D e c i s i o n Boxes'.  A d e c i s i o n box  indicates  a node a t w h i c h one o f a number o f p o s s i b l e paths c o u l d be t a k e n .  The  Hi PERT/GPM network t h e n becomes t h e d e t e r m i n i s t i c s p e c i a l form o f t h e s e p r o b a b i l i s t i c n e t w o r k s — s o - c a l l e d because o f t h e p r o b a b i l i t y o f s e l e c t i o n a t t a c h e d t o each e x i t from a d e c i s i o n box.  T h i s approach i s d i s -  c u s s e d i n g r e a t e r d e t a i l i n C h a p t e r IX.  Subnetworks Most PERT/CM systems have a maximum s i z e o f network w h i c h can be h a n d l e d .  T h i s maximum may be i n terms o f number o f a c t i v i t i e s o r o f  events o r o f a combination o f t h e s e .  I t i s sometimes n e c e s s a r y , t h e r e -  f o r e , t o decompose t h e network i n t o s m a l l e r component subnetworks.  This  i s a l s o u s e f u l i n t h e p r a c t i c a l c o n s i d e r a t i o n o f the p r o j e c t i n s o f a r as i t b r e a k s t h e p r o j e c t up i n t o more manageable segments.  The normal  a p p r o a c h t o c o n s i d e r i n g the d i v i s i o n i s t o examine t h e network b a r - c h a r t and t o f i n d t h o s e p o i n t s i n time a t which t h e r e a r e the l e a s t number o f •active' a c t i v i t i e s i . e . a c t i v i t i e s a c t u a l l y i n progress.  H a v i n g chosen  t h e s e i n t e r f a c e e v e n t s and a c t i v i t i e s , t h e subnetworks a r e p r o c e s s e d s e p a r a t e l y and are then combined, each as a s i n g l e a c t i v i t y , i n c o n j u n c t i o n with the i n t e r f a c e a c t i v i t i e s . P a r i k h and J e w e l l (53)  The network i s now a n a l y z e d as a whole.  d i s c u s s t h i s problem and t h e i r p r o p o s a l s were sub-  s e q u e n t l y amended f o r s p e c i f i c cases by B l a n n i n g and Rao (88).  Examples  o f t h i s t e c h n i q u e are the use o f 'subnets' i n the IBM PERT/Cost I I System (65)  and the ' f r a g n e t s ' o f t h e NASA PERT and Companion C o s t System  (U).  The, a s s o c i a t e d problem o f m u l t i - p r o j e c t i n t e g r a t i o n i n v o l v e s t h e s p e c i f i c a t i o n o f 'common' events between t h e p r o j e c t s .  C l e a r l y t h e r e can be  no 'common' a c t i v i t i e s as an a c t i v i t y can o n l y o c c u r i n one p r o j e c t .  15 However, dummy a c t i v i t i e s ,  or 'lead*  or ' l a g ' a c t i v i t i e s which require  a c e r t a i n time f o r completion but no other expenditure of resources, are'used to j o i n networks i f no common events e x i s t .  Numbering of Events and A c t i v i t i e s Because of the u n i d i r e c t i o n a l nature of the PERT/CPM network, a topological ordering of the event numbers can be assigned so that the i n i t i a l event of any a c t i v i t y has a lower number than i t s terminal event. This i s c a l l e d the ' i - j numbering'.  Consequently, i f each a c t i v i t y i s  assigned an ordered p a i r of numbers, representing the numbers of i t s  .  i n i t i a l and of i t s terminal events r e s p e c t i v e l y ,  too  can be t o p o l o g i c a l l y ordered.  then the a c t i v i t i e s  Some systems, such as Leavenworth's (116),  require that t h i s numbering be included i n the network description and that the a c t i v i t i e s be introduced i n ascending order. such as the Burroughs PERT/Time (69) events and a c t i v i t i e s i n t e r n a l l y . given by Kahn  Other systems,  compute the topological rank of both  Methods of performing t h i s sort are  (110) and Lasser (115). Neither ordering i s necessary i n  the analysis of a network, as i s i l l u s t r a t e d by Lass ( l l i i ) .  However they  are useful i n the computation, and e s p e c i a l l y when the connectivity matrix i s used, as t h i s reduces to a triangular matrix.  Network Errors Most systems w i l l t e s t for network errors induced by f a u l t y data. The most common e r r o r i s a loop. given by Prostick  A discussion of tracing these loops  is  (125). Whether an error can be detected depends.upon  16 the f l e x i b i l i t y permitted i n the input data.  For example, i f more than  one project s t a r t or project end nodes are allowed, then the omission of an a c t i v i t y may not be detected, where otherwise i t might have been. Similar considerations apply to the detection of duplicate a c t i v i t i e s .  Network Construction Because the time taken to analyze a network i s dependent upon i t s s i z e , i t i s necessary to have an e f f i c i e n t formulation o f the network. I n e f f i c i e n c i e s u s u a l l y occur through excessive use of dummy a c t i v i t i e s . F i s h e r , Llebman and Nemhauser present an algorithm ( 9 9 ) which constructs project networks with a minimal number of nodes d i r e c t l y from the given precedence r e l a t i o n s h i p s .  I t creates dummy a c t i v i t i e s and t o p o l o g i c a l l y  orders the arcs and nodes.  Dimsdale (26) shows that, while the normal  network for a project i s not unique, i t s dual i s .  Optimization i s achieved  by converting a network i n t o i t s dual, and then recreating i t from the dual with a minimum number of a c t i v i t i e s . f o r t h i s i n an e a r l i e r paper (95).  Dimsdale developed the basic theory Moreover, he showed that, i n general,  minimization of the number of nodes of a network does not simultaneously minimize the number of arcs.  The problem of a c t u a l l y drawing the project  network so that i t should be easy to read i s r e a l l y a problem i n the format i o n of l i n e a r graphs.  A solution to t h i s i s presented by Howden (107).  Events and a c t i v i t i e s that form a subnetwork are grouped together.  The  horizontal axis may be time-scaled and the v e r t i c a l axis i s used to emphasize the subnetworks and to minimize the number of a c t i v i t y  'crossings*.  Summary A c t i v i t i e s and Events are the b a s i c components o f a p r o j e c t . An a c t i v i t y has  a c l e a r l y defined beginning  p o i n t s i n t i m e , some e x p e n d i t u r e  and  of resources  a c t i v i t y r e q u i r e s n e i t h e r time nor r e s o u r c e s  end,  and,  must o c c u r .  A dummy  and i s p r i m a r i l y u s e d t o  i n d i c a t e a f u n c t i o n a l r e l a t i o n s h i p between a c t i v i t i e s . represented  between t h e s e  An  activity i s  by a d i r e c t e d a r c i n the p r o j e c t network and i t s i n t e r -  dependencies w i t h o t h e r a c t i v i t i e s by i t s r e l a t i v e p o s i t i o n on paths t h r o u g h the network. is  There a r e , t h e r e f o r e , no l o o p s i n a network.  d i f f i c u l t t o i n d i c a t e r e p e t i t i v e a c t i v i t i e s s a t i s f a c t o r i l y on  It  a  PERT/CPM network. An e v e n t i s a p a r t i c u l a r moment o f time u s u a l l y r e p r e s e n t i n g s t a r t o r end  o f one  o r more a c t i v i t i e s .  i z e d i n terms o f t h e i r i n i t i a l and  A l l a c t i v i t i e s can be  terminal events.  categor-  An event may  t e r m i n a l f o r a few a c t i v i t i e s and a l s o i n i t i a l f o r s e v e r a l o t h e r vities.  Most systems a l l o w more t h a n one  nodes.  The  end o f a p r o j e c t occurs  times.  Networks may  be  events as columns and  between t h e a p p r o p r i a t e A network may  a  l  a t the l a t e s t o f i t s p r o j e c t  l * i n d i c a t i n g an a c t i v i t y  actiend  end with  occurring  events.  have d e s c r i p t i o n s a t t a c h e d  to i t s a c t i v i t i e s —  • a c t i v i t y - o r i e n t e d — o r t o it's e v e n t s — ' e v e n t - o r i e n t e d ' — o r 1  The  be  p r o j e c t s t a r t and p r o j e c t  d e s c r i b e d by a b i n a r y c o n n e c t i v i t y m a t r i x  rows and  the  d u a l network has nodes r e p r e s e n t i n g a c t i v i t i e s and  the sequential r e l a t i o n s h i p .  Each a r c has  to both.  arcs i n d i c a t i n g  a duration attached t o i t  r e p r e s e n t i n g the t i m e - d i f f e r e n t i a l between the s t a r t i n g - t i m e s o f  the  18 two r e l e v a n t a c t i v i t i e s .  A PERT/GPM network can be regarded as t h e  s p e c i a l d e t e r m i n i s t i c case o f a p r o b a b i l i s t i c network i n which  there  a r e nodes, c a l l e d d e c i s i o n boxes, from which one out o f a number o f p o s s i b l e paths can be s e l e c t e d w i t h a c e r t a i n p r o b a b i l i t y . Techniques a r e a v a i l a b l e f o r decomposing networks i n t o subnetworks t o overcome t h e s i z e r e s t r i c t i o n s o f systems o r t o make t h e p r o j e c t more manageable, and a l s o t o t r e a t a number o f p r o j e c t s as one. The  i - j t o p o l o g i c a l ordering o f events r e s u l t s i n the terminal  for  any a c t i v i t y b e i n g a s s i g n e d a h i g h e r number than i t s i n i t i a l  The  most common e r r o r s i n a network a r e l o o p s .  event.  The a b i l i t y t o d e t e c t  network e r r o r s depends t o some e x t e n t upon t h e f l e x i b i l i t y the i n p u t  event  allowed i n  data.  A p r o j e c t network w i t h a m i n i m a l number o f nodes may be cons t r u c t e d from t h e p r o j e c t precedence r e l a t i o n s h i p s .  Minimization  o f the  number o f nodes does n o t s i m u l t a n e o u s l y minimize t h e number o f a r c s .  A  method i s a v a i l a b l e f o r a c t u a l l y drawing t h e p r o j e c t network s o t h a t t h e number o f a c t i v i t y ' c r o s s i n g s ' i s m i n i m i z e d .  CHAPTER I V  CRITICALTTY  Introduction The c r i t i c a l p a t h t h r o u g h a network i s d e f i n e d , and t h e e f f e c t o f changes i n a c t i v i t y d u r a t i o n s i s d i s c u s s e d .  T h i s leads t o the i n -  t r o d u t i o n o f the c r i t i c a l i t y i n d e x — t h e a c t i v i t y f l o a t and t h e event slack.  Methods f o r c a l c u l a t i n g  the c r i t i c a l p a t h l e n g t h , a c t i v i t y  f l o a t and event s l a c k a r e d e s c r i b e d .  The v a r i o u s t y p e s o f f l o a t a r e  c a t e g o r i z e d , b o t h f o r the a c t i v i t y a n d f o r t h e p r o j e c t .  A closer  e x a m i n a t i o n o f t h e i m p l i c a t i o n s o f a c t i v i t y f l o a t i s made, i n v o l v i n g both s u b - c r i t i c a l and s u p e r - c r i t i c a l  concepts.  Finally,  a process  w h i c h p e r m i t s easy network u p d a t i n g i s d e s c r i b e d .  The C r i t i c a l  Path  The most i m p o r t a n t p i e c e o f i n f o r m a t i o n t h a t management c a n obt a i n from a PERT/CPM network i s t h e e x p e c t e d c o m p l e t i o n date f o r t h e project.  I n t h e p l a n n i n g s t a g e , t h i s expected date i s compared w i t h t h e  s c h e d u l e d date and, i f n e c e s s a r y , t h e p r o j e c t i s r e p l a n n e d , e i t h e r b y altering  t h e parameters  specification  o f some o f t h e a c t i v i t i e s o r by changing t h e  o f the network, u n t i l t h e s c h e d u l e d date can be a c h i e v e d .  While t h e p r o j e c t i s i n p r o g r e s s , t h e updated network w i l l whether t h i s p l a n n e d date w i l l be met.  indicate  There a r e o f t e n c a s h p e n a l t i e s  imposed on p r o j e c t s w h i c h a r e d e l a y e d beyond t h e s c h e d u l e d d a t e . c i a t e d w i t h each a c t i v i t y i s i t s time f o r c o m p l e t i o n .  Asso-  How t h i s time i s  20 c a l c u l a t e d i s d e s c r i b e d i n d e t a i l i n the f o l l o w i n g c h a p t e r .  The d u r a -  t i o n o f a p r o j e c t i s determined by t h e l e n g t h o f t h e l o n g e s t p a t h through i t s network.  T h i s p a t h i s c a l l e d the C r i t i c a l  Path.  There may,  c o u r s e , b y more t h a n one c r i t i c a l p a t h .  A c t i v i t i e s making up  p a t h s a r e known as C r i t i c a l  K e l l e y (hh)  Activities.  c r i t i c a l a c t i v i t i e s as ' F l o a t e r s ' . if  refers to  A p r o j e c t d u r a t i o n may  of  critical non-  be s h o r t e n e d  and o n l y i f the d u r a t i o n o f a t l e a s t one a c t i v i t y on each o f i t s c r i -  t i c a l paths i s reduced. l o n g , i t may  I f t h e end o f a f l o a t e r i s d e l a y e d s u f f i c i e n t l y  become c r i t i c a l and c o u l d change the c r i t i c a l i t y  viously c r i t i c a l activities.  of the pre-  The a c t i v i t i e s i n a b a r - c h a r t c o u l d be  arranged so t h a t t h e c r i t i c a l p a t h appears as a continuous l i n e .  It is  a c t u a l l y e a s i e r t o r e a d t h e p r o j e c t d u r a t i o n o f f a b a r - c h a r t r a t h e r than off of  a network.  The c a l c u l a t i o n o f t h e c r i t i c a l p a t h was  the e a r l y PERT and CPM  the prime concern  Systems.  S l a c k and F l o a t The end o f a n o n - c r i t i c a l a c t i v i t y may of  be d e l a y e d a c e r t a i n amount  time, e i t h e r by t h e a c t i v i t y s t a r t i n g l a t e r o r by i t s d u r a t i o n b e i n g  i n c r e a s e d , without a f f e c t i n g the p r o j e c t d u r a t i o n .  Hence an a c t i v i t y i s  s a i d t o have s l a c k i . e . time t o s p a r e .  n o n - c r i t i c a l events  may  Similarly,  o c c u r anywhere w i t h i n a p a r t i c u l a r time-range w i t h o u t p o s t p o n i n g the  p r o j e c t c o m p l e t i o n date, and thus an e v e n t a l s o has s l a c k . ' s l a c k ' and  ' f l o a t ' are used i n t e r c h a n g e a b l y , a l t h o u g h B a t t e r s b y (9)  suggested t h a t to  The terms  activities.  ' s l a c k ' s h o u l d o n l y be a p p l i e d t o events and  has  'float' only  21  The c a l c u l a t i o n o f t h e c r i t i c a l p a t h i n v o l v e s o f a l l t h e a c t i v i t i e s i n the network. v i t i e s w i t h the s m a l l e s t f l o a t . for is  f i n d i n g the f l o a t  The c r i t i c a l p a t h comprises  acti-  No event can o c c u r u n t i l a l l a c t i v i t i e s  w h i c h i t i s t e r m i n a l a r e completed. c a l l e d the ' C r i t i c a l Predecessor  1  The l a t e s t a c t i v i t y t o f i n i s h  o f t h e event.  a l l a c t i v i t i e s f o r w h i c h i t i s i n i t i a l may  commence.  Once an event o c c u r s , Activity  float  and event s l a c k a r e c a l c u l a t e d i n two p a s s e s through the network.  In  the f i r s t p a s s , a l l a c t i v i t i e s are s t a r t e d as soon as p o s s i b l e , and the E a r l i e s t S t a r t T i m e — E . S . T . — a n d the E a r l i e s t F i n i s h T i m e — E . F . T . — f o r e a c h a c t i v i t y and the E a r l i e s t T i m e — E . T . — f o r e a c h event a r e determined, i n a d d i t i o n to the length o f the c r i t i c a l path.  I f a record o f the c r i -  t i c a l p r e d e c e s s o r s f o r e a c h event has been k e p t , t h e c r i t i c a l can be determined by a s i m p l e c h a i n - t r a c i n g t e c h n i q u e .  activities  The second pass  s e t s each a c t i v i t y t o s t a r t a s l a t e as p o s s i b l e and so e s t a b l i s h e s t h e Latest S t a r t Time—L.S.T. — a n d  the L a t e s t F i n i s h T i m e — L . F . T . — f o r  a c t i v i t y and the L a t e s t T i m e — L . T . — f o r each event.  The a c t i v i t y  each float  i s t h e d i f f e r e n c e between i t s E.S.T. and i t s L.S.T. o r between i t s E.F.T. and i t s L.F.T., w h i l e the e v e n t s l a c k i s c a l c u l a t e d by s u b t r a c t i n g i t s E.T. from i t s L.T.  The second pass v a l u e s a r e determined by t h e c h o i c e  o f the time f o r t h e p r o j e c t and e v e n t . f i r s t pass i s used as a r e f e r e n c e , have z e r o f l o a t . date.  I f t h e time e s t a b l i s h e d on t h e  then a l l c r i t i c a l a c t i v i t i e s  will  The a l t e r n a t i v e i s t o c o n s i d e r t h e s c h e d u l e d c o m p l e t i o n  I n t h i s case, an a c t i v i t y c o u l d have n e g a t i v e f l o a t , w h i c h  implies  t h a t i t s e a r l i e s t s t a r t i n g t i m e i s t o o l a t e t o p e r m i t the p r o j e c t t o end by t h e s c h e d u l e d d a t e .  22 An algorithm which achieves the same r e s u l t s i n one pass through the network i s described by Goldberg (37)*  The network needs to be  s p e c i f i e d as an n-dimensional l a t t i c e with the arcs representing a c t i v i t i e s directed away from the s t a r t node along the rows and down along the columns and the nodes representing events, number of a c t i v i t i e s  ' n ' i s the maximum  'entering' or ' l e a v i n g ' any event.  For much sim-  p l e r oraputations, n i s reduced to two, thereby introducing many dummy activities.  One pass through the l a t t i c e and a scan of i t s nodes w i l l  e s t a b l i s h the required information, but the work required i n the l a t t e r part i s l i k e l y to be greater than a second pass through the network. Burgess and Killebrew (16) suggest a procedure by which a network, or 'Arrow Diagram', may be converted i n t o a bar-chart.  Methods are pro-  posed for d e r i v i n g the c r i t i c a l path and f o r determining the a c t i v i t y f l o a t s from the bar-chart.  Types of Float There are various types of f l o a t which have been found useful i n some context or other, usually i n the grading of the r e l a t i v e importance of a l i s t of a c t i v i t i e s i n " o r d e r to make effective Let ' A . T . ' stand f o r the a c t i v i t y time.  Total Float—L.F.T.-E.S.T.-A.T.—  i s the f l o a t referred to i n the previous section. the largest of the f l o a t s .  scheduling decisions.  I t i s numerically  An a c t i v i t y with (Early) Free Float—  E . F . T . - E . S . T . - A . T . —occurs whenever the a c t i v i t y i s not the c r i t i c a l predecessor of i t s terminal event.  In p a r t i c u l a r , the a c t i v i t y at the  23  end o f v e r y n o n - c r i t i c a l p a t h has f r e e f l o a t . Float— are  L.F.T.-E.F.T. — a n d L a t e I n t e r f e r i n g F l o a t —  equivalent  i n i t i a l event.  t o t h e event s l a c k s Late Free F l o a t —  f e r i n g f l o a t are r a r e l y used.  Interfering  L.S.T.-E.S.T.—  of the a c t i v i t y ' s terminal  and  L.F.T.-L.S.T.-A.T. — a n d l a t e i n t e r -  I n t e r f e r i n g f l o a t i s equal to the  d i f f e r e n c e between t o t a l f l o a t and f r e e f l o a t . maximum ( 0 , E.F.T.-L.S.T.-A.T.) — d s nor a f f e c t i n g , other a c t i v i t i e s . free  (Early)  Independent  Float—  t h e o n l y f l o a t not a f f e c t e d  I t i s always l e s s t h a n or e q u a l t o  float. E v e r y p a t h i n t h e network has a f l o a t a t t a c h e d t o i t .  the s m a l l e s t  This i s  o f the t o t a l f l o a t s o f t h e a c t i v i t i e s c o m p r i s i n g t h e p a t h .  Sometimes t h i s f l o a t i s d i v i d e d t i o n to t h e i r durations.  amongst the a c t i v i t i e s , o f t e n i n p r o p o r -  T h i s i s known as ' A l l o c a t e d F l o a t ' .  i s c a l c u l a t e d r e l a t i v e t o a m i l e s t o n e , r a t h e r t h a n the p r o j e c t it  by,  i s c a l l e d 'Secondary  Float'.  'External  Float' or 'Project  If float end, Float'  i s the d i f f e r e n c e between the e x p e c t e d completion t i m e , a c c o r d i n g t o the network p l a n ,  and the s c h e d u l e d c o m p l e t i o n t i m e .  C r i t i c a l and N o n - c r i t i c a l A c t i v i t i e s The o b j e c t i v e  i n categorizing  a c t i v i t i e s as c r i t i c a l i s t o i n -  d i c a t e t o management w h i c h a c t i v i t i e s a r e the most i m p o r t a n t , so any spare r e s o u r c e s may a r e completed  that  be a l l o c a t e d t o ensure t h a t t h e s e a c t i v i t i e s  on s c h e d u l e .  t i c a l i t y q f a c t i v i t i e s may  However, as has been shown above, the c r i vary.  Consequently, a c t i v i t i e s w i t h s m a l l  2h t o t a l f l o a t must a l s o be watched c a r e f u l l y . called 'sub-critical'.  These a c t i v i t i e s  U s i n g t h i s c o n c e p t , Handa (105)  are  proposes  a  method f o r e l i m i n a t i n g c e r t a i n n o n - c r i t i c a l a c t i v i t i e s i n o r d e r t o s i m p l i f y t h e network.  B a t t e r s b y ( 9 ) notes t h a t B r i t i s h R a i l r e g a r d  any a c t i v i t y whose t o t a l f l o a t i s l e s s t h a n one-tenth o f t h e p r o j e c t d u r a t i o n as  sub-critical.  A f u r t h e r r e f i n e m e n t o f c r i t i c a l i t y i s suggested by Welsh  (133).  A l l the p a t h s i n t h e network a r e o r d e r e d a c c o r d i n g t o d e c r e a s -  ing duration.  The  ' k i - S u p e r c r i t i c a l A r c s ' comprise  the s e t o f a r c s  t h a t are common t o each o f t h e f i r s t k paths i n t h e o r d e r i n g .  A l l these  ^ a r c s a r e c r i t i c a l , as t h e s e t o f k - s u p e r c r i t i c a l a r c s i s c o n t a i n e d i n the s e t o f ( k - l ) - s u p e r c r i t i c a l a r c s , and the s e t o f 1 - s u p e r c r i t i c a l a r c s i s the s e t o f c r i t i c a l a c t i v i t i e s .  Reducing  t h e d u r a t i o n a t t a c h e d t o any  k - s u p e r c r i t i c a l a r c w i l l reduce t h e l e n g t h o f each o f the k l o n g e s t p a t h s and t h e r e f o r e i s most l i k e l y t o reduce the a c t u a l p r o j e c t d u r a t i o n . A f t e r a network has been a n a l y z e d , the d u r a t i o n s o f some a c t i v i t i e s a r e o f t e n changed. network.  T h i s n o r m a l l y r e s u l t s i n t h e r e a n a l y z i n g o f the  A method by which the e f f e c t o f t h e s e changes on the  activity  f l o a t s and c r i t i c a l p a t h l e n g t h i s e a s i l y shown, b e s i d e s a l s o i n d i c a t i n g where t h e s e changes may b e s t be made, i s d e s c r i b e d by Thompson ( 6 l ) .  A  p a t h i s d e f i n e d as a connected s e r i e s o f a c t i v i t i e s having the same t o t a l f l o a t , w h i c h i s not i n t e r s e c t e d by a p a t h o f l e s s e r f l o a t , except a t i t s ends.  Whenever two p a t h s merge i n t o one common p a t h a t e i t h e r end,  change o f f l o a t a l o n g the common p a t h r e s u l t s i n the i d e n t i c a l  float  a  25 change i n b o t h p a t h s .  Consequently, paths are l i s t e d by e v e n t s , and  i n t e r s e c t i o n s w i t h o t h e r paths are shown.  Summary A time f o r c o m p l e t i o n i s a s s o c i a t e d w i t h e v e r y a c t i v i t y o f the network.  The c r i t i c a l p a t h i s the l o n g e s t p a t h t h r o u g h t h e network.  Activities  on the c r i t i c a l p a t h a r e c a l l e d  c r i t i c a l a c t i v i t i e s are c a l l e d  floaters.  critical activities. Activity  Non-  f l o a t o r event s l a c k  i s the amount by which an a c t i v i t y s t a r t o r an e v e n t o c c u r r e n c e can be d e l a y e d w i t h o u t e x t e n d i n g the p r o j e c t d u r a t i o n . c r i t i c a l path length, a c t i v i t y f l o a t s  earliest  p a t h l e n g t h o r t h e s c h e d u l e d completion time.  bar-charts. ant  end.  Use  Other a l g o r i t h m s i n v o l v e  be the  float i s float relative  critical  o f t h e l a t t e r may i n -  ' l a t t i c e ' networks o r  There a r e v a r i o u s t y p e s o f a c t i v i t y f l o a t .  a r e t h e t o t a l f l o a t , the f r e e f l o a t ,  Secondary  times and the s e c -  R e f e r e n c e s f o r t h e second pass may  volve negative f l o a t s .  o f the  and event s l a c k s i s u s u a l l y p e r -  formed i n two p a s s e s — t h e f i r s t c a l c u l a t i n g ond l a t e s t t i m e s .  The c a l c u l a t i o n  The most i m p o r t -  and the independent  float.  t o a m i l e s t o n e , r a t h e r than the p r o j e c t  The d i f f e r e n c e between t h e p l a n n e d and s c h e d u l e d c o m p l e t i o n date  i s the p r o j e c t o r e x t e r n a l  float.  A c t i v i t i e s w h i c h are n e a r l y c r i t i c a l are termed  sub-critical.  A c t i v i t i e s w h i c h are common t o t h e k l o n g e s t paths i n t h e network a r e called k-supercritical. it  A method i s a v a i l a b l e w h i c h w i l l i n d i c a t e where  i s b e s t t o reduce an a c t i v i t y d u r a t i o n , and what the r e s u l t s  any change a r e , w i t h o u t r e q u i r i n g r e - a n a l y s i s o f the network.  o f making  CHAPTER V TIME  Introduction Traditionally, time estimates for any activity were extremely inaccurate.  The reason for this i s illustrated i n the following  excerpt from an address given by Ernest 0. Codier of the General Electric Company to the American Management Association Briefing Session on  'New  Techniques for Management Control' i n July 1961, as quoted by Evarts (76): There i s a traditional man/manager gaming model for time estimating. This game runs approximately as follows: you pick up the phone and with casual and disarming friendliness inquire, "Say, George, just off the top of your head, how long w i l l i t take to get out lli frabastats?" Now George i s a very competent fellow and he knows instinctively that i t would take eight weeks to fabricate lh frabastats provided everything happened the way i t should happen. But he does not give you this answer immediately because he has to stop and figure this thing out. In the f i r s t place he knows that there w i l l be some normal amount of unforeseen d i f f i c u l t y involved, and he w i l l add a factor to take into account this average uncertainty, say, two weeks. Now he has ten weeks. But George i s not misled by the informality of your request; he knows that ultimately in one form or another this i s going to show up as a commitment to you, to provide llj frabastats i n a specified time. Furthermore, George knows the nature of the business, and he knows you. He knows that time estimates are traditionally too long, and that somewhere along the line he can expect to get cut back. So at this point, he adds the fat, which i s his considered opinion as to how severely this time estimate w i l l get cut later on—say three weeks. He has now arrived at 13 weeks, and this i s the figure you get. When this proves to be too long, as i t invariably does, since i t i s in fact too long, the time w i l l get cut as George expects. By one or another mechanism you w i l l work the problem backwards and ask for performance two and a half weeks sooner. You have done pretty well as a manager, and have cut out only approximately the amount that George anticipated; he makes routine grumbling  27  n o i s e s , because t h i s i s p r o p e r form, and 10§ weeks becomes h i s commitment. Now p r e c i s e l y what do you know a t t h i s p o i n t ? Not v e r y much. You do not know how l o n g the j o b s h o u l d take under i d e a l c i r c u m s t a n c e s , and you have no measure o f p o t e n t i a l t r o u b l e sources which y o u as George's manager might be a b l e t o h e l p him smooth o u t . Furthermore, i f George a c t u a l l y d e l i v e r s Ik f r a b a s t a t s i n 10§ weeks, you a r e n o t r e a l l y s u r e whether t h i s was an o u t s t a n d i n g e f f o r t , or whether i t was j u s t an average performance... Considerably greater r e l i a b i l i t y  i s r e q u i r e d by time e s t i m a t e s f o r  PERT/CPM network a n a l y s i s . The  d i f f e r e n c e between the CFM  one  time e s t i m a t e and the PERT  t h r e e time e s t i m a t e f o r a c t i v i t y d u r a t i o n i s d e s c r i b e d .  The  assumption  o f t h e b e t a d i s t r i b u t i o n f o r the l a t t e r i s d i s c u s s e d and t h e  probability  concept  criticality.  i s extended t o c o n s i d e r a t i o n o f p r o j e c t d u r a t i o n and  E r r o r s , i n h e r e n t i n the d i s t r i b u t i o n assumptions a t v a r i o u s l e v e l s ,  are  exposed and a l t e r n a t i v e approaches d e s c r i b e d .  are  F i n a l l y , conclusions  drawn about the u s e f u l n e s s o f the t h e o r y i n t h i s  Time  area.  Estimates One  o f t h e main d i f f e r e n c e s between the. o r i g i n a l CPM  and PERT  systems l i e s i n t h e i r approach towards e s t i m a t i n g the expected of  an a c t i v i t y .  CPM  was  designed  f o r i n d u s t r i e s , s u c h as c o n s t r u c t i o n ,  where t h e same a c t i v i t i e s occur i n many p r o j e c t s . accurate estimate Conversely, developed, expected  duration  Consequently, an  can be made, and t h i s i s what i s used i n the network.  i n t h e r e s e a r c h and  development p r o j e c t s , f o r which PERT  an a c t i v i t y d u r a t i o n i s h i g h l y i n d e t e r m i n a t e , and so  time f o r completion  was  the  i s d e r i v e d from t h r e e time e s t i m a t e s .  These  28  t h r e e e s t i m a t e s r e p r e s e n t t h e most o p t i m i s t i c , most l i k e l y and most p e s s i m i s t i c times f o r completion o f the a c t i v i t y .  The most o p t i m i s t i c  and most p e s s i m i s t i c t i m e s are each r e g a r d e d as o n l y l i k e l y t o o c c u r one p e r c e n t o f the t i m e .  The e x p e c t e d time i s c a l c u l a t e d by summing  o n e - s i x t h o f the most o p t i m i s t i c t i m e , t w o - t h i r d s o f the most l i k e l y time and o n e - s i x t h o f t h e most p e s s i m i s t i c t i m e . c o n s i d e r a t i o n o f the b e t a d i s t r i b u t i o n . i n Appendix 'A'.  This results  from  The a n a l y s i s i n v o l v e d i s g i v e n  T h i s c h o i c e i s d i s c u s s e d i n the o r i g i n a l PERT Summary  Report, Phase I (5),  as f o l l o w s :  Our problem i s t o e s t i m a t e t h e expected v a l u e and v a r i a n c e o f an a c t i v i t y time from the l i k e l y , o p t i m i s t i c , and p e s s i m i s t i c t i m e s d i s c u s s e d above. Furthermore, we f e e l f r e e t o use a non-normal model o f the d i s t r i b u t i o n o f a c t i v i t y t i m e s as a t o o l i n t h i s e s t i m a t i o n . We r e c a l l t h a t f o r unimodal f r e q u e n c y d i s t r i b u t i o n s , t h e s t a n d a r d d e v i a t i o n can be e s t i m a t e d r o u g h l y as o n e - s i x t h o f the range. Hence, i t seems r e a s o n a b l e t o e s t i m a t e the s t a n d a r d d e v i a t i o n o f an a c t i v i t y time as o n e - s i x t h of t h e d i f f e r ence between the p e s s i m i s t i c and o p t i m i s t i c time e s t i m a t e s . The e s t i m a t e o f the e x p e c t e d v a l u e o f an a c t i v i t y time i s more d i f f i c u l t . We do n o t a c c e p t the l i k e l y time as t h e exp e c t e d v a l u e . We f e e l t h a t an a c t i v i t y time w i l l more o f t e n exceed t h a n be l e s s than an e s t i m a t e d l i k e l y t i m e . Hence, i f l i k e l y times were a c c e p t e d as expected v a l u e s , an u n d e s i r a b l e b i a s would be i n t r o d u c e d . Our a p p r e h e n s i o n o f t h i s b i a s i s s u p p o r t e d by e s t i m a t e s a l r e a d y o b t a i n e d . I n many c a s e s the l i k e l y time i s n e a r e r t h e o p t i m i s t i c t h a n t h e p e s s i m i s t i c t i m e . I n such a s i t u a t i o n one f e e l s t h a t the e x p e c t e d time s h o u l d exceed the l i k e l y t i m e . We s h a l l i n t r o d u c e an e s t i m a t e o f the e x p e c t e d time which would seem t o a d j u s t a t l e a s t c r u d e l y f o r the b i a s t h a t would be p r e s e n t i f l i k e l y times were a c c e p t e d as expected t i m e s . As a model o f t h e d i s t r i b u t i o n o f an a c t i v i t y t i m e , we i n t r o duce t h e b e t a d i s t r i b u t i o n whose mode i s a t t h e l i k e l y time, whose range i s the i n t e r v a l between t h e o p t i m i s t i c and p e s s i m i s t i c t i m e s , and whose s t a n d a r d d e v i a t i o n i s o n e - s i x t h o f t h e range.  29 A l t h o u g h t h e r e p o r t r e f e r s t o 'the b e t a d i s t r i b u t i o n ' , Grubbs ( 3 8 ) has shown t h a t t h e r e are a c t u a l l y p r e c i s e l y t h r e e b e t a d i s t r i b u t i o n s t h a t s a t i s f y the g i v e n c o n d i t i o n s .  The i m p l i c a t i o n o f the d i s t r i b u -  t i o n i s t h a t the a c t u a l a c t i v i t y d u r a t i o n has a p r o b a b i l i t y o f p e r c e n t o f b e i n g e q u a l t o o r b e t t e r t h a n the expected The f i r s t a r t i c l e on PERT (50), C l a r k and F a z e r , i n no way distribution. Clark ( 2 1 ) , was  attempted  fifty  time.  by i t s a u t h o r s Malcolm,  Roseboom,  t o j u s t i f y the c h o i c e o f the b e t a  The c o n t r o v e r s y t h a t a r o s e f i n a l l y brought a l e t t e r  from  i n w h i c h he s t a t e d t h a t , i n e f f e c t , the b e t a d i s t r i b u t i o n  the f i r s t  d i s t r i b u t i o n t o o c c u r t o him t h a t s a t i s f i e d h i s r e q u i r e -  m e n t s — n a m e l y , t h a t the d i s t r i b u t i o n curve be unimodal, u s u a l l y w i t h a skew mode b u t , i n the case when the most l i k e l y time o c c u r s c a l l y between the extreme t i m e s , w i t h a symmetric approximated  symmetri-  mode which c l o s e l y  t h e normal d i s t r i b u t i o n , continuous and w i t h two  non-  negative abcissa i n t e r c e p t s .  Extensions of the P r o b a b i l i t y  Concept  The p r o b a b i l i t y concept i s extended beyond t h a t o f t h e to  activity  i n c l u d e the p r o b a b i l i t y o f the p r o j e c t c o m p l e t i o n time b e i n g reached,  the p r o b a b i l i t y o f an a c t i v i t y h a v i n g z e r o o r n e g a t i v e s l a c k and the p r o b a b i l i t y o f a path being  critical.  The e x p e c t e d time o f an event i s t h e expected c o m p l e t i o n time o f its to  c r i t i c a l p r e d e c e s s o r i . e . o f the l o n g e s t p a t h from t h e s t a r t event the g i v e n e v e n t .  The v a r i a n c e o f t h i s time i s the sum  of the  30 v a r i a n c e s o f t h e a c t i v i t i e s on the l o n g e s t p a t h . assumed, w i t h o u t  j u s t i f i c a t i o n , t o be normal.  The d i s t r i b u t i o n i s  As t h e o c c u r r e n c e  of  t h e f i n a l event i n d i c a t e s t h e end o f the p r o j e c t , t h e l i k e l i h o o d o f a c e r t a i n s c h e d u l e d date b e i n g a c h i e v e d i s i n d i c a t e d by the r a t i o o f t h e a r e a under t h e d i s t r i b u t i o n curve t o the ' l e f t ' o f the s c h e d u l e d  date  t o t h a t of t h e whole a r e a under the c u r v e .  infinite  p o r t i o n s of the curve.  A l l a r e a s i g n o r e the  I f t h e s c h e d u l e d date was  the l a t e s t end  o f the a c t i v i t y , t h i s r a t i o , a p p l i e d t o t h e a c t i v i t y time w i l l i n d i c a t e the p r o b a b i l i t y slack.  time  distribution,  o f an a c t i v i t y having z e r o o r n e g a t i v e  I n t h i s case, any a c t i v i t y w i t h a p r o b a b i l i t y  f i f t y p e r c e n t s h o u l d be r e g a r d e d as  of l e s s  than  critical.  E a c h p a t h has a d i s t r i b u t i o n a s s o c i a t e d w i t h i t s expected l e n g t h . Because o f t h e l a r g e v a r i a n c e a t t a c h e d t o t h i s l e n g t h , the r e l a t i v e b a b i l i t y o f a path being c r i t i c a l i s a v a l i d consideration. c r i t i c a l ! t y concept o f Welsh  The  pro-  super-  (133)> d e s c r i b e d i n t h e p r e v i o u s c h a p t e r ,  may be used e f f e c t i v e l y i n c o n j u n c t i o n w i t h t h i s p a t h c r i t i c a l i t y Charnes, Cooper and Thompson (18)  attempt  index.  t o c h a r a c t e r i z e the  r e s u l t i n g d i s t r i b u t i o n of p r o j e c t completion times, considering a l l paths and assuming some p a r t i c u l a r d i s t r i b u t i o n — n o t n e c e s s a r i l y f o r e a c h b r a n c h o f the network.  I n the r e s u l t i n g  distribution  beta—  "multi-  m o d a l i t y i s t o be expected whenever t h e r e a r e p a r a l l e l l i n k s o r c h a i n s t h a t a l t e r n a t e i n c r i t i c a l i t y and t h a t i n v o l v e s u f f i c i e n t l y d i f f e r e n t times." An a l t e r n a t i v e  approach t o f i n d i n g the p r o b a b i l i t i e s  o f paths  and  31 o f a c t i v i t i e s b e i n g c r i t i c a l , and o f p r o j e c t completion times b e i n g a c h i e v e d , i s t o use the Monte C a r l o sampling t e c h n i q u e , as d e s c r i b e d by Van S l y k e ( 6 2 ) .  A sample v a l u e o f e a c h a c t i v i t y t i m e from i t s i n -  d i v i d u a l d i s t r i b u t i o n i s t a k e n and the network a n a l y s i s  performed.  T h i s i s r e p e a t e d many t i m e s , and the r e q u i r e d p r o b a b i l i t y c u r v e s be determined  from t h i s  may  data.  H a r t l e y and Worthara (39)  attempt  t o d e s c r i b e the e x a c t  distribu-  t i o n of the p r o j e c t completion times by c l a s s i f y i n g networks.  The  dis-  t r i b u t i o n i s d e r i v e d by n u m e r i c a l i n t e g r a t i o n .  However, independent  t r i b u t i o n s a r e assumed.  are used t o s o l v e i n -  v o l v e d networks.  Monte C a r l o procedures  dis-  The method i s u s e f u l i n a s s e s s i n g the r e l a t i v e m e r i t s  o f two o r more networks and i n a s s e s s i n g the p r o g r e s s o f a p r o j e c t .  E r r o r s i n h e r e n t i n t h e D i s t r i b u t i o n Assumptions E r r o r s i n the assumption times o c c u r a t v a r i o u s l e v e l s .  of beta d i s t r i b u t i o n s f o r the  activity  There a r e e r r o r s caused by t h e c h o i c e  o f b e t a , r a t h e r than some o t h e r d i s t r i b u t i o n , by t h e l i n e a r  assumption  f o r t h e c a l c u l a t i o n o f t h e expected time and by t h e assumed independence o f t h e v a r i o u s branches o f t h e network, b e s i d e s the e r r o r i n t h e t h r e e time e s t i m a t e s  themselves.  Grubbs (38),  i n a c h a l l e n g e t o the t h e o r y o f the  statistical  b a s i s , n o t e s t h a t t h e t h r e e b e t a d i s t r i b u t i o n s t h a t s a t i s f y the g i v e n formulae  f o r e x p e c t e d time and v a r i a n c e a r e a l l f a t , f l a t c u r v e s , a s i d e  from t h e e n d - p o i n t s .  MacCrimmon and Ryavec (u8),  i n the major work on  e r r o r a n a l y s i s i n the PERT assumptions, found the e r r o r bounds on the b e t a d i s t r i b u t i o n compared w i t h t h r e e o t h e r d i s t r i b u t i o n s , the w o r s t o f which was  one-third  o f t h e d i f f e r e n c e between the most o p t i m i s t i c  and p e s s i m i s t i c t i m e s f o r the expected time and o n e - t h i r d  o f t h i s range  f o r the v a r i a n c e .  gave the same  e r r o r margin. cubic  They showed t h a t the l i n e a r assumption  T h i s l i n e a r assumption  comes from the s o l u t i o n t o a  e q u a t i o n d e r i v e d i n t h e a n a l y s i s g i v e n i n t h e PERT Summary Report,  Phase I (5).  The a n a l y s i s  g i v e n i n Appendix  i t s o r i g i n and i t s i m p l i c a t i o n s .  'A' shows more c l e a r l y  MacCrimraon and Ryavec a l s o  indicated  t h a t the e r r o r bounds f o r d i f f e r e n t d i s t r i b u t i o n s a r e a p p r o x i m a t e l y the  same f o r the t r i a n g u l a r d i s t r i b u t i o n , f o r which the e x p e c t e d time  can be c a l c u l a t e d e x a c t l y , as i t i s the average o f t h e t h r e e time mates.  Archibald  and V i l l o r i a  (6) note t h a t , i n p r a c t i c e , a poor  time e s t i m a t e f o r a s h o r t a c t i v i t y i s n o t as dangerous a c t i v i t y when the problem may serious time  esti-  as f o r a l o n g  n o t be apparent u n t i l the p r o j e c t i s i n  trouble.  The a n a l y s i s g i v e n i n Appendix  'A*  shows t h a t t h e e r r o r i n the  s o - c a l l e d l i n e a r assumption f o r t h e d e r i v a t i o n o f t h e expected time c o u l d be reduced by v a r y i n g  the weight a s s i g n e d t o t h e most l i k e l y  a c c o r d i n g t o i t s p o s i t i o n r e l a t i v e t o the extreme t i m e s . the  In  addition,  analysis i n d i c a t e s t h a t , i f a constant weighting i s required,  s t a n d a r d a p p r o x i m a t i o n w i l l minimize the maximum e r r o r i n the  time  the  variance.  I n t h e i r paper on PERT (f>0), the team t h a t developed i t admits t h a t the e x p e c t e d p r o j e c t d u r a t i o n  i s always o p t i m i s t i c .  However, the  3 3  v a r i a n c e i s u s u a l l y g r e a t e r than t h e t r u e v a l u e . t r o d u c e d by o n l y c o n s i d e r i n g t h e c r i t i c a l p a t h ,  This e r r o r i s i n r a t h e r than a l l p a t h s  t h r o u g h t h e network, and b y t h e assumption o f normal d i s t r i b u t i o n . The Monte C a r l o method d e s c r i b e d above g i v e s an unbiased  estimate.  MacCrimmon and Ryavec ( I | 8 ) show t h a t t h e e r r o r i s p r o p o r t i o n a l t o t h e degree o f p a r a l l e l i s m i n t h e network and t h a t , i n the few s m a l l p r o j e c t s s t u d i e d , t h e e r r o r ranged from t e n t o t h i r t y p e r c e n t .  Activities  s h a r e d b y two o r more p a t h s tend t o o f f s e t t h e e r r o r due t o p a r a l l e l i s m . B i l d s o n and G i l l e s p i e ( 8 6 ) note t h a t a change i n c r i t i c a l i t y  can r e -  s u l t i n a r e d u c t i o n o f t h e average d u r a t i o n f o r a p r o j e c t b u t an i n c r e a s e i n t h e time a t s p e c i f i c p r o b a b i l i t y l e v e l s .  T h i s p o i n t was made  i n t h e i r a n a l y s i s o f t h e e f f e c t o f the i m p l i c i t r e s t r i c t i o n s o f the r a t i o s o f t h e t h r e e time e s t i m a t e s t i c a l path  i n K e l l e y ' s a l g o r i t h m (hh)  for c r i -  scheduling.  A method b y which the degree o f optimism i n t h e expected c a l p a t h l e n g t h may b e reduced i s d e s c r i b e d by F u l k e r s o n a c t i v i t i e s a r e grouped t o g e t h e r i n t o 'bundles'.  (35).  critiCertain  Joint probability  dis-  t r i b u t i o n on a c t i v i t y times w i t h i n a bundle a r e c o n s i d e r e d , b u t bundles a r e assumed independent o f each o t h e r . creases  The number o f c a l c u l a t i o n s i n -  e x p o n e n t i a l l y w i t h t h e maximum bundle s i z e , i n c o n t r a s t w i t h  the normal i n c r e a s e w i t h network s i z e . An a n a l y s i s o f t h e v a r i a n c e o f t h e maximum o f a number o f depende n t v a r i a b l e s , each w i t h normal d i s t r i b u t i o n , was made by C l a r k  (20).  3U However, f o r more t h a n two dependent v a r i a b l e s , t h e t h e o r y i s cumbersome, and i t becomes v e r y complex f o r non-normal d i s t r i b u t i o n s . A r e a s o n a b l e b u t n o t optimum s o l u t i o n i s o b t a i n e d f o r use w i t h PERT networks• The e f f e c t o f s u b d i v i d i n g a c t i v i t i e s on the computed p r o b a b i l i t i e s was  s t u d i e d b y Healy  (111).  S e r i e s and  series-and-parallel,  but n o t s o l e l y p a r a l l e l , s u b - d i v i s i o n s a r e examined and i t i s found t h a t e r r o r s are i n c r e a s e d .  However, as C l a r k remarks, the problem i s  e s s e n t i a l l y i r r e l e v a n t , as t h e d e p t h o f d e t a i l i n v o l v e d i n the network s p e c i f i c a t i o n depends upon management r a t h e r than t h e a n a l y s t .  Conclusions I n an a r e a n o t a b l y l a c k i n g i n mathematical c o n t e n t , the to  a p p l y a n a l y s i s g i v e n b y the p r o b a b i l i t y concept  has  l e d to  e v e r y f a c e t o f t h e c o n s i d e r a t i o n o f time d i s t r i b u t i o n b e i n g examined.  chance almost  closely  The p r a c t i c a l u t i l i t y o f t h i s has f r e q u e n t l y been c h a l l e n g e d .  I t i s apparent  t h a t a p p l y i n g t h e p r o b a b i l i t y approach when f i n d i n g  t i c a l paths and a c t i v i t i e s i s more e f f e c t i v e than t h e simple use total float.  However, the b e t a d i s t r i b u t i o n assumption and t h e  e r r o r i n v o l v e d i n the t h r e e time e s t i m a t e s do make a l l the based on t h i s i m p r a c t i c a l .  cri-  of likely  analyses  I t i s i r o n i c t o r e a l i z e t h a t , i n PERT n e t -  works f o r p r o j e c t s i n a r e s e a r c h and development environment where any time e s t i m a t e i s somewhat meaningless,  c o n s i d e r a b l e t h e o r y i s based  an assumed d i s t r i b u t i o n dependent upon t h r e e time e s t i m a t e s , w h i l e ,  on  35 i n CPM  networks f o r p r o j e c t s w i t h many common a c t i v i t i e s f o r w h i c h an  a c c u r a t e time e s t i m a t e may  be made and, c o n s e q u e n t l y , f o r which the  time d i s t r i b u t i o n i s a c t u a l l y known, t h e time e s t i m a t e i s r e g a r d e d as d e t e r m i n a t e and no t h e o r y has been d e v e l o p e d .  Summary CPM  networks are designed f o r p r o j e c t s whose composite  activi-  t i e s are o f t e n s i m i l a r , and t h e r e f o r e a c c u r a t e time e s t i m a t e s may made f o r them.  PERT was  be  developed f o r r e s e a r c h and development p r o -  j e c t s , f o r w h i c h an a c t i v i t y d u r a t i o n may  be i n d e t e r m i n a t e .  Consequently,  most p e s s i m i s t i c , most l i k e l y and most o p t i m i s t i c time e s t i m a t e s are r e quired.  The  extreme e s t i m a t e s supposedly r e p r e s e n t a one p e r c e n t  hood o f o c c u r r e n c e .  The e x p e c t e d time i s a weighted  t h r e e t i m e s , i n the r a t i o o f o n e - t o - f o u r - t o - o n e .  likeli-  average o f t h e  T h i s i s d e r i v e d from  an assumed b e t a d i s t r i b u t i o n f o r each a c t i v i t y d u r a t i o n .  Probability  c o n s i d e r a t i o n s a r e made o f the p r o j e c t c o m p l e t i o n t i m e , c r i t i c a l  paths  and a c t i v i t y f l o a t , by combining the a c t i v i t y time d i s t r i b u t i o n s .  An  a l t e r n a t i v e approach i s t o use t h e Monte C a r l o sampling t e c h n i q u e . B e s i d e s t h e e r r o r s i n t h e a c t u a l time e s t i m a t e s , t h e e r r o r i n the e x p e c t e d time c a u s e d by the c h o i c e o f t h e b e t a d i s t r i b u t i o n i s o f t h e o r d e r o f twenty p e r c e n t o f t h e d i f f e r e n c e between the most o p t i m i s t i c and p e s s i m i s t i c t i m e s , and a f u r t h e r e r r o r o f t h e same o r d e r o c c u r s because o f the l i n e a r i t y assumption  i n the f o r m u l a f o r the e x p e c t e d t i m e .  36  The l a t t e r e r r o r may  be reduced by changing the weight a t t a c h e d t o  t h e most l i k e l y time a c c o r d i n g t o i t s v a l u e .  The t r i a n g u l a r d i s t r i -  b u t i o n , f o r w h i c h t h e e x p e c t e d time i s t h e average o f the t h r e e t i m e estimates,  causes the same e r r o r bounds i n i t s c h o i c e .  I n the PERT  a n a l y s i s , t h e e x p e c t e d completion time f o r t h e p r o j e c t i s always mistic.  T h i s i s due m a i n l y t o the assumption o f independence  through the network.  opti-  o f paths  Methods e x i s t t o c a t e g o r i z e and reduce t h i s e r r o r .  C o n s i d e r a t i o n o f c r i t i c a l i t y i n terms o f p r o b a b i l i t y i s more e f f e c t i v e t h a n t o t a l f l o a t , but i m p r a c t i c a l i n the e x t r e m e l y i n d e t e r minate case o f r e s e a r c h and development used f o r CPM networks, be  determined.  projects.  However, i t c o u l d be  f o r which t r u e a c t i v i t y time d i s t r i b u t i o n s c a n  CHAPTER VI  Cost  Introduction The problem o f d e r i v i n g t h e t i m e / c o s t t r a d e o f f curve i s d e f i n e d and mention i s made o f t h e papers g i n a l PERT/CPM model. assumption  s u g g e s t i n g improvements t o t h e o r i -  The a c t i v i t y t i m e / c o s t r e l a t i o n s h i p i s a b a s i c  f o r most s o l u t i o n s .  s h i p might take a r e d i s c u s s e d .  The v a r i o u s forms t h a t t h i s  A l t h o u g h t h e r e a r e many methods o f  s o l u t i o n , some o f w h i c h a r e quoted,  t h e y a l l i m p l y t h e same s e r i e s o f  o p e r a t i o n s on t h e network and i t i s t h i s t h a t i s d e s c r i b e d . t h e s e methods have been m o d i f i e d i n o r d e r t o decrease complexity.  relation-  Some o f  computational  A d e s c r i p t i o n o f two d i f f e r e n t approaches t o t h e s o l u t i o n  i s f o l l o w e d b y an examination  o f some r e l a t e d problems.  F i n a l l y , the  i n h e r e n t and s t a t e d assumptions i n the v a r i o u s s o l u t i o n s a r e a n a l y z e d .  The C o s t Problem A l t h o u g h t h e advent  o f PERT/CPM networks marked a g r e a t advance  i n p r o j e c t management, i t s l i m i t a t i o n s were immediately  apparent.  B o t h methods made u s e o f o n l y one time e s t i m a t e o r d i s t r i b u t i o n f o r each a c t i v i t y , c o s t and o t h e r r e s o u r c e s were i g n o r e d and a l l a c t i v i t i e s were s c h e d u l e d t o s t a r t as e a r l y as p o s s i b l e .  I t i s apparent t h a t the  d u r a t i o n o f most a c t i v i t i e s may be d e c r e a s e d up t o a c e r t a i n p o i n t ,  with the penalty of higher cost, that cost, manpower and other r e sources are important considerations, and that advantage may be taken of a c t i v i t y f l o a t i n reducing resource requirements by delaying the s t a r t of c e r t a i n a c t i v i t i e s , without increasing the project duration. The time/cost tradeoff curve indicates the minimum project cost f o r each possible project time* Certain overhead and resource costs increase with the length o f the p r o j e c t , and, consequently, there exists a schedule with m i n i mum cost which i s not necessarily found by s e t t i n g each a c t i v i t y to i t s smallest cost conditions.  The true project cost/tirae curve i s  therefore a combination of "a d i r e c t cost curve—the d i r e c t cost i s the sum of the costs of a l l the a c t i v i t i e s — a n d a nondirect, or overhead, cost curve. 1  Consideration i s r a r e l y made of the l a t t e r curve.  The a v a i l a b i l i t y of resources i s ' assumed unlimited and therefore resource requirements are regarded only i n terms of cost.  The  case of l i m i t e d resources i s considered i n Chapter V I I . The i n i t i a l CPM paper actually proposed ah approach to the problem of a costwise optimum schedule and suggested methods of a p p l l cation.  Freeman (31)* i n ' a l e t t e r concerning methods' of general!-"  zing PERT, performance.  considered hot only cost but also d e t a i l s of t e c h n i c a l Freeman (32) expanded t h i s approach i n a l a t e r paper,  but the d e t a i l involved makes the computational effort too great f o r the size of network encountered i n p r a c t i c e .  39 The  A c t i v i t y Time/Cost R e l a t i o n s h i p Conventionally,  each a c t i v i t y i s assumed t o have a 'normal'  d u r a t i o n and a s s o c i a t e d c o s t , and a ' c r a s h ' d u r a t i o n , w h i c h i s l e s s than t h e normal d u r a t i o n b u t a t a l a r g e r c o s t .  The f a s t e r time might  be a c h i e v e d , f o r example, b y a s s i g n i n g more men t o t h e a c t i v i t y . r e l a t i o n s h i p o f time and c o s t between t h e s e two d u r a t i o n s  The  i s presumed  e i t h e r t o be c o n t i n u o u s o r t o c o n s i s t o f a s e r i e s o f d i s c r e t e p o i n t s . A l t h o u g h l i n e a r i t y would be convenient and i s o c c a s i o n a l l y assumed, most systems t h a t c o n s i d e r  a continuous t i m e / c o s t  relation-  s h i p , o n l y assume t h e f u n c t i o n t o be convex and n o n - i n c r e a s i n g . systems do p e r m i t t h e f u n c t i o n t o b e concave.  However, i n p r a c t i c e , i t  i s a l i n e a r o r p i e c e w i s e - l i n e a r approximation t h a t i s used. c a t i o n s o f t h e s e assumptions a r e a n a l y z e d  Some  The i m p l i -  below.  D i s c r e t e p o i n t s a r e u n r e s t r i c t e d b u t a r e more complex, computationally.  A disadvantage i s t h a t t h e amount o f time o r c o s t change  t h a t can be made i s one o f a f i x e d s e t o f v a l u e s . has  s u g g e s t e d a compromise whereby t h e d i s c r e t e p o i n t s a r e r e g a r d e d  as b r e a k - p o i n t s  The  Rosenbloom (127)  o f a piecewise-linear  1  approximation.  Solution There i s a wide v a r i e t y o f a n a l y t i c a l s o l u t i o n s a v a i l a b l e .  Kelley  (W0» and K e l l e y and Walker (li5), produced t h e f i r s t mathema-  t i c a l formulation  o f the time/cost  t r a d e o f f problem and used a network  It©  f l o w a l g o r i t h m t o o b t a i n the c u r v e . s h o r t l y a f t e r w a r d s by F u l k e r s o n c e r t a i n improvements t o i t . as d i d J e w e l l  (1*2).  A s i m i l a r s o l u t i o n was  suggested  (3l*)> and Roper (126) l a t e r proposed  C l a r k (19) p r e s e n t e d a d i f f e r e n t  approach,  As a m o d i f i c a t i o n o f the p a r a m e t r i c l i n e a r p r o -  gramming f o r m u l a t i o n , an i n t e g e r l i n e a r programming t e c h n i q u e i s o f f e r e d by Meyer and S h a f f e r (51).  A l l t h e s e methods assume convex,  c o n t i n u o u s , n o n i n c r e a s i n g , independent a c t i v i t y c o s t / t i m e f u n c t i o n s . Nonconvex f u n c t i o n s and d i s c r e t e p o i n t s are handled by p r o c e d u r e s p r o (2),  posed by t h e Department o f Defence Moder and P h i l l i p s  (119).  by A l p e r t and Orkand (85)  and by  The mathematical models and t e c h n i q u e s a r e  discussed i n Chapter V I I I . Examples o f a c t u a l programs t h a t f i n d t h e p r o j e c t t i m e / c o s t t r a d e o f f curve a r e the IBM PERT/Cost System (65) of B r i g g s ( 9 0 ) .  and t h e a l g o r i t h m  An i n t e r e s t i n g f e a t u r e o f the IBM system i s t h a t i t  p e r m i t s g r o u p i n g o f s e v e r a l a c t i v i t i e s i n t o one package f o r c o s t i n g purposes. A l t h o u g h t h e mathematical approach and the assumptions d i f f e r w i d e l y i n t h e above methods, r e a c h e d i s e s s e n t i a l l y t h e same. or minimum c o s t , c o n d i t i o n s .  t h e way  made  i n w h i c h the s o l u t i o n i s  A l l a c t i v i t i e s a r e s e t a t normal,  The c r i t i c a l p a t h i s c a l c u l a t e d and i s  t h e n s h o r t e n e d by d e c r e a s i n g the d u r a t i o n o f t h e 'most e c o n o m i c a l  1  c r i t i c a l a c t i v i t y — t h a t a c t i v i t y w i t h the s m a l l e s t d e r i v a t i v e on i t s cost/time curve a t i t s present s e t t i n g .  T h i s i s r e p e a t e d , changing  t h e c r i t i c a l p a t h as o t h e r branches become c r i t i c a l , u n t i l a l l c r i t i c a l  Ul  a c t i v i t i e s are s e t a t c r a s h  conditions.  T h i s produces the minimum  cost schedule f o r a l l p o s s i b l e p r o j e c t durations. w i l l have any  f l o a t unless  may  be,  float.  activity  i t i s s t i l l s e t a t normal c o n d i t i o n s , f o r  o t h e r w i s e i t s c o s t c o u l d be amount o f the  C l e a r l y no  reduced by  extending i t s d u r a t i o n by  Whatever t h e assumptions concerning  o n l y d i s c r e t e p o i n t s on the p r o j e c t t i m e / c o s t  the  activity  cost  t r a d e o f f curve  are c a l c u l a t e d .  Modifications There i s c o n s i d e r a b l e  c o m p u t a t i o n a l e f f o r t r e q u i r e d i n the  solu-  t i o n by network f l o w , w h i c h i s the t e c h n i q u e most f r e q u e n t l y u s e d . s i d e r a b l e data storage ject.  T h i s was  reduction i s achieved  examined by P a r i k h and  by s u b d i v i s i o n o f the  J e w e l l (53) > who  p i e c e w i s e - l i n e a r convex f u n c t i o n t h a t i s o b t a i n e d can a c t as the t i m e / c o s t  show t h a t  f o r each  t i o n o f t h i s p r o c e d u r e f o r c e r t a i n s p e c i a l cases was and Rao  (88).  i s p r e s e n t e d by Handa  An  who  A  acti-  by  the network  Much e f f o r t i s  spent i n the c a l c u l a t i o n o f c r i t i c a l p a t h s , as t h i s needs t o be a f t e r each r e d u c t i o n i n the p r o j e c t d u r a t i o n . (36)  sub-  i n d i c a t e s a technique f o r e l i m i n a t i n g  from c o n s i d e r a t i o n c e r t a i n n o n - c r i t i c a l a c t i v i t i e s .  by F u l k e r s o n  the  modifica-  suggested  a l t e r n a t i v e method o f r e d u c i n g  (105),  the  t o these  v i t i e s produces the t r a d e - o f f curve f o r the whole p r o j e c t .  pro-  subproject,  c u r v e f o r an a c t i v i t y r e p r e s e n t i n g  p r o j e c t , and t h a t a second a p p l i c a t i o n o f the a l g o r i t h m  Blanning  Con-  i n which successive  An approach i s  done described  c r i t i c a l p a t h s are e a s i l y f o u n d .  Other Solutions; An e n t i r e l y d i f f e r e n t form o f s o l u t i o n i s proposed b y Berman (l!>).  I n t h i s a l g o r i t h m , e v e n t s a r e b a l a n c e d by e q u a t i n g , i n a b s o l u t e  v a l u e s , the sums o f t h e t i m e / c o s t s l o p e s o f t e r m i n a l a c t i v i t i e s and initial activities. continuous  A c t i v i t y t i m e / c o s t f u n c t i o n s a r e assumed t o be  and concave.  The u n d e r l y i n g p r i n c i p l e i s b e s t i l l u s t r a t e d  by c o n s i d e r i n g a s i m p l e event w i t h o n l y one i n i t i a l terminal a c t i v i t y .  of  a c t i v i t y and  one  An i n c r e a s e i n the time f o r the s t e e p e r s l o p e and  an e q u a l d e c r e a s e i n t h e time f o r the f l a t t e r s l o p e w i l l always r e s u l t in  a r e d u c t i o n i n the combined c o s t — p r o v i d e d t h a t t h e e n d - p o i n t s  the c u r v e s a r e n o t i n v o l v e d .  of  The minimum combined c o s t o c c u r s a t those  p o i n t s where the s l o p e s a r e e q u a l .  The e v e n t s a r e b a l a n c e d one a t a  t i m e , w i t h o t h e r e v e n t s h e l d c o n s t a n t , i n an i t e r a t i v e p r o c e s s w h i c h t e r m i n a t e s when a l l e v e n t s are s i m u l t a n e o u s l y b a l a n c e d . is  The  procedure  r e p e a t e d t o d e r i v e the complete p r o j e c t c o s t / t i m e c u r v e . An a l t e r n a t i v e approach,  not mentioned i n t h e l i t e r a t u r e , i s  to  r e v e r s e t h e s t a n d a r d t e c h n i q u e by i n i t i a l l y  to  crash conditions.  T h i s immediately  setting a l l activities  determines  t h e minimum p r o j e c t  time, i n c o n t r a s t w i t h the u s u a l minimum p r o j e c t c o s t . f l o a t s are reduced  A l l activity  t o z e r o i n t h e most e c o n o m i c a l o r d e r , t h e r e b y  a c h i e v i n g the minimum c o s t f o r t h i s d u r a t i o n .  The p r o j e c t time i s  i n c r e a s e d by l e n g t h e n i n g c r i t i c a l a c t i v i t y times most e c o n o m i c a l l y . Management may  o f t e n n o t be i n t e r e s t e d i n t h e e n t i r e t r a d e o f f c u r v e , e i  t h e r because o f p o l i c y o r because o f the shape o f the overhead r e q u i r e d segment would determine  curve,  w h i c h procedure i s a p p r o p r i a t e .  1*3 D i f f e r e n t Problems J e w e l l (U2) c o n s i d e r e d an approach b y w h i c h t h e e f f o r t t o maint a i n event t i m e s , n e c e s s i t a t e d b y t h e d i f f e r e n c e between a c t u a l and expected a c t i v i t y t i m e s , i s minimized.  The event times which r e s u l t e d  from t h e i n i t i a l a n a l y s i s a r e regarded as f i x e d .  As a c t u a l d u r a t i o n s  are r e p o r t e d , t h e a c t i v i t i e s a r e r e - s c h e d u l e d most e c o n o m i c a l l y i n o r d e r t o a c h i e v e t h e s e event t i m e s . Sampling  t e c h n i q u e s a r e used by Beckwith  a p r o j e c t c o s t overrun i s l i k e l y t o take p l a c e .  ( l h ) t o e s t i m a t e when I t i s assumed t h a t t o -  t a l c o s t i s t h e sum o f l a b o u r c o s t , w h i c h i s time dependent, and some n o n d i r e c t c o s t , assumed c o n s t a n t .  Random samples a r e taken from each  a c t i v i t y c o s t d i s t r i b u t i o n , w h i c h i s i t s e l f d e r i v e d from t h e a c t i v i t y time d i s t r i b u t i o n .  The method may be u s e d b o t h i n t h e p l a n n i n g s t a g e  and w h i l s t t h e p r o j e c t i s i n p r o g r e s s . is  However, no c o r r e c t i v e  action  specified. Some c o s t a c c o u n t i n g problems encountered  a r e c o n s i d e r e d b y H i l l (106).  i n PERT/Cost  (2)  A l t h o u g h t h e weakness i n t h e assump-  t i o n o f u n l i m i t e d r e s o u r c e s i s mentioned, most emphasis i s p l a c e d on t h e d i f f i c u l t y i n o b t a i n i n g a c c u r a t e i n f o r m a t i o n , b o t h b e f o r e and d u r i n g the p r o j e c t . accounting. ing.  C l e r i c a l e r r o r s b y foremen c a n s e r i o u s l y d i s t o r t t h e The model i t s e l f cannot d e a l e f f e c t i v e l y w i t h s u b c o n t r a c t -  hh Assumptions D a v i s (210  examined the v a r i o u s assumptions, b o t h i n h e r e n t  s t a t e d , i n t h e s o l u t i o n s t o the t i m e / c o s t t r a d e o f f problem. d i f f i c u l t i e s i n c o n s i d e r i n g t e c h n i c a l performance, suggested (31)  were c o n s i d e r e d above.  y  (100)  Fondahl  observed,  by Freeman of  As  a c t i v i t y d u r a t i o n and r e s o u r c e a l l o c a t i o n i s  o f t e n dependent upon the parameters o f o t h e r a c t i v i t i e s . c o n s i d e r e d , some methods can a l l o w f o r i t , tional  The  However, assuming a c o n s t a n t l e v e l  performance i s l i k e l y t o c o n t r i b u t e e r r o r s t o the a n a l y s i s .  and  If this i s  a t t h e expense o f computa-  complexity. There i s i n s u f f i c i e n t e m p i r i c a l data t o e s t a b l i s h w h i c h assump-  t i o n , w i t h r e g a r d t o a c t i v i t y t i m e / c o s t c u r v e s , i s most a c c u r a t e . c r e t e p o i n t s i n t u i t i v e l y appear more r e a l i s t i c . apparent valent  i n what way  'slope'.  However, i t i s n o t  i t i s b e s t t o evaluate i n t e r i m p o i n t s or the equi-  I t i s an open q u e s t i o n whether the assumption o f con-  t i n u i t y might l e a d , i n p r a c t i c e , t o an i n f e a s i b l e s o l u t i o n . p l e , t h e time an a c t i v i t y t a k e s may assigned to i t . man  Dis-  Consequently  depend upon the number o f  a s c h e d u l e may  F o r exammen  require a f r a c t i o n of a  and would t h e r e f o r e , t e c h n i c a l l y , be i n f e a s i b l e .  I t i s a tribute  t o the b a s i c i n a c c u r a c y o f the d a t a t h a t t h e n e c e s s a r y r o u n d i n g i s a p p a r e n t l y u n n o t i c e d and t h a t the q u e s t i o n has y e t t o be r e s o l v e d .  As  automation and u n i o n piecework c o n t r o l i s i n c r e a s e d , i t i s r e a s o n a b l e t o presume t h a t more a c c u r a t e i n f o r m a t i o n w i l l become a v a i l a b l e .  The  e f f e c t o f c o n s i d e r i n g manpower and o t h e r r e s o u r c e s which are f u n c t i o n s  o f the a c t i v i t y d u r a t i o n on the a c t i v i t y t i m e / c o s t curve i s a n a l y z e d i n Appendix 'B'.  I t i s shown t h a t c a r e f u l examination o f the p a r a -  meters d e f i n i n g each o f t h e s e c u r v e s s h o u l d be made, as i t i s p o s s i b l e f o r a c u r v e t o have a l o c a l maximum o r minimum between i t s extreme points.  The c o n c l u s i o n reached i s t h a t a n a l y s i s o f the c o s t components  i s n e c e s s a r y f o r t r u e u n d e r s t a n d i n g o f the c o s t / t i m e r e l a t i o n s h i p .  Summary The t i m e / c o s t t r a d e - o f f problem  i s d e r i v i n g t h e minimum p r o -  j e c t c o s t f o r each p o s s i b l e p r o j e c t d u r a t i o n .  The a v a i l a b i l i t y o f  r e s o u r c e s i s assumed u n l i m i t e d , and t h e y are t h e r e f o r e regarded units of cost.  T e c h n i c a l performance  n o t be r e a s o n a b l y i n c o r p o r a t e d  i s a v a l i d v a r i a b l e , but i t can-  i n t o the model.  Each a c t i v i t y i s assumed t o have 'normal' and a s s o c i a t e d c o s t s .  and  'crash'  Between t h e s e extreme p o i n t s , the  s h i p can be c o n t i n u o u s o r d i s c r e t e . The  times  relation-  L i n e a r i t y i s r a r e l y assumed.  curve can be convex, concave o r p i e c e w i s e - l i n e a r .  t h a t i s used i n p r a c t i c e .  as  I t i s the  latter  D i s c r e t e p o i n t s are u n r e s t r i c t e d but r e -  q u i r e more c o m p u t a t i o n a l e f f o r t .  The curve c o u l d be c o n s i d e r e d as  l i n e a r between t h e s e p o i n t s . There i s a wide v a r i e t y o f a n a l y t i c a l s o l u t i o n s a v a i l a b l e , g e n e r a l l y d i f f e r i n g o n l y i n t h e i r assumption relationship.  of a c t i v i t y time/cost  Some o f t h e s e methods use r e a l and i n t e g e r  linear  U6 programming t e c h n i q u e s . on t h e network. cal  Most s o l u t i o n s p e r f o r m  t h e same o p e r a t i o n s  A l l a c t i v i t i e s a r e s e t t o normal c o n d i t i o n s .  Criti-  a c t i v i t i e s a r e reduced most e c o n o m i c a l l y , u n t i l t h e minimum c o s t  schedule f o r the minimum p r o j e c t d u r a t i o n i s found. Reduction i n computational of  e f f o r t i s a c h i e v e d by s u b d i v i s i o n  t h e p r o j e c t , b y e l i m i n a t i n g from c o n s i d e r a t i o n c e r t a i n  a c t i v i t i e s o r by r a p i d e v a l u a t i o n o f s u c c e s s i v e c r i t i c a l  non-critical paths.  An a l t e r n a t i v e approach i s t o equate t h e sum o f t h e c o s t / t i m e s l o p e s o f a c t i v i t i e s e n t e r i n g and o f a c t i v i t i e s l e a v i n g each event. The  c o s t i s minimum when a l l events a r e s i m u l t a n e o u s l y  Another method i s t o r e v e r s e t h e s t a n d a r d procedure activities initially  balanced.  by crashing a l l  and working backwards towards t h e minimum c o s t  s c h e d u l e f o r t h e maximum t i m e . I f a l l event times a r e assumed f i x e d a t t h e i r planned technique  dates, a  e x i s t s f o r m i n i m i z i n g t h e e f f o r t t o a c h i e v e t h e s e times i n  the l i g h t o f a c t u a l a c t i v i t y durations.  Sampling t e c h n i q u e s may be  u s e d t o e s t i m a t e when a p r o j e c t c o s t o v e r r u n i s l i k e l y t o o c c u r . a c c o u n t i n g problems a r e encountered  because o f l i m i t e d  Cost  resources,  c l e r i c a l e r r o r s , and i n a c c u r a t e i n f o r m a t i o n . The of  assumptions i n h e r e n t i n t h e systems a r e a c o n s t a n t  standard  t e c h n i c a l performance and t h e independence o f a c t i v i t y d u r a t i o n s and  resource requirements.  What form o f a c t i v i t y t i m e / c o s t r e l a t i o n s h i p i s  b e s t i s n o t known.  The assumption o f c o n t i n u i t y might l e a d t o an i n -  feasible solution.  Deeper a n a l y s i s o f t h e c o s t components s h o u l d be made.  CHAPTER V I I  MANPOWER AND  OTHER RESOURCES  Introduction There a r e two b a s i c problems t o be examined when r e s o u r c e s considered.  The  f i r s t i s r e s o u r c e l e v e l i n g and the second r e s o u r c e -  c o n s t r a i n e d s c h e d u l e s w i t h minimum d u r a t i o n . of s o l u t i o n  A f t e r the various types  have been a n a l y z e d , t h e g e n e r a l s o l u t i o n ,  the c r i t e r i a used  and t h e methods t h a t use them are d e s c r i b e d f o r b o t h c a s e s . lytic  are  Two  ana-  approaches a r e i n t r o d u c e d , and t h e s e a r e f o l l o w e d by an attempt  to c l a s s i f y feasible  s c h e d u l e s and by the s p e c i f i c a t i o n o f a f u n c t i o n  w h i c h approximates the r e s o u r c e p r o f i l e .  The  Problems I n the p r e v i o u s c h a p t e r , i t was  l i m i t e d s u p p l y o f r e s o u r c e s and, s o u r c e s was  considered.  assumed t h a t t h e r e i s an  consequently,  un-  only the c o s t o f r e -  W h i l e , f o r a l l p r a c t i c a l purposes,  t h i s may  be  t r u e f o r many r e s o u r c e s , o t h e r r e s o u r c e s , n o t a b l y manpower, are d i s t i n c t l y scarce.  The  demand f o r these r e s o u r c e s i s an i m p o r t a n t f a c t o r .  quote Berman (l5)«  "No  l o a d i n g , a s e r i o u s and maker.  The  To  c o n s i d e r a t i o n has been g i v e n here t o manpower difficult  problem f o r t h e r e a l - w o r l d d e c i s i o n -  f a i l u r e t o c o n s i d e r t h i s problem undoubtedly l i m i t s the  u s e f u l n e s s o f t h i s model." Two  d i f f e r e n t problems are c o n s i d e r e d .  The  f i r s t assumes t h a t  U8 t h e p r o j e c t d u r a t i o n i s f i x e d , and, w i t h i n t h i s c o n s t r a i n t , r e q u i r e s t h a t the resource p r o f i l e , which i n d i c a t e s the resource  requirements  f o r the d u r a t i o n o f the p r o j e c t , be as l e v e l as p o s s i b l e . t i o n t o t h i s problem may off analysis.  The  The  solu-  be u s e d i n c o n j u n c t i o n w i t h a t i m e / c o s t t r a d e -  second approach i s t o attempt the m i n i m i z a t i o n  t h e p r o j e c t d u r a t i o n , s u b j e c t t o a g i v e n l i m i t e d s u p p l y o f the  of  resource.  B o t h m u l t i - p r o j e c t s and m u l t i - r e s o u r c e s need a l s o t o be examined. cause o f i t s importance,  Be-  manpower i s f r e q u e n t l y t h e s c a r c e r e s o u r c e  r e f e r r e d t o by the a u t h o r s i n d e s c r i b i n g t h e i r methods of  solution;  Types o f S o l u t i o n I n sharp c o n t r a s t w i t h the a n a l y t i c approach t o t h e t i m e / c o s t t r a d e - o f f problem, a l l p r a c t i c a l s o l u t i o n s t o t h e r e s o u r c e problems are h e u r i s t i c i n nature.  C e r t a i n a n a l y t i c procedures  have been des-  c r i b e d , but these are e i t h e r h i g h l y r e s t r i c t i v e or computationally prohibitive. R e d u c t i o n i n r e s o u r c e requirements start of certain a c t i v i t i e s . i s o f the o r d e r o f the p r o d u c t activities.  i s a c h i e v e d by d e l a y i n g t h e  The number o f p o s s i b l e s c h e d u l e s i n v o l v e d o f the f l o a t s o f a l l the  non-critical  I n f a c t , i t i s o n l y l e s s t h a n t h i s p r o d u c t because o f the  r e s t r i c t i o n s on the o t h e r a c t i v i t i e s imposed by the c h o i c e o f s t a r t i n g time f o r each a c t i v i t y . activities. excessive.  T h i s i g n o r e s the added c o m p l i c a t i o n o f  split  The magnitude o f the a n a l y t i c s o l u t i o n i s t h e r e f o r e The number o f e q u a t i o n s i n the m a t h e m a t i c a l model can  be  U9 considerably reduced by imposing severe r e s t r i c t i o n s upon a c t i v i t y durations and resource requirements, but t h i s i s not a p r a c t i c a l consideration f o r the PERT/CPM network.  The General Solution to the Leveling Problem I f each a c t i v i t y i s assigned a duration, there i s a minimum rectangular resource p r o f i l e , usually unattainable, determined by the r a t i o of resource-hours  whose height i s  to project duration.  Each  schedule i s given a value, determined by i t s deviation from t h i s minimum, according t o one of a number of different c r i t e r i a .  A solution to  the l e v e l i n g problem i s found by choosing an a r b i t r a r y schedule and c a l culating i t s value. value.  A new schedule i s then derived, with a smaller  The procedure i s repeated u n t i l no further reduction i n value  can be achieved.  The optimum i s not necessarily reached and, i n f a c t ,  different r e s u l t s are obtained from different i n i t i a l schedules. of solutions i s derived and the best i s  A set  selected.  Criteria For t h e i r method, Dewitte (25) and Wagner, G i g l i o and Glaser (132) minimize the sum of the absolute values of the manpower fluctuations from the mean, while Tate (130) variations i n work f o r c e .  considers the variance of the  Burgess and Killebrew (16) propose a solu-  t i o n which i s a combination of a combinatorial and graphical approach, implementing a Gantt-chart for ease of a c t i v i t y ' s h i f t i n g ' .  Their  c r i t e r i o n , also used by Petrovic (55)* involves the sum of the squares  50  o f the resource  requirements.  Wagner a l s o suggested m i n i m i z i n g t h e  maximum change i n manpower, the sura o f t h e a b s o l u t e  changes i n man-  power o r the peak manpower requirement.  Other S o l u t i o n s The  c r i t e r i o n o f peak manpower requirement i s used by Levy,  Thompson and W i e s t (1*7) i n t h e i r a l g o r i t h m f o r h a n d l i n g s e v e r a l p r o j e c t s simultaneously.  Each p r o j e c t has a r e s o u r c e p r o f i l e and, i f  p o s s i b l e , t h e peaks f o r a l l the p r o f i l e s a r e reduced a t t h e same t i m e . The b a s i c procedure i s t h e same as t h a t d e s c r i b e d above--the o n l y  dis-  t i n c t i o n between t h i s and t h e s i n g l e p r o j e c t s o l u t i o n i s t h a t e a c h p r o j e c t r e c e i v e s a p r i o r i t y r a t i n g , w h i c h i s used when the simultaneous r e d u c t i o n has been completed and i n d i v i d u a l l e v e l i n g i s s t a r t e d . M a t h e m a t i c a l programming s o l u t i o n s t o b o t h the l e v e l i n g and t h e resource-constrained (55)  problems a r e f o r m u l a t e d  by Wagner  (132), P e t r o v i c  and, u s i n g a n e l e c t r i c a l network a n a l o g u e , by Ghare (10l*).  i m p r a c t i c a l ! t y has a l r e a d y been d i s c u s s e d above.  Their  However, P e t r o v i c  shows t h a t s u b d i v i s i o n o f t h e p r o j e c t and u s e o f optimum c o n t r o l t h e o r y can c o n s i d e r a b l y reduce the c o m p u t a t i o n a l  The  effort.  General S o l u t i o n t o the Resource-Constrained The  Problem  r e s o u r c e - c o n s t r a i n e d minimum-duration s c h e d u l i n g problem  may be s o l v e d b y a h e u r i s t i c , l o c a l s t r a t e g y i n w h i c h a c t i v i t i e s a r e scheduled  t o s t a r t as soon as p o s s i b l e , p r o v i d e d t h a t the r e s o u r c e  s t r a i n t i s n o t exceeded.  I f i t i s , then a c t i v i t i e s are s t a r t e d  con-  51 a c c o r d i n g t o some p r i o r i t y system, w h i l e r e s o u r c e s a r e a v a i l a b l e . The  remaining  alleviate  a c t i v i t i e s have t h e i r s t a r t s  delayed.  In order t o  t h e i n e v i t a b l e b o t t l e n e c k , some form o f look-ahead f e a t u r e  i s usually incorporated.  T h i s procedure i s c o n t i n u e d u n t i l a l l a c t i -  v i t i e s have been s c h e d u l e d . different  priority  The r e s u l t  i s n o t always optimum, and  systems w i l l u s u a l l y produce d i f f e r e n t  results.  P r i o r i t y Systems A  'serial' allocation  procedure l i s t s a l l a c t i v i t i e s  t o some p r i o r i t y and a l l o c a t e s a v a i l a b l e r e s o u r c e s order.  On t h e o t h e r hand, a p a r a l l e l ' a l l o c a t i o n 1  s e v e r a l a c t i v i t i e s a t one time. ' e x t r u s i o n ' method. tem  i s 'static'.  procedure,  according  according t o t h i s procedure  schedules  I t i s sometimes r e f e r r e d t o as an  I f t h e l i s t i n g remains c o n s t a n t , the p r i o r i t y  sys-  C o n v e r s e l y , i f t h e o r d e r changes d u r i n g t h e s c h e d u l i n g  a 'dynamic* p r i o r i t y system i s b e i n g used.  Most s o l u t i o n s employ an o r d e r e d s e t o f p r i o r i t y systems t o s o r t a c t i v i t i e s w h i c h would have e q u a l p r i o r i t y under one system. Martino  F o r example,  ( 1 0 ) , i n h i s M u l t i - r e s o u r c e A l l o c a t i o n P r o c e d u r e , known as MAP,  l i s t s r e l e v a n t a c t i v i t i e s a c c o r d i n g t o one dynamic and t h r e e p r i o r i t y systems,  static  They a r e , i n o r d e r , L e a s t Remaining T o t a l F l o a t ,  L a r g e s t Need o f Resource-Hours, L a r g e s t Resource Requirement and S m a l l e s t Sequence Code.  The Sequence Code p r i o r i t y assumes an i - j o r d e r i n g and  sorts a c t i v i t i e s , f i r s t  on ' j ' ,  code i t s e l f i s t h e o r d e r e d p a i r  then on ' i ' . (j,i).  Consequently, t h e sequence  52 Pascoe (5U)  conducted an e x p e r i m e n t a l  comparison o f  h e u r i s t i c methods f o r a l l o c a t i n g r e s o u r c e s .  these  U s i n g networks w i t h  c e r t a i n c h a r a c t e r i s t i c s , such as a p a r t i c u l a r a c t i v i t y - t o - e v e n t r a t i o or a s i m i l a r p o s i t i o n o f r e s o u r c e p r o f i l e peak, Pascoe i n v e s t i g a t e d t e n p r i o r i t y systems f o r e a c h o f f i v e o b j e c t i v e f u n c t i o n s , u s i n g b o t h p a r a l l e l and  s e r i a l methods and  and r e s o u r c e - c o n s t r a i n e d networks.  The  o b j e c t i v e f u n c t i o n s i n c l u d e d the  l e v e l i n g c r i t e r i a o f T a t e , o f Dewitte and conclusions of K e l l e y  considering time-constrained  o f Levy.  C o n t r a r y t o the  (111), Pascoe showed t h a t p a r a l l e l methods a l -  ways produced b e t t e r r e s u l t s .  He  observed  t h a t the c r i t e r i a o f  and o f Dewitte a c h i e v e d t h e same r e s u l t s f o r any network w i t h priority  system.  However, t h e peak requirement  was  Tate  any  b e s t minimized by  c o n s i d e r i n g t h e L a t e s t F i n i s h Time.  U n f o r t u n a t e l y , o n l y one network  w i t h more than twenty a c t i v i t i e s was  c o n s i d e r e d , and t h i s  seriously  l i m i t s the v a l i d i t y o f the c o n c l u s i o n s .  Methods o f S o l u t i o n Martino  (10) makes t h e q u e s t i o n a b l e c l a i m t h a t no procedure w i l l  " o f f e r any b e t t e r s o l u t i o n " than h i s system. t i e s and v a r i a b l e r e s o u r c e requirements more t h a n one r e s o u r c e .  MAP  permits  and i s designed  split  activi-  to deal with  I t s p r i o r i t y procedure i s d e s c r i b e d above.  B o t h t h e l a c k o f any look-ahead f e a t u r e and t h e l i m i t a t i o n t o o n l y  one  s e t o f p r i o r i t y systems l e s s e n s i t s e f f e c t i v e n e s s . A s l i g h t m o d i f i c a t i o n o f t h i s a l g o r i t h m i s g i v e n i n Chapter  XIV.  53 K e l l e y ( U l ) presents  a very s i m i l a r routine t o that o f  Burgess and K i l l e b r e w (16), d i f f e r i n g i n i t s c o n s i d e r a t i o n o f s p l i t a c t i v i t i e s and o f a t i m e / r e s o u r c e a c t i v i t y duration. b e s t schedule  r e l a t i o n s h i p that allows v a r i a b l e  D i f f e r e n t p r i o r i t y systems a r e c o n s i d e r e d and t h e  i s selected.  A d i f f e r e n t approach, a t t r i b u t e d t o G.H. B r o o k s , i s d e s c r i b e d by Moder and P h i l l i p s  (119).  durations are f i x e d .  However, v e r y good r e s u l t s a r e o b t a i n e d  A c t i v i t i e s may n o t be s p l i t and t h e i r on t h e  f i r s t pass t h r o u g h t h e a l g o r i t h m . RAMPS—Resource A l l o c a t i o n and M u l t i - P r o j e c t a comprehensive program, i n t r o d u c e d i n 1962 p r o j e c t s and m u l t i - r e s o u r c e s .  Scheduling—is  ( 3 ) , which handles  There i s t e c h n i c a l l y v e r y l i t t l e  ence between s c h e d u l i n g one p r o j e c t and a few p r o j e c t s .  raultidiffer-  The r e s o u r c e  p r o f i l e s o f t h e p r o j e c t s a r e combined i n t o one p r o f i l e f o r s c h e d u l i n g p u r p o s e s , as i t i s assumed t h a t t h e p r o j e c t s share a common p o o l o f r e sources.  Resources a r e s p e c i f i e d as one Lead Resource, w h i c h i s t h e  only resource f o r which a resource/time and s e v e r a l T r a i l i n g R e s o u r c e s . corporated resources  r e l a t i o n s h i p may be s p e c i f i e d ,  A l t h o u g h a look-ahead f e a t u r e i s i n -  t o a l l e v i a t e b o t t l e n e c k s , Premium R e s o u r c e s , w h i c h a r e e x t r a o n l y a v a i l a b l e a t g r e a t e r c o s t , s u c h as overtime,  i f a b s o l u t e l y necessary. f a c t o r y schedules  are used  However, as Lambourn (U6) comments, u n s a t i s -  due t o b o t t l e n e c k s s t i l l may o c c u r , and then  data  changes a r e r e q u i r e d i n o r d e r t o produce a f e a s i b l e s c h e d u l e .  Although  no p r e c i s e c l a i m t o t r u e m i n i m i z a t i o n  o f RAMPS,  i s made b y the c o m p i l e r s  Moshman, Johnson and L a r s e n , i n an o p t i m i s t i c d e s c r i p t i o n o f t h e  a l g o r i t h m (52),  do i m p l y t h a t o n l y f a u l t y d a t a s t o p s t h e system  f i n d i n g an optimum s c h e d u l e .  from  They a l s o c r e d i t G.J. F i s h e r o f E . I .  du Pont de Nemours w i t h the o r i g i n a l p r o p o s i t i o n o f the problem. Management a s s i g n s p r i o r i t i e s t o v a r i o u s f a c t o r s , such as t h e m i n i m i z a t i o n o f c o s t o r o f time o r of i d l e r e s o u r c e s , o r t h e maximizat i o n o f the t o t a l number o f a c t i v i t i e s i n o p e r a t i o n a t any one  time.  Work e f f i c i e n c y , s p l i t a c t i v i t i e s and c o s t p e n a l t i e s f o r d e l a y e d p r o j e c t completion are considered.  The h e u r i s t i c procedure  reduces r e -  s o u r c e r e q u i r e m e n t s , where t h e y a r e i n excess o f t h e c o n s t r a i n t , by r e d u c i n g t h e demand o f a c o n t r i b u t i n g a c t i v i t y most e c o n o m i c a l l y , i f this i s possible.  Otherwise, an i n f e a s i b l e s c h e d u l e i s produced  the output w i l l i n d i c a t e w h i c h c o n s t r a i n t i s too low. Klein (H3)  and  An a r t i c l e  by  n o t e s t h a t t h e network does n o t r e q u i r e t o p o l o g i c a l o r d e r -  i n g and d e s c r i b e s i t s e x t e n s i v e e r r o r d e t e c t i o n . A n o t h e r approach and M a r k a r i a n (1*9)• relationships,  The  t o m u l t i - p r o j e c t s c h e d u l i n g i s p r e s e n t e d by McGee system assumes l i n e a r man/time and manhour/time  the l a t t e r r e p r e s e n t i n g c o s t .  The assumptions  i t y a r e n o t e s s e n t i a l and a r e made f o r c o m p u t a t i o n a l ease.  of l i n e a r -  The  arguments  w i t h reg;ard t o t h e a c t u a l form o f t h e s e curves would r u n p a r a l l e l w i t h t h o s e used f o r t h e t i m e / c o s t t r a d e o f f c u r v e , c o n s i d e r e d i n Chapter  VI.  A l l a c t i v i t i e s are s c h e d u l e d w i t h minimum manpower requirements and r e s u l t a n t p r o j e c t d u r a t i o n compared w i t h t h e r e q u i r e d c o m p l e t i o n  the  time.  I f t h i s d u r a t i o n needs t o be reduced, t h i s i s done a t minimum expense. Where r e s o u r c e c o n s t r a i n t s a r e exceeded, n o n - c r i t i c a l a c t i v i t i e s are r e s c h e d u l e d h e u r i s t i c a l l y t o reduce the e x c e s s .  For the m u l t i - p r o j e c t  55 case, each project i s handled separately and i t s p r i o r i t y f o r r e source a l l o c a t i o n i s determined by the difference between the projected and required completion dates.  A n a l y t i c Approaches The Generalized Lagrange M u l t i p l i e r Method of Everett  III  (30) does not guarantee a solution to every problem, but any solution that i t does f i n d i s optimum. on i t a real-valued ' P a y - O f f valued 'Resource'  A set of f e a s i b l e schedules has defined function and, f o r each resource,  function and a constraint.  No r e s t r i c t i o n s  a realsuch as  continuity or d i f f e r e n t i a b i l i t y need be applied to these functions. Lagrange m u l t i p l i e r s are used to maximize the pay-off function, subject to the constraints  on the resource  functions.  Besides introducting a s l i g h t modification of Levy's algorithm (ii7) which e s s e n t i a l l y produces the resource/time tradeoff curve by a dynamic programming method, Wilson (61*) compared the resource l e v e l i n g problem with the assembly l i n e balancing problem and showed that some of the theory of the l a t t e r can be applied to the former. considerable r e s t r i c t i o n s that a l l a c t i v i t i e s  However, the  should have the same  duration and the same resource requirements need to be imposed. illustrate by  To  t h i s , Wilson showed how some l i n e balancing theory, developed  Hu (108), may be applied to give a lower bound on the maximum r e -  source requirement for a p r o j e c t .  I f i t i s assumed that a c t i v i t i e s may  be interrupted at w i l l , they may be divided i n t o u n i t time i n t e r v a l s and the a c t i v i t y duration r e s t r i c t i o n may be l i f t e d .  This sub-division  56 o f a c t i v i t i e s i s used b y B l a c k (8?)  i n a technique f o r generating a l l  f e a s i b l e s c h e d u l e s , under r e s o u r c e c o n s t r a i n t s , w h i c h i s a l s o based  on  l i n e balancing theory.  Other Approaches P r o p e r t i e s o f f e a s i b l e s c h e d u l e s under r e s o u r c e c o n s t r a i n t s are examined by W i e s t (63).  S e t s o f l e f t - j u s t i f i e d and  right-  j u s t i f i e d s c h e d u l e s and t h e one-to-many mapping between them a r e def i n e d , and t h i s l e a d s t o a new the r e s o u r c e l i m i t a t i o n . analytic or'heuristic.  concept o f s l a c k w h i c h i n c o r p o r a t e s  G e n e r a t i o n o f t h e s e s c h e d u l e s may  be  either  Whichever method i s u s e d , though, t o d e r i v e a l l  s c h e d u l e s , and hence t o choose t h e optimum, would take a p r o h i b i t i v e amount o f time.  A non-rigorous set of r u l e s to f i n d  schedules i s d e v i s e d .  c e r t a i n o f the  However, no r e s u l t s a r e g i v e n and so the e f f e c t -  i v e n e s s o f the p r o c e d u r e i s n o t  determined.  A p r o f i l e f u n c t i o n d e s c r i b e s a continuous curve that, may f i t t e d t o a resource p r o f i l e .  Norden (75)  be  c l a i m s t o have found a  g e n e r a l type o f p r o f i l e f u n c t i o n w h i c h has been " e n c o u r a g i n g l y s t a b l e o v e r a wide range o f p r o j e c t s and a number o f y e a r s . "  The  function  2 is y  a  K a t e " * , where y i s the manpower employed a t time t , K i s the a  t o t a l number o f manhours f o r the p r o j e c t and a i s an 'urgency  factor",  w h i c h c o u l d i r r e v e r e n t l y be r e f e r r e d t o as a 'fudge f a c t o r ' and which c o n t r o l s t h e h e i g h t o f t h e peak. knowledge o f t h i s f u n c t i o n ,  Whatever v a l u e t h e r e might be i n  o t h e r t h a n t o v e r y c r u d e l y approximate  the  57 d i s c o n t i n u o u s r e s o u r c e p r o f i l e w i t h a continuous and i s n o t i n d i c a t e d b y the  curve, i s not  obvious  author.  Suimnary Some r e s o u r c e s , s u c h as manpower, are l i m i t e d i n s u p p l y t h e demand f o r them must be c o n s i d e r e d . lems a s s o c i a t e d w i t h t h i s .  The  first  There a r e two  concerned w i t h m i n i m i z i n g are maintained  d i f f e r e n t prob-  i n v o l v e s l e v e l i n g the  p r o f i l e , while the p r o j e c t d u r a t i o n i s kept constant.  resource  The second i s  the d u r a t i o n , w h i l e the r e s o u r c e  within their constraints.  and  requirements  P r a c t i c a l s o l u t i o n s are a l l  h e u r i s t i c , as t h e number o f p o s s i b l e s c h e d u l e s A s o l u t i o n t o the l e v e l i n g problem may  i s so l a r g e . be found  by s e l e c t i n g a  s e r i e s o f s c h e d u l e s w i t h d e c r e a s i n g v a l u e s , determined b y some c r i t e r i o n . D i f f e r e n t s o l u t i o n s r e s u l t from d i f f e r e n t i n i t i a l  schedules.  The  t e r i a i n d i c a t e t h e degree o f f l u c t u a t i o n o f t h e r e s o u r c e p r o f i l e ing  from the  h a n d l i n g one s o l u t i o n s may mum  schedule.  reduced  result-  There i s l i t t l e t e c h n i c a l d i f f e r e n c e between  and many p r o j e c t s . be  cri-  The  excessive e f f o r t involved i n a n a l y t i c  by s u b d i v i d i n g the p r o j e c t and by u s i n g  opti-  r e s o u r c e - c o n s t r a i n e d problem i s s o l v e d by a h e u r i s t i c  local  c o n t r o l theory. The  s t r a t e g y i n w h i c h t h e s t a r t o f an a c t i v i t y i s o n l y d e l a y e d i f t h e r e i n s u f f i c i e n t resources.  The p r i o r i t y system determines which a c t i v i t i e s  have t h e g r e a t e r demand on t h e a v a i l a b l e r e s o u r c e s . scheduled  are  A c t i v i t i e s may  be  i n d i v i d u a l l y by a s e r i a l a l l o c a t i o n procedure o r i n groups by  58 a p a r a l l e l a l l o c a t i o n procedure.  A f i x e d o r d e r , such a s s o r t i n g by-  r e s o u r c e requirement, i n d i c a t e s a s t a t i c p r i o r i t y , w h i l e an o r d e r w h i c h v a r i e s w i t h t i m e , such a s s o r t i n g b y r e m a i n i n g t o t a l f l o a t , i s dynamic. D i f f e r e n t p r i o r i t y systems a r e u s e d t o break t i e s . shown t o be always b e t t e r t h a n s e r i a l p r o c e d u r e s .  P a r a l l e l methods a r e Some systems c o n s i d e r  s p l i t a c t i v i t i e s and r e s o u r c e requirements t h a t v a r y w i t h a c t i v i t y duration.  RAMPS i s a comprehensive  and m u l t i - r e s o u r c e s .  program w h i c h handles m u l t i - p r o j e c t s  A s o l u t i o n s i m i l a r t o that f o r the time/cost trade-  o f f curve i s used i n c o n s i d e r i n g t h e r e s o u r c e / t i m e r e l a t i o n s h i p .  However  a h e u r i s t i c method i s s t i l l needed t o keep r e s o u r c e requirements w i t h i n their  constraints. Lagrange m u l t i p l i e r s a r e used i n an a n a l y t i c method which g i v e s  optimum s o l u t i o n s , b u t does n o t always f i n d a s o l u t i o n . assembly  The t h e o r y o f  l i n e b a l a n c i n g may be used f o r t h e r e s o u r c e l e v e l i n g  However, i t s a r e a s o f a p p l i c a t i o n a r e h i g h l y  problem.  restrictive.  The c l a s s i f i c a t i o n o f f e a s i b l e s c h e d u l e s l e a d s t o a new d e f i n i t i o n o f slack which incorporates resource c o n s t r a i n t s .  A continuous gen-  e r a l p r o f i l e f u n c t i o n has been found, which v e r y c r u d e l y approximates the r e s o u r c e p r o f i l e .  CHAPTER V I I I MODELS AND  SOLUTIONS  Introduction M a t h e m a t i c a l s o l u t i o n s have o n l y been l i g h t l y touched on i n p r e vious chapters.  I n t h i s c h a p t e r , models and s o l u t i o n s f o r t h e t i m e / c o s t  t r a d e o f f problem and f o r the r e s o u r c e a l l o c a t i o n problem a r e d e s c r i b e d , and an a d a p t a t i o n o f t h e b a s i c model f o r a d i f f e r e n t t y p e o f network i s explained.  A f t e r c o n s i d e r i n g t h e use o f analogue computers i n s o l v i n g  t h e problems,  o t h e r analogue d e v i c e s a r e i n t r o d u c e d , which a r e u s e f u l  e i t h e r i n a c t u a l s o l u t i o n o r i n c l a r i f i c a t i o n o f t h e s e problems.  M a t h e m a t i c a l Models K e l l e y (hS) t i o n , w h i c h was  developed a p a r a m e t r i c l i n e a r programming f o r m u l a -  t h e f i r s t f o r m u l a t i o n o f the f u n c t i o n a l r e l a t i o n s h i p  tween p r o j e c t c o s t and d u r a t i o n .  be-  T h i s model i s d e s c r i b e d i n the f o l l o w -  i n g e x t r a c t from an a r t i c l e by K e l l e y  (UJ).  L e t E be a f i n i t e p a r t i a l l y o r d e r e d s e t o f (n+1) elements c a l l e d E v e n t s . There a r e two d i s t i n g u i s h e d events i n E, O r i g i n and Terminus, r e s p e c t i v e l y , w i t h t h e p r o p e r t y t h a t o r i g i n p r e cedes and terminus f o l l o w s e v e r y e v e n t i n E . E a c h event i s denoted b y a nonnegative i n t e g e r , i t s l a b e l . S i n c e E i s p a r t i a l l y o r d e r e d , we may assume t h a t t h e events a r e l a b e l e d such t h a t i f event i precedes event j then i < j. I n p a r t i c u l a r , o r i g i n i s g i v e n the l a b e l 0 and t e r minus i s g i v e n t h e l a b e l n. A l s o a s s o c i a t e d w i t h event i i s a nonnegative number, t§i, which r e p r e s e n t s t h e time a t which the e v e n t o c c u r s . Thus i f  event i precedes event j then tj_^ t.*. l e t t 0 - 0.  We w i l l always  An a c t i v i t y i s an element, ( i , j ) , of E X E , such that i<j.  Associated with each a c t i v i t y i s a nonnegative num-  ber, y y , i t s Duration.  I t i s assumed that a c t i v i t y  (i,j)  must be performed sometime between the occurrences of event i  and event j .  Thus we must have  7 ± 3  * t ± - t j * 0.  A P r o j e c t , P, i s a set of events and a c t i v i t i e s with the property that i f event k i s i n P then k i s either o r i gin or terminus, or else there e x i s t events i and j i n P such that a c t i v i t i e s ( i , k ) and ( k , j ) are both i n P. An assignment of durations, y ^ j , to a c t i v i t i e s and occurrence times, t± to events i n P i s c a l l e d a Schedule. t  A schedule w i l l be denoted by ( Y , T ) , where Y and T are vectors whose coordinates are the y ^ and t ^ , r e s p e c t i v e l y , which define the schedule.  I f there are m a c t i v i t i e s i n  P, (Y,T) may be interpreted as a vector i n an (m+n+1)dimensional Euclidean space. Sometimes the duration of an a c t i v i t y i s a matter of management decision subject to c e r t a i n r e s t r i c t i o n s .  The  simplest r e s t r i c t i o n s , and the only ones with which we w i l l d e a l , are that y ^ be bounded above and below f o r each a c t i v i t y i n P.  That i s , there are number d y and D ^  such that  for a l l ( i , j )  i n P.  of a c t i v i t y ( i , j ) j Crash Duration.  We w i l l c a l l D ^ the Normal Duration d j j w i l l be c a l l e d the Expedited or  61 A schedule satisfying (1) and ( 2 ) with t » 0 i s called a Feasible Schedule* 0  The duration actually selected for each activity when forming a feasible schedule i s made to depend upon i t s U t i l i t y . For the moment we w i l l assume that the u t i l i t y of an activity i s a linear function of i t s duration on the closed interval defined by ( 2 ) and has the form: a  ij^ij  +  **iy  w h e r e  OSa^<oo and - oo<bij<oo.  The U t i l i t y of a schedule i s defined as the sum of the u t i l i t i e s of the individual activities i n P, v i z . : (3)  The Duration of a Schedule i s L »• tn. I t i s clear that among a l l feasible schedules having a given duration, L, there i s at least one which has maximum u t i l i t y , i.e., maximizes (3). Such a feasible schedule w i l l be called Optimal. We denote this value of (3) for this schedule by U(L). Whenever the measure of u t i l i t y i s cost, loss, etc., which require minimization, we simply take the negative of the function and then maximize* Considered as a function of L, U(L) w i l l be called the Project U t i l i t y Function. Our main objective i s to find an algorithm for generating U(L) and optimum feasible schedules that define i t * Although the u t i l i t y of an activity i s assumed to be linear i n equation (3)y later i n the article, Kelley shows that i t i s only necessary that i t be "piece-wise linear, nondecreasing, and concave between i t s crash and normal durations." The same assumptions are made i n a similar  62  model d e v e l o p e d by F u l k e r s o n Optimization  (3U).  o f r e s o u r c e a l l o c a t i o n i s c o n s i d e r e d i n a model (55),  p r e s e n t e d by P e t r o v i c  i n which the scheduling  regarded as a d i s c r e t e multistage d e c i s i o n process. the  d e s c r i p t i o n by P e t r o v i c  of his project  of a project i s The f o l l o w i n g i s  model.  C o n s i d e r the o v e r - a l l p r o j e c t i n terms o f i t s component activities. L e t p r o j e c t A c o n s i s t of M separate a c t i v i t i e s (-'i, j ) 6 A . The s t a t e s o f a c t i v i t i e s a r e f u n c t i o n s o f t i m e . Denote by X j j ( t ) t h e amount o f work a t time t n e c e s s a r y t o complete a c t i v i t y ( l , j ) . Generally, a c t i v i t i e s r e q u i r e d i f f e r e n t r e s o u r c e s and are vector-time functions. It  i s accepted that the resource a l l o c a t i o n s u ^ j ( t ) to  each a c t i v i t y a r e v a r i a b l e s under t h e c o n t r o l o f management.  Under t h e assumptions o f c o n t i n u i t y and homogeneity  o f r e s o u r c e s t h e management d e c i s i o n s U j i ( t ) , each a c t i v i t y ( i , j ) e A , a r e a l s o v e c t o r - t i m e whose components sources.  concerning  functions,  correspond t o the d i f f e r e n t types o f r e -  Relations  between r e s o u r c e a l l o c a t i o n s u ^ j and  amounts o f work x-jj f o r each a c t i v i t y e x i s t . u^j t o corresponding a c t i v i t y ( i , j ) , x - j j ( t ) become n o n i n c r e a s i n g f u n c t i o n s  By  applying  t h e components  of  o f time.  To be more r e a l i s t i c , l e t us suppose t h a t management d e c i s i o n s have t o be made a t a f i n i t e s e t W o f t i m e . we t u r n t o t h e d i s c r e t e v e r s i o n o f t h e p r o j e c t model. s t a t e o f e a c h a c t i v i t y can be d e s c r i b e d  Then The  by a f a m i l y o f  transformations, x  1 ; j  (n+l)=T j(x (n),u j(n)), i ;  i 5  i ;  | n - 0 , . . . , N - l , ( i , j ) € A]  where t h e i n i t i a l s t a t e o f t h e p r o j e c t i s a known s e t x  ij(cO » °ij> x  * o r each a c t i v i t y ( i , j ) e A .  minal state i s also prescribed. a l l x-j^, ( i , j ) € A .  The t e r -  I t has t o be z e r o f o r  ..  ( 1 )  A s s o c i a t e d w i t h each a c t i v i t y ( i , j ) e A cost function  J  The  ij*4n=0  g  i j  (  x  i j  (  n  )  '  "li™)*  g  ij °> s  f  i s a scalar  o  r  u  i5  (  n  )  =  0  '  •••  t o t a l c r i t e r i o n index of the p r o j e c t e v a l u a t i o n i s then  The f i r s t term e x p r e s s e s t h e c o s t t h a t i s due t o r e s o u r c e s a l l o c a t e d t o a l l a c t i v i t i e s . The second term d i s p l a y s a p a r t o f c o s t t h a t i s due t o p r o j e c t l e n g t h . I n t h e second term ^ ( n ) i s a n o n d e c r e a s i n g f u n c t i o n o f n, and L i s a m u l t i p l i e r expressing the weighting attached t o timeduration cost. Assuming t h a t t h e p r o j e c t g r a p h i s c o n s t r u c t e d and t h a t the node numbering i n t h e p r o j e c t network i s performed, i . e . , i f ( i , j ) ^ A , ( j , k ) € A , then i < j < k , t h e p r e d e c e s s o r - s u c c e s s o r r e l a t i o n s i n t h e network a r e s a t i s f i e d i f , f o r any time yields, Uj sO  forXy/O,  u^sO  f o r x.,,/0, f o r every i , j , k such t h a t (i,j)€A, (j,k)*A. .  k  J  J  K  The l i m i t e d a v a i l a b i l i t y o f r e s o u r c e s p a r t o f t h e model by s t a t i n g ,  S(i,j)£A ^ijWGV*  i s made an e x p l i c i t  (n»0, ...,N-1)  where U i s a n a d m i s s i b l e s e t . The f a c t t h a t r e s o u r c e l i m i t s may v a r y i n time, as men, machines, money, e t c . , a r e added or removed d u r i n g t h e p r o j e c t e x e c u t i o n c a n be i n t r o d u c e d s i m p l y by t a k i n g i n t o account t h a t t h e a d m i s s i b l e s e t U i s a g i v e n f u n c t i o n o f time o r t h e s t a t e o f the p r o j e c t i t s e l f .  6k The purpose o f management d e c i s i o n s can be s t a t e d as follows:  ...,N-1, s u b j e c t t o t r a n s f o r m a t i o n s (1) and  Uij(n),(i,j)€A,n=0,  choose t h e d e c i s i o n v a r i a b l e s  and c o n s t r a i n t s (k)  (5), w h i c h w i l l t r a n s f o r m t h e p r o j e c t i n i t i a l s t a t e  X i j ( O ) i n t o z e r o s t a t e so as t o minimize t h e t o t a l index The  criterion  (3).  t r a n s f o r m a t i o n s , T ^ j , a r e l a t e r assumed t o i n d i c a t e a l i n e a r  r e l a t i o n s h i p between a c t i v i t y d u r a t i o n and r e s o u r c e a l l o c a t i o n . r e s o u r c e l e v e l i n g problem i s a l s o c o n s i d e r e d , as was Chapter  The  mentioned i n  VI.  S o l u t i o n s t o t h e Time/Cost Problem A f t e r showing t h e e q u i v a l e n c e o f t h e Gass-Saaty p a r a m e t r i c programming a l g o r i t h m (103)  w i t h the p r i m a l - d u a l procedures,  c u s s e d by D a n t z i g , F o r d and F u l k e r s o n (23), n i q u e s o f b o t h i n h i s own  K e l l e y (U3)  dis-  uses t h e t e c h -  s o l u t i o n o f the t i m e / c o s t t r a d e o f f problem.  Fundamental t o t h i s a l g o r i t h m i s the s o l u t i o n o f a maximum network f l o w problem w i t h b o t h p o s i t i v e upper and lower bound c a p a c i t y r e s t r i c t i o n s . The network f l o w i n t e r p r e t a t i o n i s a c h i e v e d by c o n s i d e r i n g t h e d u a l o f the model f o r m u l a t e d above. l a b e l l i n g technique  (101)  A s l i g h t v a r i a t i o n of the  Ford-Fulkerson  i s implemented.  A s i m i l a r but d i f f e r e n t network f l o w s o l u t i o n i s d e s c r i b e d by F u l k e r s o n (3k)•  H i s a l g o r i t h m s e q u e n t i a l l y determines  a l l the b r e a k -  p o i n t s o f the convex, p i e c e - w i s e l i n e a r t i m e / c o s t t r a d e o f f c u r v e . v a r i a t i o n o f t h i s i s d e t a i l e d by B r i g g s (90).  An improvement t o  A  65  Fulkerson's method i s suggested by Roper (126), based on a combinat i o n of K e l l e y ' s ideas and some o r i g i n a l s i m p l i f i c a t i o n . The f i r s t model described above was modified by Meyer and Shaffer  (51) f o r an integer l i n e a r programming technique which  handles a wide v a r i e t y of a c t i v i t y time/cost functions.  Discrete  points and dependent curves, as w e l l as a l l continuous functions, can be handled by t h i s formulation.  However, the computational e f f o r t  involved i s even greater than that for network flow solutions, and the authors state that networks with more than f i f t y a c t i v i t i e s are too large f o r consideration by t h e i r method. An approach, which only handles the c r i t i c a l path analysis but which could be expanded to handle additional constraints, i s by Charnes and Cooper (17).  presented  The network i s converted i n t o a graph by  assigning lengths representing a c t i v i t y durations to the branches.  The  c r i t i c a l path may then be i d e n t i f i e d with a spanning tree that contains a maximal chain. Petrovic recognizes that his mathematical formulation, described above, " y i e l d s a t y p i c a l c o n t r o l problem," and consequently techniques of modern control theory may be applied to solve i t .  Other t h e o r e t i c a l  solutions by Wagner (132) and by Ghare (10ii) were mentioned i n Chapter VI.  Another Model Decision C r i t i c a l Path Method i s a method by which a l l the possible ways of planning a project may be portrayed on a single diagram.  66 Job s e t s a r e u s e d by Crowston and Thompson (22) i n t h e i r d e s c r i p t i o n o f the mathematical model f o r t h i s problem.  Every  element—represent-  i n g an a c t i v i t y — o f a j o b s e t i s a s s i g n e d a b i n a r y c o n s t a n t i n c l u s i o n o r e x c l u s i o n i n the schedule.  indicating  The c o n v e n t i o n a l network  would have u n i t j o b s e t s w i t h a l l c o n s t a n t s i n d i c a t i n g i n c l u s i o n . s e t o f constants  The  r e p r e s e n t m a n a g e r i a l d e c i s i o n s , and complex i n t e r -  dependencies o f t h e s e w h i c h i s otherwise  d e c i s i o n s may be i n c o r p o r a t e d i n t o the model,  similar to that o f Kelley.  Analogues The a c t u a l s o l u t i o n o f l i n e a r programming problems by analogue computers i s d i s c u s s e d b y F i r t h ( 9 8 ) .  The a c c u r a c y i s comparable w i t h  t h a t o f t h e d a t a , and, i n c l u d i n g v a r i o u s c h e c k s , t h e c a l c u l a t i o n o f the s o l u t i o n b y analogue computer i s f a s t e r t h a n i t i s by d i g i t a l computer.  The l i m i t i n g f a c t o r i s s i z e .  One a m p l i f i e r i s r e q u i r e d f o r  each v a r i a b l e , and two f o r each c o n s t r a i n t .  There a r e o n l y a few  machines i n e x i s t e n c e w i t h g r e a t e r t h a n f i v e hundred a m p l i f i e r s . P r a g e r (56) d e s c r i b e d a s t r u c t u r a l i n t e r p r e t a t i o n o f F u l k e r s o n ' s algorithm.  Each a c t i v i t y i s represented  by a s t r u c t u r a l member c o n s i s t -  i n g o f a r i g i d s l e e v e , whose l e n g t h r e p r e s e n t s t a i n s a compressible  c r a s h time and w h i c h con-  r o d o f n a t u r a l l e n g t h r e p r e s e n t i n g normal t i m e .  The  r o d w i l l o n l y compress under a f o r c e g r e a t e r t h a n t h e c o s t p e r u n i t decrease i n time f o r t h e a c t i v i t y , and i t cannot compress f u r t h e r t h a n the o u t s i d e end o f t h e s l e e v e .  Each s l e e v e i s a t t a c h e d t o a d i s c ,  r e s e n t i n g t h e i n i t i a l event o f t h e a c t i v i t y .  The whole model i s  rep-  c o n s t r a i n e d a t each end by an a d j u s t a b l e w a l l .  The d i s t a n c e between  these w a l l s i n d i c a t e s the d e s i r e d p r o j e c t d u r a t i o n . p o s i t i o n i s the s o l u t i o n .  The e q u i l i b r i u m  An a l g o r i t h m i s d e s c r i b e d which r e q u i r e s  l e s s e l a b o r a t e bookkeeping than F u l k e r s o n ' s , b u t no r e s u l t s f o r n e t works o f any  s i z e are g i v e n .  Another mechanical i s p r e s e n t e d by C l a r k (19).  analogue f o r t h e t i m e / c o s t t r a d e o f f curve Events  but s p r i n g s r e p l a c e a c t i v i t i e s .  The  are a g a i n r e p r e s e n t e d by  discs,  a c t i v i t y d u r a t i o n corresponds  to  t h e l e n g t h o f t h e s p r i n g , and t h i s i s c o n s t r a i n e d by the s p r i n g c h a r a c teristics.  The n e g a t i v e o f t h e p o t e n t i a l energy i n d i c a t e s c o s t .  The  e q u i l i b r i u m p o s i t i o n w h i c h r e p r e s e n t s the optimum schedule i s shown t o be  unique. B a t t e r s b y (9)  n o t e d t h a t models o f b a r c h a r t s have been  s u c c e s s f u l l y used t o manually perform c r i t i c a l path a n a l y s e s and r e source' l e v e l i n g . each o t h e r — o n  The model has u n i t cubes, and these are hooked t o  the h o r i z o n t a l plane t o represent the durations o f  v i t i e s , and on the v e r t i c a l p l a n e t o show r e s o u r c e r e q u i r e m e n t s . c o n n e c t i n g hooks i n d i c a t e i n t e r d e p e n d e n c i e s . onto w i r e s f o r e a s y  The cubes are  actiCross-  threaded  manipulation.  Summary. The  f i r s t mathematical d e s c r i p t i o n o f t h e p r o j e c t t i m e / c o s t  r e l a t i o n s h i p i n v o l v e d a p a r a m e t r i c l i n e a r programming f o r m u l a t i o n . The model assumed t h a t t h e a c t i v i t y t i m e / c o s t f u n c t i o n i s p i e c e w i s e l i n e a r , non-decreasing  and concave between the c r a s h and normal  68 extreme p o i n t s . represents  A model f o r the  optimization  of resource a l l o c a t i o n  the problem as a d i s c r e t e m u l t i s t a g e  decision process.  A  l i n e a r a c t i v i t y t i m e / r e s o u r c e f u n c t i o n i s assumed. The  s o l u t i o n o f the t i m e / c o s t model i s a c h i e v e d by  considering  i t s d u a l , which becomes a maximum network f l o w problem w i t h b o t h p o s i t i v e upper and  lower bound c a p a c i t y r e s t r i c t i o n s .  cedures are used t o s o l v e t h i s . may  be u s e d t o handle any  c r i t i c a l p a t h a n a l y s i s may  Primal-dual  I n t e g e r l i n e a r programming t e c h n i q u e s  type of a c t i v i t y time/cost f u n c t i o n .  The  be accomplished by f i n d i n g the maximal  i n the s p a n n i n g t r e e o f a graph.  The  work, showing a l l p o s s i b l e ways o f p l a n n i n g w h i c h i n d i c a t e whether or not  chain  s o l u t i o n t o the r e s o u r c e a l l o c a -  t i o n model i s o b t a i n e d by implementing modern c o n t r o l t h e o r y .  in  pro-  a p r o j e c t , uses job  A netsets,  an a c t i v i t y i s t o be i n c l u d e d i n a schedule,  i t s mathematical d e s c r i p t i o n . Analogue computers t o s o l v e l i n e a r programming problems f o r  networks a r e more e f f i c i e n t t h a n d i g i t a l computers but are too to handle reasonably-sized be  projects.  The  small  t i m e / c o s t t r a d e o f f curve  may  found by e s t a b l i s h i n g the e q u i l i b r i u m p o s i t i o n o f c e r t a i n m e c h a n i c a l  systems under a c o n s t r a i n t r e p r e s e n t i n g s i m i l a r t o an abacus, may a n a l y s e s and  project duration.  A model,  be used f o r manual s o l u t i o n of c r i t i c a l p a t h  o f r e s o u r c e a l l o c a t i o n problems.  CHAPTER IX  VARIATIONS ON A THEME  Introduction A l t h o u g h t h e t h e o r y o f p r o j e c t c o n t r o l t e c h n i q u e s has been d e s c r i b e d , l i t t l e mention has been made o f t h e systems t h a t a c t u a l l y implement i t .  I n t h i s c h a p t e r , t h e s e systems a r e i n t r o d u c e d w i t h  some o f t h e i r c h a r a c t e r i s t i c s . t i o n , o n l y acronyms a r e used. meaning.  In order t o f a c i l i t a t e the presentaT a b l e I l i s t s t h e s e acronyms and t h e i r  A d e s c r i p t i o n o f a g e n e r a l i z e d network approach,  t h e o r y and i m p l e m e n t a t i o n ,  and i t s  i s f o l l o w e d b y i n t r o d u c t i o n o f o t h e r areas  i n w h i c h t h e PERT/CPM concepts have been used.  PERT and CPM The d i f f e r e n c e s between PERT and CPM have been d i s c u s s e d i n e a r l i e r chapters.  E s s e n t i a l l y , CPM i s a t e c h n i q u e , developed  by i n -  d u s t r y f o r such p r o j e c t - o r i e n t e d areas a s c o n s t r u c t i o n , w h i c h u s e s a d e t e r m i n i s t i c a p p r o a c h t o time e s t i m a t i o n , w h i l e PERT i s a system, developed by t h e U n i t e d S t a t e s Navy f o r r e s e a r c h and development p r o j e c t s , which c o n s i d e r s p r o b a b l e time d i s t r i b u t i o n s .  A combined approach  was made w i t h NASA/PERT ( 8 ) , which i s the PERT system m o d i f i e d t o u s e a s i n g l e a c t i v i t y time e s t i m a t e . without  The U n i t e d S t a t e s A i r F o r c e  attempted  success t o promote t h e i r acronym, PEP (70) ( 1 0 9 ) , as an a l t e r -  n a t i v e t o PERT.  The o r i g i n a l PERT t e c h n i q u e was r e p l a c e d by PERT I I ,  70 TABLE I  MANAGEMENT PROJECT CONTROL SYSTEMS  Acronym AGREE AMPERE COMET CPA CPM CPPS CPS CRAM DeLTA REPP ICON IMPACT LESS LOB LOPS MAPS MCI MOST MPACS NASA/PERT PACE PACT PAR PEP PERGO PERT PERTCO PES PLANNET PREPARES PRISM PROMPT RAMPS RITE RPSM  Meaning A d v i s o r y Group on t h e R e l i a b i l i t y o f E l e c t r o n i c Equipment. APL Management P l a n n i n g and E n g i n e e r i n g Resources Evaluation. Computer Operated Management E v a l u a t i o n Technique. C o s t P l a n n i n g and A p p r a i s a l . C r i t i c a l P a t h Method. C r i t i c a l P a t h P l a n n i n g and S c h e d u l i n g . C r i t i c a l Path Scheduling. C o n t r a c t u a l Requirements R e c o r d i n g , A n a l y s i s and Management System. D e t a i l e d Labour and Time A n a l y s i s . Hoffman E v a l u a t i o n Program and P r o c e d u r e . Integrated Control. Implementation, P l a n n i n g and C o n t r o l T e c h n i q u e . L e a s t C o s t E s t i m a t i n g and S c h e d u l i n g . L i n e o f B a l a n c e Technique. L a t e r a l O p e r a t i o n a l P l a n n i n g and S c h e d u l i n g . M u l t i v a r i a t e A n a l y s i s and P r e d i c t i o n o f Schedule. Minimum C o s t E x p e d i t i n g . Management O p e r a t i o n a l System Technique. Management P l a n n i n g and C o n t r o l System. N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n PERT. Performance and C o s t A n a l y s i s . Production A n a l y s i s C o n t r o l Technique. P r o j e c t Audit Report. Program E v a l u a t i o n P r o c e d u r e . P r o j e c t E v a l u a t i o n and Review w i t h Graphic Output. Program E v a l u a t i o n and Review T e c h n i q u e . PERT/Cost. Program E v a l u a t i o n Systems. P l a n n i n g Network. Program E v a l u a t i o n Procedure and R e s o u r c e . Program R e l i a b i l i t y I n f o r m a t i o n "System f o r Management. Program Management P l a n n i n g T e c h n i q u e s . Resource A l l o c a t i o n and M u l t i - P r o j e c t S c h e d u l i n g . R a p i d I n f o r m a t i o n Technique f o r E v a l u a t i o n . Resources P l a n n i n g and S c h e d u l i n g Method.  71 TABLE I ( C o n t i n u e d )  Acronym  SCANS SCOPE SKED SPECTROL SPERT SVS TTPAC TOES TOPS TRACE WAP ZPA  Meaning  S c h e d u l i n g and C o n t r o l By Automated Network Systems. S y s t e m a t i c C o n t r o l o f O p e r a t i o n s and Program Evaluation. Computer Program f o r S c h e d u l i n g Time and D i s t r i b u t i n g Cost. S c h e d u l i n g , P l a n n i n g , E v a l u a t i o n , C o s t and C o n t r o l . Schedule Performance E v a l u a t i o n and Review Technique. Schedule V i s i b i l i t y System. Texas Instrument Programming and C o n t r o l . T r a d e - O f f E v a l u a t i o n System. The O p e r a t i o n a l PERT System. Task R e p o r t i n g and C u r r e n t E v a l u a t i o n . Work Assignment P r o c e d u r e . Zeus Program A n a l y s i s .  72 PERT I I I and PERT IV, a l t h o u g h no major changes were made. PERT U  (8)  F o r example,  r e q u i r e d a l l a c t i v i t i e s t o have d i s c r e t e i n i t i a l and  minal e v e n t s — a s  ter-  d i d the Minuteman PERT ( 6 7 ) — a n d u s e d b o t h event  a c t i v i t y o r i e n t e d networks.  and  Some companies i n c o r p o r a t e d o t h e r m o d i f i c a (73),  t i o n s , such as the s i n g l e g e n e r a l i z e d network o f Aerospace's TOPS the t i m e - s c a l e d network i n MOST (8)  which was  developed  by Avro,  and  the f i r m l y committed event times used by G e n e r a l E l e c t r i c i n t h e i r SPERT (79)  system.  Other s i m i l a r v a r i a t i o n s a r e TIPAC (8)  and PROMPT  (66).  Cost  Considerations The  f i r s t e x t e n s i o n t o the b a s i c s y s t e m s — a c t u a l l y suggested  the o r i g i n a l CPM  paper—involved  the t i m e / c o s t t r a d e o f f c u r v e .  c o s t and  concerned the d e r i v a t i o n o f  PERT and CPM—both CPS  (66)  and CPPS  are e q u i v a l e n t acronyms f o r the l a t t e r — m e r g e , i n s o f a r as the t i m e / c o s t f u n c t i o n i s concerned w i t h s i n g l e time e s t i m a t e s . Navy i n t r o d u c e d PERT/Cost (2), and c a l l e d i t PERTCO (11).  in  (8)  activity The  U.S.  and the A i r Force promptly adopted i t  Less d e t a i l e d c o s t i n f o r m a t i o n i s r e p o r t e d  by the NASA PERT and Companion C o s t System ( h ) .  PAR  (79)  i s Burroughs'  m o d i f i c a t i o n o f PERT/Cost. I n 1961,  I n t e r n a t i o n a l B u s i n e s s Machines i n t r o d u c e d LESS  (102), which a l s o e s t a b l i s h e s the most economical  method o f  (97)  completing  a p r o j e c t f o r a g i v e n time, which i s o f t e n t h e minimum d u r a t i o n , and which i n i t i a l l y  considers three time estimates per a c t i v i t y .  The  73 minimum c o s t s c h e d u l e i s a c h i e v e d  i n t h r e e phases*  Node Numbering d e v e l o p s t h e network, t h e S c h e d u l i n g  Arrow Diagram phase performs  the c r i t i c a l p a t h a n a l y s i s , and Schedule Compression a t Minimum C o s t f i n d s the r e q u i r e d schedule.  MPACS (79) i s used t o communicate f i n a n -  c i a l and manpower d a t a , i n c l u d i n g budgeted c o s t / t i m e time c u r v e s ,  and a c t u a l c o s t /  t o t h e program manager w h i l e t h e p r o j e c t i s i n p r o g r e s s .  S i m i l a r systems a r e MCX (8), PACE (11),  SCOPE (8), SKED (79) and  TOES (8). The proving  U.S. A i r F o r c e developed CPA (79) s p e c i f i c a l l y f o r i m -  c o n t r a c t management by i n t e g r a t i n g d a t a on c o s t and time and  on t e c h n o l o g y , and CRAM (66) t o handle t h e p u r c h a s i n g o p e r a t i o n  f o r con-  tracts.  cost  S y l v a n i a E l e c t r o n i c s u s e ICON (79) i n t h e p r e p a r a t i o n ,  e v a l u a t i o n and r e - e v a l u a t i o n o f b i d s and c o n t r a c t s . HEPP (8) u s e s an e l a b o r a t e  output which i n c o r p o r a t e s  onto a  s i n g l e c h a r t a work breakdown, budget and a c t u a l c o s t c u r v e s ,  and a  time-scaled  Another  PERT network.  T h i s chart i s f r e q u e n t l y updated.  system w h i c h emphasizes a d e t a i l e d output i s PERGO (93).  Milestones  are a f e a t u r e o f SPECTROL (72), w h i c h u t i l i z e s a t e c h n i q u e t h a t p e r m i t s m o d i f i c a t i o n o f t h e network w h i l e t h e p r o j e c t i s i n p r o g r e s s and t h a t w i l l then i n d i c a t e the e f f e c t o f t h e m o d i f i c a t i o n .  Resource Requirements Subsequent t o c o s t c o n s i d e r a t i o n i n t h e development o f p r o j e c t management systems was r e s o u r c e  requirement c o n t r o l .  RAMPS (3), i n i t i a l l y  7U known as RPSM, was d e s c r i b e d i n d e t a i l i n Chapter V I I .  AMPERE (8)  and TRACE (8) a r e b o t h PERT systems m o d i f i e d t o c o n s i d e r r e s o u r c e s . TRACE, developed b y Chance Vought, i d e n t i f i e s t a s k s w i t h i n a c t i v i t i e s t o improve e v a l u a t i o n o f a c t i v i t y p r o g r e s s .  WAP  c o n j u n c t i o n w i t h a PERT system t o e s t i m a t e r e s o u r c e S i m i l a r t e c h n i q u e s a r e employed i n DELTA COMET (66),  (11)  (8) i s u s e d i n requirements.  and PREPARES  (66)).  developed b y t h e U.S. Army, was t h e f i r s t  t o be a p p l i e d t o l o g i s t i c s .  system  I t handles t h e a c q u i s i t i o n , r a t h e r t h a n  the development, o f equipment.  A s t a n d a r d network i s used f o r any  type o f equipment procurement and a c t i v i t i e s n o t i n t h e p r o j e c t become dummy a c t i v i t i e s . Another comprehensive system i s SCANS (71)* w h i c h was i n t r o duced b y Systems Development C o r p o r a t i o n i n 1961.  U s i n g one time  e s t i m a t e and v a r i o u s t r a d e o f f f u n c t i o n s , SCANS c o n s i d e r s time, c o s t , m a n p o w e r — c a l l e d human r e s o u r c e s — a n d called f a c i l i t i e s .  other p h y s i c a l r e s o u r c e s —  I t i s an outgrowth o f SPECTROL.  I n a comparison  o f t h e system w i t h PERT, F r y (33) notes t h a t SCANS uses t i m e - s c a l e d networks and m i l e s t o n e s . a budget-planning  The method was designed t o be compatible  with  system.  Other V a r i a t i o n s I n a d d i t i o n t o PERT, i n v e l o p e d PRISM bility.  I960  t h e U.S. Navy Department a l s o de-  (79)—a t e c h n i q u e f o r m o n i t o r i n g and measuring r e l i a -  One o f t h e p r o d u c t s o f c o n s i d e r i n g q u a l i t y c o n t r o l , was a  75 R e l i a b i l i t y Maturity Index, described by Malcolm (117).  This concept  of r e l i a b i l i t y of physical hardware may be integrated with PERT by introduction of a ' r e l i a b i l i t y e v e n t 1 , which s i g n i f i e s  the s t a r t or  completion of some form of design t e s t , required i n the development to enhance the r e l i a b i l i t y of the finished product.  AGREE (8) con-  siders the r e l i a b i l i t y of electronic equipment. PACT (60) combines features of PERT and LOB to forecast where production slippage i s l i k e l y to occur.  Introduced i n 1962 by the  U.S. Navy, i t was l a t e r discarded i n favour of the non-computer technique LOB. Other variations include IMPACT (79), which considers time, cost and resource requirements i n the preparation and i n s t a l l a t i o n of computer programs, and RITE (8), which produces as output a detailed description of a weapon or space system configuration, and SVS (11), which considers v i s i b i l i t y for aeroplanes.  Pan American Airways uses  PLANNET (8), which achieves v i s u a l scheduling by displaying a timeoriented chart which i s a s e r i a l and p a r a l l e l combination of  bar-charts.  A Generalized Network Approach Since the i n i t i a l introduction of the project network technique, various methods of generalization have been suggested.  Eisner (27)  proposed a generalized network i n which an a d d i t i o n a l degree of freedom i s given to the basic network by replacing events with 'Decision Boxes'. A decision box i s an event from which the branching a c t i v i t i e s may or  76 may  not be  s t a r t e d a c c o r d i n g t o some p r o b a b i l i t y — h e n c e  p r o b a b i l i s t i c networks. m i n a l e v e n t , and  There i s u s u a l l y more t h a n one  game t r e e s .  d e c i s i o n boxes.  There a r e two A  made as a consequence o f the p l a n n i n g , and A  only  kinds o f decision that 'Personal'  may  decision i s usually  so e a c h b r a n c h i s a s s i g n e d  The  method i s p r i m a r i l y used f o r r e s e a r c h  development p r o j e c t s , i n w h i c h a l t e r n a t e p r o c e s s e s may i n g upon the p r e v i o u s r e s u l t s .  and  be u s e d , depend-  C r i t i c a l p a t h t e c h n i q u e s are  applied  to  network. Eisner introduces  t i o n " i n a system. bility  (78)  Hence  ' R e s u l t a n t ' d e c i s i o n a s s i g n s an e s t i m a t e d p r o -  b a b i l i t y t o each e x i t .  As  and  s p e c i a l case o f a network w i t h  be made a t a d i s j u n c t i v e d e c i s i o n box.  equal p r o b a b i l i t y .  A  o f t h e p o s s i b l e e x i t s are t o be made.  t h e PERT/CPM network becomes the  the  be made.  i s ' C o n j u n c t i v e * i f a l l branches a r e t o be t a k e n ,  ' D i s j u n c t i v e ' i f o n l y one  conjunctive  project ter-  a p r o b a b i l i t y e s t i m a t e o f each outcome may  A. c l o s e s i m i l a r i t y e x i s t s between t h e s e networks and d e c i s i o n box  t h e name o f  E n t r o p y as a measure o f the  I t i s f o u n d by  o f each f i n a l outcome and  summing the p r o d u c t s o f the  the  a measure o f r e l a t i v e u n c e r t a i n t y define  R e l a t i v e E n t r o p y as the  t o t h e maximum e n t r o p y o f the f i n a l The  algebra  by Elmaghraby (29)» feedback l o o p s and  "Average Informa-  logarithm  proba-  of t h i s probability.  i n a system, Goode and  Machol  r a t i o o f the E n t r o p y o f the  system  outcomes.  f o r the a n a l y s i s o f t h e s e networks was  developed  i n c l u d i n g more c o m p l i c a t e d r e l a t i o n s h i p s s u c h as the  l o g i c a l o p e r a t o r s AND,  inclusive-OR  and  exclusive-OR.  Elmaghraby (28)  extended  the concept o f g e n e r a l i z e d  networks t o i n c l u d e c o n d i t i o n a l p a r a l l e l p r o g r e s s . ted  He a l s o e l a b o r a -  on t h e a l g e b r a w i t h an a n a l y s i s o f networks composed e n t i r e l y o f  exclusive-OR  nodes.  A procedure  f o r implementing  l u a t i o n and Review Technique,  this concept—the  G r a p h i c a l Eva-  GERT—which uses t h e a l g e b r a  by Elmaghraby, i s p r e s e n t e d by P r i t s k e r and Happ (57). a n a l y s i s f o r AND  o p e r a t o r s , such as 'minimum' and  The c o n s i d e r a t i o n o f o t h e r l o g i c a l ' i n v e r t ' i s suggested.  a p p l i c a t i o n s o f GERT a r e d e s c r i b e d by P r i t s k e r (58),  c o l l a b o r a t i o n w i t h Whitehouse.  Some p r a c t i t h i s time i n  The method has been used t o a n a l y z e  a s p a c e - v e h i c l e countdown and t h e manufacture o f material.  The method o f  and i n c l u s i v e - O R nodes, u s i n g random v a r i a b l e t i m e s ,  i s not computationally f e a s i b l e .  cal  developed  semi-conductor  I t has been shown t h a t the system can a l s o be used t o study  i n v e n t o r y , queuing and r e l i a b i l i t y  problems.  D i f f e r e n t Approaches D e c i s i o n CPM,  d e s c r i b e d by Crowston and Thompson (22),  is a  method which f o r m a l l y c o n s i d e r s t h e i n t e r a c t i o n between the p l a n n i n g and s c h e d u l i n g phases o f a p r o j e c t .  A l l p o s s i b l e p l a n s are i n c l u d e d  i n t h e p r o j e c t graph.  C o s t m i n i m i z a t i o n i s a c h i e v e d by choosing  l e a s t expensive p l a n .  As some p l a n s are i n t e r c o n n e c t e d , the  may b e u s e d w h i l e the p r o j e c t i s i n p r o g r e s s .  the  procedure  As a c t u a l a c t i v i t y  a r e r e c o r d e d , a d i f f e r e n t p l a n , w h i c h can s t i l l be executed, may  times become  optimum.  The model f o r t h i s procedure was  d i s c u s s e d i n the  previous  chapter. How  PERT may  be used as an a i d to L o g i c Design i s e x p l a i n e d  K i r k p a t r i c k and C l a r k ( 1 1 2 ) .  An a c t i v i t y - o r i e n t e d network  l o g i c blocks as a c t i v i t i e s .  Each l o g i c b l o c k has  two  d i s t r i b u t i o n s — i n d i c a t i n g the l i k e l y d e l a y f o r the ON o f the b l o c k .  probable and OFF  time states  T h i s i s a simple b u t u s e f u l means o f  s t a t i s t i c a l l y designing l o g i c  systems.  PERT i s combined w i t h the p r o d u c t i o n technique I n h i s d e s c r i p t i o n o f t h i s , Digman (9h) t o i n c l u d e r e p e t i t i v e a c t i v i t i e s and the t i m e - s c a l e d network and be  represents  In c o n t r a s t w i t h PERT, the e a r l i e s t time i n t o an event  i s u s u a l l y i t s s t a r t time.  may  by  output  LOB  i n PERT/LOB.  showed t h a t PERT i s broadened  d e l i v e r y i n f o r m a t i o n , and t o  r e p o r t s o f LOB.  Cost  use  considerations  s i m p l y i n t e g r a t e d i n t o the system, u s i n g PERT/Cost. A combination o f p r o j e c t network t e c h n i q u e s  t h e o r y may  and  be a p p l i e d t o the f i e l d o f communications.  c r i b e d by N o e t t l and Brumbaugh ( 1 2 1 ) .  The  information T h i s i s des-  concept o f u n c e r t a i n t y o r  the q u a n t i t y o f i n f o r m a t i o n i s compared w i t h s l a c k , and the p a t h a n a l y s i s i s performed, u s i n g u n c e r t a i n t y v a l u e s ,  critical  f o r communica-  t i o n s networks.  Summary There have been many systems compiled PERT and CPM.  that are s i m i l a r t o  T h e i r acronyms and meanings are g i v e n i n T a b l e  I.  79 Most o f t h e s l i g h t m o d i f i c a t i o n s t o t h e o r i g i n a l system a p p l y t o t h e network s p e c i f i c a t i o n . d u r a t i o n estimate  When c o s t i s c o n s i d e r e d ,  i s required.  the same c o s t m i n i m i z a t i o n it.  o n l y one a c t i v i t y  A l t h o u g h t h e a l g o r i t h m s may d i f f e r ,  i s a c h i e v e d by a l l systems w h i c h attempt  C o n t r a c t management has produced some p r o c e d u r e s o f i t s own.  E l a b o r a t e g r a p h i c a l output i s o c c a s i o n a l l y o f f e r e d . Armed F o r c e s control.  The U n i t e d  States  c o n t r i b u t e d c o n s i d e r a b l y t o t h e development o f p r o j e c t  Techniques c o n s i d e r i n g r e s o u r c e requirements were i n t r o d u c e d  l a t e i n t h i s development. by some systems.  R e l i a b i l i t y o f p h y s i c a l hardware i s examined  O t h e r s s p e c i a l i z e i n t h e p r e p a r a t i o n and i n s t a l l a t i o n  o f computer programs, t h e d e s c r i p t i o n o f weapon c o n f i g u r a t i o n s , v i s i b i l i t y , and manual s c h e d u l i n g by b a r - c h a r t s . A g e n e r a l i z e d network approach i s p r e s e n t e d w h i c h i n v o l v e s r e p l a c i n g e v e n t s by d e c i s i o n boxes.  E x i t s from a d e c i s i o n box may be  t a k e n w i t h an a s s i g n e d p r o b a b i l i t y , which c o u l d i n d i c a t e c e r t a i n t y . P r o b a b i l i t y estimates  o f p o s s i b l e outcomes a r e macle.  The c h o i c e s i n p e r -  s o n a l d e c i s i o n s , made a c c o r d i n g t o t h e p r o j e c t a n a l y s i s , a r e a s s i g n e d equal p r o b a b i l i t y .  C r i t i c a l path techniques  d e f i n e d a s a measure o f u n c e r t a i n t y .  are used.  Entropy i s  The a l g e b r a f o r t h e s e  networks  i n c l u d e s feedback l o o p s , l o g i c a l r e l a t i o n s h i p s and c o n d i t i o n a l p a r a l l e l progress.  A system has been s u c c e s s f u l l y designed  t o implement  these  concepts. A n o t h e r network, i n c o r p o r a t i n g a l l p o s s i b l e p l a n s o f a p r o j e c t ,  80  i s used t o f o r m a l l y c o n s i d e r t h e i n t e r a c t i o n between the and s c h e d u l i n g phases.  planning  PERT, w i t h an a c t i v i t y - o r i e n t e d network,  be u s e d as an a i d t o s t a t i s t i c a l l y been combined b o t h w i t h LOB,  d e s i g n l o g i c systems.  I t has  may also  t o be a p p l i e d t o p r o d u c t i o n problems,  with information theory, t o analyze  communications networks.  and  CHAPTER X  CONCLUSIONS  Introduction C o n c l u s i o n s a r e drawn from the r e s u l t s o f t h e a p p l i c a t i o n o f network p l a n n i n g t e c h n i q u e s nesses a r e noted possibility  to project control.  Some o f i t s weak-  and t h e t r e n d o f p r e s e n t systems i s d i s c u s s e d .  t h a t some o f t h e remaining  The  problems may be s o l v e d i s a l s o  examined.  PERT/CPM Network p l a n n i n g i s now a w e l l - e s t a b l i s h e d management t o o l f o r project control.  Many a p p l i c a t i o n s have been s u c c e s s f u l l y implemented,  and t h e r e a r e undoubtedly s t i l l areas f o r w h i c h t h e t e c h n i q u e to be c o n s i d e r e d and where i t can be o f g r e a t v a l u e . the panacea o f a l l i l l s  t h a t many t h i n k i t i s .  has y e t  I t i s n o t however  As C l a y t o n and Glenn  (91) remark, i t s c o n s i d e r a b l e c o s t i s r a r e l y d i s c u s s e d , and t h e r e i s a tendency t o f i t t h e p r o j e c t t o t h e system, r a t h e r than t h e r e v e r s e . The is  r e s u l t o f t h i s i s t o c u r t a i l c r e a t i v e thought.  The use o f computers  f r e q u e n t l y i n d i s c r i m i n a t e , i n t h a t s m a l l p r o j e c t s may o f t e n be more  e f f i c i e n t l y handled The  manually.  g r e a t e s t v a l u e o f PERT/CPM procedures  i s t h e demand i t makes  on management t o o r g a n i z e t h e i r thoughts about t h e e x e c u t i o n o f a p r o j e c t before i t begins. (12li).  Although  The management v i e w p o i n t  i s p r e s e n t e d b y Pocock  b o t h " p a y o f f s and problems" a r e d i s c u s s e d , most problems  82 t h a t a r e mentioned are caused b y the i n a b i l i t y o f management t o d e a l w i t h the g r e a t e r d i s c i p l i n e r e q u i r e d by network p l a n n i n g . c r i t i c i s m o f PERT/CPM procedures  i s r a r e l y made.  Unfortunately,  This i s probably f o r  the r e a s o n t h a t f a i l u r e s are a t t r i b u t e d t o poor a p p l i c a t i o n s , r a t h e r t h a n t o i n h e r e n t weaknesses i n the  The  technique.  Program The  programs t h a t implement p r o j e c t network t e c h n i q u e s v a r y i n  complexity,  from t h e s i m p l e a l g o r i t h m o f Eisenman and S h a p i r o  the comprehensive system RAMPS (3).  The  directions.  to  numerous f e a t u r e s t h a t d i f f e r -  e n t i a t e them a r e d e s c r i b e d by P h i l l i p s (123). i n two  (96)  The  t r e n d appears t o be  The m a j o r i t y o f systems a r e o f f e r i n g more complex  a n a l y s e s b u t , f o r a l l t h e i r v a r i e d acronyms, t h e programs a r e becoming essentially similar.  On t h e o t h e r hand, a few s p e c i a l i z e d systems a r e  being compiled which are only a p p l i c a b l e i n a p a r t i c u l a r f i e l d  of  endeavour.  The  Problems The b a s i c problem i s t h a t o f time e s t i m a t i o n .  This i s not  t h e o r e t i c a l but i s a q u e s t i o n o f t h e a c c u r a c y o f data and t h e p r a c t i c a l approach o f t h e o r y t o i t .  I t s importance i s emphasized i n t h a t e v e r y  a n a l y s i s i s dependent upon time e s t i m a t e s .  Automation and a t r e n d  towards u n i o n piecework w i l l a l l e v i a t e i n a c c u r a c y , but t h i s w i l l always be a s e r i o u s weakness i n network p l a n n i n g  techniques.  83  The and  time/cost  t r a d e o f f problem has been s o l v e d  t h e s o l u t i o n i s o f a c c e p t a b l e magnitude.  s o l u t i o n s t o resource  theoretically,  This contrasts with the  a l l o c a t i o n problems which w i l l o n l y become p r a c -  t i c a l w i t h f a s t e r computers o r when a more e f f i c i e n t s o l u t i o n t o i n t e g e r l i n e a r programming models i s d e r i v e d .  That t h i s  c h a r a c t e r i z e s an  o c c a s i o n when h e u r i s t i c approaches a r e u s e d i s e x p l a i n e d b y Tonge (.113)'. The  h e u r i s t i c p r o c e d u r e used i n v o l v e s a l o c a l s c h e d u l i n g s t r a t e g y and  i t s r e s u l t s are f a r from optimum.  T h i s i s h a r d l y s u r p r i s i n g , as t h e  problem i s e s s e n t i a l l y g l o b a l i n n a t u r e . t o s o l v e h e u r i s t i c a l l y one r e s o u r c e  P a r t 'B« d e s c r i b e s an attempt  a l l o c a t i o n problem w i t h a g l o b a l  approach.  Summary Network p l a n n i n g i s a s u c c e s s f u l and a c c e p t e d technique  f o r project control.  i n g o f c r e a t i v e thought. small projects.  management  I t s weaknesses i n c l u d e c o s t and c u r t a i l -  Manual a n a l y s i s i s o f t e n more e f f i c i e n t f o r  The procedure r e q u i r e s c o n s i d e r a b l e m a n a g e r i a l d i s c i p l i n e  i n t h e p r e p a r a t i o n and e x e c u t i o n o f t h e p r o j e c t . rarely criticized.  PERT/CPM methods a r e  F a i l u r e s may be caused by i n h e r e n t weaknesses as  w e l l a s by poor a p p l i c a t i o n s . Most systems a r e becoming e s s e n t i a l l y s i m i l a r . specialized.  A few a r e h i g h l y  Time e s t i m a t i o n i s a problem o f i n a c c u r a t e data t h a t can  never be s o l v e d .  The o n l y p r a c t i c a l s o l u t i o n s , a t p r e s e n t , t o r e s o u r c e  a l l o c a t i o n problems a r e h e u r i s t i c . their effectiveness.  The l o c a l s t r a t e g i e s u s e d  limit  P A R T  »B«  CHAPTER X I  AH  The  INTRODUCTION TO THE  PROBLEM  Problem There a r e many problems t o be s o l v e d i n r e s o u r c e - c o n s t r a i n e d p r o -  jects.  The ndnimax problem c o n s i d e r e d here i s t h a t o f f i n d i n g t h e  bound on the maximum r e s o u r c e requirement  lower  f o r the duration o f a p r o j e c t .  The  s o l u t i o n i n v o l v e s the l e v e l i n g o f t h e r e s o u r c e p r o f i l e w i t h t h e maxi-  mum  r e s o u r c e requirement  The  Approach I n Chapter  VH,  s o l u t i o n are h e u r i s t i c .  as the c r i t e r i o n t o be  i t was  minimized.  shown t h a t the o n l y p r a c t i c a l forms o f  However, t h e h e u r i s t i c a l g o r i t h m s t h a t have been  d e v e l o p e d produce r e s u l t s t h a t a r e o f t e n f a r from optimum.  One  o f the  drawbacks o f t h e s e problems i s t h a t t h e r e i s no t h e o r y t o e s t a b l i s h what i s optimum, and the enumeration of a l l p o s s i b l e s c h e d u l e s i n v o l v e s a p r o h i b i t i v e amount o f e f f o r t .  The o n l y a v a i l a b l e method t o t e s t r e s u l t s i s  t o compare them w i t h t h e r e s u l t s o f o t h e r a l g o r i t h m s .  These s o l u t i o n s  a r e e s s e n t i a l l y s i m i l a r i n t h a t a l o c a l approach i s used.  Activities  are  s c h e d u l e d a c c o r d i n g t o some p r i o r i t y , w h i l e t h e r e a r e any a v a i l a b l e r e sources.  Otherwise,  ahead f e a t u r e may  t h e i r s t a r t i s delayed.  Although  some form o f l o o k -  be i n c o r p o r a t e d , b o t t l e n e c k s are i n e v i t a b l e .  n o t s u r p r i s i n g as t h e problem i s b a s i c a l l y g l o b a l . approach i s c o n s i d e r e d i n the f o l l o w i n g c h a p t e r s .  This i s  A heuristic global  86 The S o l u t i o n The r e s o u r c e p r o f i l e i s m o d i f i e d t o show e x c e s s i v e r e s o u r c e r e quirements,  by a s s i g n i n g r e s o u r c e s t o e a c h a c t i v i t y f o r the e n t i r e p e r i o d  between i t s e a r l i e s t and l a t e s t s t a r t i n g t i m e s , i n o r d e r t o make more meaningful  d e c i s i o n s about s c h e d u l i n g the a c t i v i t i e s .  Reducing  o f a r e s o u r c e p r o f i l e i n t u r n i s not an e f f e c t i v e procedure,  e a c h peak  as i t might  be n e c e s s a r y , depending upon t h e i n i t i a l s c h e d u l e , t o i n c r e a s e a peak before decreasing i t further.  However, i n t h e m o d i f i e d r e s o u r c e  profile,  a peak r e p r e s e n t s an upper bound on the minimax v a l u e and so must be r e duced, i f p o s s i b l e . resource  These peaks i n d i c a t e the g r e a t e s t d e n s i t y o f p o s s i b l e  requirements. H e u r i s t i c s a r e u s e d t o d e c i d e how  the c u r r e n t peak w i l l be r e -  d u c e d — c o n s i d e r i n g a l l p o s s i b l e a l t e r n a t i v e s i s a g a i n not  practical.  Having made a d e c i s i o n , a c h a i n o f l o g i c a l i m p l i c a t i o n s may t h e r automatic r e d u c t i o n t o be made elsewhere The  standard procedures  fur-  profile.  'move' a c t i v i t i e s about the  p r o f i l e and t e s t the r e s u l t i n g minimax v a l u e . resource-hour  i n the  enable  resource  T h i s approach 'chops'  excess  ' s l i c e s ' o f f the m o d i f i e d r e s o u r c e p r o f i l e , and t h i s i s  e s s e n t i a l l y a simpler  procedure.  The P r e s e n t a t i o n B e f o r e any e x p e r i m e n t a l r e s u l t s can be o b t a i n e d , networks have t o be c o n s t r u c t e d — t h i s i s done w i t h a random network g e n e r a t o r — a n d c r i t i c a l p a t h a n a l y s e s have t o be performed.  the  A l g o r i t h m s t o do t h i s ,  and  87 supporting theory, are given i n Chapter X I I .  The following chapter  shows how to f i n d upper and lower minimax resource bounds, and the theory i s developed further to indicate what l o g i c a l implications may be drawn from the network.  The h e u r i s t i c procedure f o r approximating  the minimax conditions, and a standard h e u r i s t i c solution used to compare r e s u l t s , are both described i n Chapter XIV.  F i n a l l y , results ob-  tained from a program, which i s l i s t e d i n Appendix ' D ' , are examined and conclusions are drawn about the value of the global approach and possible future developments.  The d e t a i l e d r e s u l t s are tabled i n Appendix  CHAPTER X I I  CRITICAL PATH ANALYSIS  Introduction B e f o r e c o n s i d e r i n g the r e s o u r c e minimax problem, t o p e r f o r m the b a s i c c r i t i c a l p a t h a n a l y s i s . by the use o f a random network g e n e r a t o r .  i t i s necessary  Examples a r e c o n s t r u c t e d  F o r b o t h g e n e r a t o r and  a n a l y s i s i t i s b e s t t o d e f i n e the terms used and t o d e r i v e the theorems on w h i c h t h e p r o c e d u r e s depend, b e f o r e d e s c r i b i n g the a l g o r i t h m s . Reference i s made t o the i m p l e m e n t a t i o n o f t h e s e a l g o r i t h m s i n F o r t r a n programs, l i s t e d i n Appendix  'D'.  Data P r e p a r a t i o n Definition 1 :  An ACTIVITY i s a time span d u r i n g which some e x p e n d i t u r e o f r e s o u r c e s o c c u r s . The l e n g t h o f t h e time span i s the DURATION o f the a c t i v i t y .  Definition 2  An EVENT i s a moment o f t i m e , u s u a l l y r e p r e s e n t i n g the s t a r t o r c o m p l e t i o n o f one o r more a c t i v i t i e s . I f an event r e p r e s e n t s the end o f some a c t i v i t i e s , t h e n t h e event cannot o c c u r u n t i l a l l the a c t i v i t i e s w h i c h end a t i t have done s o .  :  Definition 3 s  A NETWORK (E,A) i s comprised o f a nonempty s e t o f events E and a mapping from E i n t o E x E. The range o f the mapping i s the s e t o f a c t i v i t i e s A. Two non-empty subsets o f E , Eg and E f , which r e p r e s e n t p r o j e c t s t a r t events and p r o j e c t end events r e s p e c t i v e l y , are d i s t i n g u i s h e d such t h a t , f o r any i , j € E , i f ( i , j ) 6 A, then j ^ E s and i ^ E . f  D e f i n i t i o n I*  :  A PROJECT i s the achievement o f a s p e c i f i c a c c o m p l i s h ment. I t i s composed o f a s e r i e s o f t a s k s . The s t a r t  89 o f most t a s k s i s dependent upon t h e completion o f o t h e r t a s k s . The p r o j e c t i s r e p r e s e n t e d b y a network and t h e t a s k s b y a c t i v i t i e s . The DURATION o f a p r o j e c t i s t h e time d i f f e r e n t i a l between t h e b e g i n n i n g o f t h e e a r l i e s t t a s k t o be s t a r t e d and t h e end o f t h e l a t e s t t a s k t o be completed. 5 «  Definition  Lemma 1-1  :  A DUMMY ACTIVITY i s an a c t i v i t y w i t h z e r o d u r a t i o n and r e q u i r i n g no e x p e n d i t u r e o f r e s o u r c e s . I t i s used t o e s t a b l i s h a p a r t i a l o r d e r i n g o f two e v e n t s .  There a r e no l o o p s i n a network.  Proof C o n s i d e r a network ( E , A ) . t i e s b y one.  Increment t h e d u r a t i o n s o f a l l a c t i v i -  ( T h i s i n no way a l t e r s t h e i n t e r d e p e n d e n c i e s o f t h e  a c t i v i t i e s i n A, and i s i n t r o d u c e d t o a v o i d h a v i n g t o c o n s i d e r dummy a c t i v i t i e s s e p a r a t e l y . )  A l o o p would o c c u r i n t h e network  i f t h e r e were a sequence o f e v e n t s e ] 6 2 . . . e e i , where e-j £ E f o r n  j»l,2,...,n, such t h a t ( e _ ^ , e ^ J e A f o r k=2,3>...,n , and k  ( e , e i ) e A. n  Because o f t h e d u r a t i o n o f ( e ^ - l * e ^ ) , t h e o c c u r r e n c e  o f ejj i s l a t e r than t h a t o f e^.^. e  n  i s l a t e r t h a n t h a t o f e^.  Consequently,  t h e occurrence o f  B u t ( e , e ^ ) ^ A and, b y t h e same n  r e a s o n i n g , t h e o c c u r r e n c e o f e ^ i s l a t e r than t h a t o f e ^  Hence  t h e r e i s a c o n t r a d i c t i o n and t h e r e c a n be no l o o p s i n a network.  Theorem 1 :  There i s a numbering o f t h e events o f a network (E,A) such that, f o r i , j e E , i f ( i , j ) e A ,  then i < j .  Proof C o n s i d e r a l l events e & E c: E . s  fl  A s s i g n t o these events  s e c u t i v e numbers s t a r t i n g w i t h one, i n an a r b i t r a r y  con-  order.  90 For  any event j e E , remove a l l a c t i v i t i e s ( e , j ) from A. s  Continue r e p e a t i n g t h i s p r o c e d u r e w i t h t h e m o d i f i e d n e t w o r k — and a new s e t E — m a i n t a i n i n g t h e sequence o f numbers, s  all  e v e n t s i n E f have been numbered.  until  This i s the required  ordering of events.  T h a t a l l events have been a s s i g n e d a number i s e s t a b l i s h e d by c o n s i d e r i n g t h a t any e v e n t i s on a t l e a s t  one p a t h from  an event i n E , and so w i l l i t s e l f b y a member o f a s  set E  different  a t some s t a g e i n the p r o c e d u r e , and hence w i l l be  s  numbered.  Assume t h a t , f o r some a c t i v i t y ing  o f events i > j .  (i,j)<£A, under t h e new o r d e r -  Because o f t h e a c t i v i t y ( i , j ) ,  1 and j n e v e r o c c u r s i m u l t a n e o u s l y i n any s e t E .  Therefore  s  as i  j n e i t h e r can be i n t h e o r i g i n a l s e t E  s  events  and t h e r e must  be a p a t h from an event i n t h i s s e t t o event i .  Consequently,  the network (E,A) may be s l i g h t l y m o d i f i e d by t h e a d d i t i o n o f the a c t i v i t y ( j , k ) w i t h o u t a l t e r i n g cedure.  o f the p r o -  However the m o d i f i e d network would have the two-  a c t i v i t y l o o p (j,±) ( i , j ) , impossible. activity  the r e s u l t s  and Lemma 1-1  shows t h i s  t o be  T h i s c o n t r a d i c t i o n i s o n l y a v o i d e d i f t h e r e i s no  ( i , j ) s A f o r which i > j .  91  The following algorithm f o r finding a random network, with a given number of events NE, i s j u s t i f i e d by Theorem 1.  I t appears as  the subroutine SB1-PERT i n Appendix ' D * . 1)  Assign the number ' l ' to the network start event.  2)  Find a random number of a c t i v i t i e s maximum of C.  3)  For each a c t i v i t y , find a random terminal event, with a number greater than the event number but not greater than NE.  ii)  For each a c t i v i t y , determine nonzero random parameters, up to the given maxima.  5)  Consider the next event number i n sequence. NE, go to step (2).  6)  Count the number of a c t i v i t i e s which end at each event. I f , for any event except the f i r s t , the count i s z,ero, add an additional a c t i v i t y f o r which t h i s event i s terminal and which has a random i n i t i a l event, whose number i s less than the given event number.  7)  Stop.  Notes :  to s t a r t a t t h i s event, up to a  I f t h i s i s less than  For convenience, only one network s t a r t and end events are allowed. Otherwise, step (1) should be amended and step (6) omitted. The c r i t e r i o n C, used i n step (2), controls the t o t a l number of a c t i v i t i e s i n the network by l i m i t i n g the maximum number of a c t i v i t i e s leaving any event. I t s effect i s shown i n Figure 1, from r e s u l t s i n Table IX on Page lli8 . An additional control may l i m i t the maximum number of a c t i v i t i e s leaving an event to the difference between NE and the event number. The a c t i v i t y parameters, i n step (U), include duration, cost, manpower, e t c . Maximum values are required f o r these. Dummy a c t i v i t i e s may be included by permitting a zero parameter. In t h i s case, a l l parameters are then set to zero.  93 F i n d i n g the C r i t i c a l  Path  Definition  6 :  The CRITICAL PATH i s t h e p a t h o f l o n g e s t d u r a t i o n through t h e network. T h i s d u r a t i o n i s t h e CRITICAL PATH LENGTH. I t r e p r e s e n t s the s h o r t e s t time i n which the p r o j e c t m a y b e completed.  Definition  7  :  The EARLIEST TIME (E.T.) o f an event i s t h e e a r l i e s t time a t w h i c h t h e event can o c c u r , assuming t h a t a l l p r e v i o u s events o c c u r and a l l p r e v i o u s a c t i v i t i e s s t a r t as soon as p o s s i b l e .  Definition  8  :  The LATEST TIME (L.T.) o f an event i s t h e l a t e s t time a t which the event can o c c u r , assuming t h a t a l l subsequent events o c c u r and a l l subsequent a c t i v i t i e s s t a r t as l a t e as p o s s i b l e w i t h o u t e x t e n d i n g the c r i t i c a l p a t h .  Definition  9 :  Definition  10 s The EARLIEST STARTING TIME (E.S.T.) o f an a c t i v i t y i s the e a r l i e s t time a t which t h e a c t i v i t y can s t a r t , assuming t h a t a l l p r e v i o u s e v e n t s o c c u r and a l l p r e v i o u s a c t i v i t i e s s t a r t as soon as p o s s i b l e . A s i m i l a r meaning i s g i v e n t o t h e EARLIEST FINISHING TIME ( E . F . T . ) .  Definition  11 : The LATEST STARTING TIME (L.S.T.) o f an a c t i v i t y i s the l a t e s t time a t which the a c t i v i t y can s t a r t , assuming t h a t a l l subsequent e v e n t s o c c u r and a l l subsequent a c t i v i t i e s s t a r t a s l a t e as p o s s i b l e w i t h o u t e x t e n d i n g the c r i t i c a l p a t h . A s i m i l a r meaning i s g i v e n t o t h e LATEST FINISHING TIME ( L . F . T . ) .  Definition  12 : The TOTAL FLOAT o f an a c t i v i t y i s t h e d i f f e r e n c e between the L.S.T. and the E.S.T. o f t h e a c t i v i t y . This i s e q u i v a l e n t t o the d i f f e r e n c e between t h e E.F.T. and the L.F.T. o f t h e a c t i v i t y .  Definition  13  The SLACK o f an event i s the d i f f e r e n c e between t h e L.T. and t h e E.T. o f the e v e n t .  The DELAY o f an a c t i v i t y i s the amount by w h i c h t h e s t a r t o f the a c t i v i t y i s t o be d e l a y e d beyond i t s E.S.T. T h i s i s l e s s t h a n the t o t a l f l o a t o f t h e a c t i v i t y .  Definition  lit s  The FLOAT o f an a c t i v i t y i s the d i f f e r e n c e between the t o t a l f l o a t and the d e l a y o f t h e a c t i v i t y .  Definition  1$ : A CRITICAL ACTIVITY i s any a c t i v i t y which has z e r o f l o a t .  9h D e f i n i t i o n 16 : The WATTING LIST at time t i s the l i s t of a c t i v i t i e s whose E . S . T . i s less than t , but which have yet t o be scheduled to s t a r t , and t h e i r remaining delays. D e f i n i t i o n 17 t The ACTIVE LIST at time t i s the l i s t of a c t i v i t i e s which are i n progress at time t , and t h e i r remaining durations. D e f i n i t i o n 18 j The RESOURCE REQUIREMENTS of an a c t i v i t y are the constant supply of a p a r t i c u l a r resource t h a t i s r e quired for p r e c i s e l y the duration of the a c t i v i t y . D e f i n i t i o n 19 : The RESOURCE PROFILE for a p a r t i c u l a r resource i n a project i s a histogram which shows the resource requirements of a l l a c t i v i t i e s throughout the duration of the p r o j e c t .  Theorem 2 :  A Project i s completed i f and only i f both the Active L i s t and the Waiting L i s t are empty.  Proof I f the project i s finished then a l l a c t i v i t i e s i n i t must be completed.  Hence both the Active L i s t and the Waiting L i s t  must be empty.  I f the Active L i s t and the Waiting L i s t are empty, then there are no more a c t i v i t i e s to be completed. any further a c t i v i t i e s ,  For, i f there were  there could be no r e s t r a i n t  on t h e i r  E . S . T . occurring, and they could be placed at l e a s t on the Waiting L i s t . The algorithm, given below and derived from Theorem 2, i s  designed  to calculate a resource p r o f i l e while performing the c r i t i c a l path analysis.  95 Allowance i s made for activity delays, which are likely to occur once minimal-cost or constrained-resource schedules are considered. In addition to the profile, event slacks and activity floats are, derived.  The program which implements this algorithm i s the subroutine  SB3-CP i n Appendix 'D'.  In order to simplify the frequent search for  the activities entering and leaving each event, a simple l i s t structure with minimum space requirements i s used, which i s constructed prior to the c r i t i c a l path analysis i n subroutine SB2-TAB i n Appendix  'D*.  1) Set the clock to zero. 2) Consider those activities starting at the i n i t i a l event. Add those with zero delay to the Active List, "and the remainder to the Waiting L i s t . Record the E.S.T. for a l l the activities. 3) Find the resource requirements at this time by summing the requirements of a l l activities on the Active List. h) Find the smallest time remaining to any activity on either l i s t . $) Move the clock forward by this amount. Subtract the amount from the delay times of the activities on the Waiting List and from the durations of the activities on the Active List. Record the resource requirements for this time interval. 6) I f any activity on the Waiting L i s t i s ready to start, transfer i t to the Active L i s t . 7) I f no activity on the Active List has finished, go to step (h)» 8) Remove a completed activity from the Active L i s t . Consider i t s terminal event. I f not a l l the activities ending at this event have done so, go to step (11). 9 ) Record the E.T. of this terminal event and add any activities, for which i t i s the i n i t i a l event, to the lists—those with no delay to the Active List and the remainder to the Waiting List—and record their E.S.T.  96 10) I f t h i s i s the f i n a l event o f the p r o j e c t , go to step (13). 11) I f there are s t i l l completed a c t i v i t i e s on the Active L i s t , go to step (8). 12) Go to step (3). 13) Add a l l a c t i v i t i e s terminating at the f i n a l event to the empty Active L i s t , leaving the clock at i t s f i n a l s e t t i n g . Ik)  Find the smallest duration remaining to a c t i v i t i e s on the Active L i s t .  1 5 ) Move the clock backward by t h i s amount. Subtract the amount from the duration o f the a c t i v i t i e s on the Active L i s t . 16) Remove an a c t i v i t y with zero remaining duration from the Active L i s t and record i t s L . S . T . Consider i t s i n i t i a l event. I f not a l l a c t i v i t i e s s t a r t i n g at t h i s event have been placed on and removed from the Active L i s t , go to step (19). " 17) Record the L . T . of t h i s i n i t i a l event. Add any a c t i v i t i e s for which i t i s the terminal event, t o the Active L i s t . 18) I f t h i s i s the s t a r t event of the p r o j e c t , go to step (21). 19) I f there are s t i l l a c t i v i t i e s with zero remaining duration on the Active L i s t , go to step (16). 20) Go to step (1U). 21) Calculate event slacks and a c t i v i t y f l o a t s . 22) Stop.  CHAPTER X I I I  BOUNDS AND RESOURCE PROFILES  Introduction Upper and lower bounds on the maximum r e s o u r c e requirements f o r a p r o j e c t a r e u s e f u l r e f e r e n c e p o i n t s i n f i n d i n g a minimax s c h e d u l e . An upper bound and two lower bounds a r e d e f i n e d .  The t o t a l  P r o f i l e , t h a t d e f i n e s t h e upper bound, i s a v i t a l  concept f o r t h e g l o b a l  r e d u c t i o n procedure d e s c r i b e d i n t h e next c h a p t e r . i n c o n j u n c t i o n w i t h an E x c e s s Resource  Resource  The lower bound,  P r o f i l e , may e s t a b l i s h some  l o g i c a l i m p l i c a t i o n s t h a t can be used t o reduce t h e t o t a l number o f decisions necessary t o f i n d a f e a s i b l e schedule.  A chain of i m p l i c a -  t i o n s may r e s u l t from a d e c i s i o n c o n c e r n i n g the s c h e d u l i n g o f an a c t i vity,  each i m p l i c a t i o n o f w h i c h needs t o be checked,  as i t i s p o s s i b l e  f o r a s i t u a t i o n l e a d i n g t o an u n f e a s i b l e s c h e d u l e t o d e v e l o p . A l l s u b r o u t i n e s t h a t a r e r e f e r r e d t o a r e l i s t e d i n Appendix  'D'.  Bounds D e f i n i t i o n 20 :  A SCHEDULE f o r a p r o j e c t i s an assignment o f s t a r t i n g times t o a l l t h e a c t i v i t i e s i n t h e p r o j e c t .  D e f i n i t i o n 21 :  A FEASIBLE SCHEDULE f o r a p r o j e c t i s a s c h e d u l e which does n o t extend t h e c r i t i c a l p a t h l e n g t h .  D e f i n i t i o n 22 :  The COMMON RESOURCE PROFILE i s t h e r e s o u r c e p r o f i l e t h a t i n d i c a t e s a c t i v i t y r e s o u r c e requirements o n l y o v e r those time p e r i o d s d u r i n g which t h e a c t i v i t i e s w i l l always be i n p r o g r e s s however t h e y a r e s c h e d u l e d .  D e f i n i t i o n 23 s  The TOTAL RESOURCE PROFILE i s t h e r e s o u r c e p r o f i l e t h a t r e s u l t s from assuming t h a t t h e r e s o u r c e s o f each  98  a c t i v i t y i n t h e network a r e r e q u i r e d f o r t h e e n t i r e time span between t h e E.S.T. and t h e L.F.T. o f t h e activity. D e f i n i t i o n 2h :  The RESOURCE-HOURS r e q u i r e d b y a n a c t i v i t y i s t h e p r o d u c t o f i t s r e s o u r c e requirement and i t s d u r a t i o n .  Lemma 3-1 There e x i s t s a p e r i o d o f time d u r i n g w h i c h an a c t i v i t y always be i n p r o g r e s s , however i t i s scheduled, i t s d u r a t i o n i s g r e a t e r than i t s t o t a l  will  i f and o n l y i f  float.  Proof C o n s i d e r an a c t i v i t y w i t h d u r a t i o n 'y* and t o t a l f l o a t ' f . Let  'ti'  and 't2' be t h e E.S.T. and t h e L.S.T. o f t h e a c t i v i t y ,  res-  pectively. By D e f i n i t i o n 12, If y>f,  then y > t or  f • t2 - t ^  2  -  ^  y + t^>t2  But, b y D e f i n i t i o n 10, y + t  2  - t h e E.F.T. o f t h e a c t i v i t y .  T h e r e f o r e , t h e E.F.T. o f t h e a c t i v i t y i s g r e a t e r t h a n i t s L.S.T. T h e r e f o r e , t h e a c t i v i t y w i l l always be i n p r o g r e s s between t and 2  (t-j+y).  C o n v e r s e l y , i f t h e r e i s some time d u r i n g which t h e a c t i v i t y  will  always be i n p r o g r e s s , i t s E.F.T. must be l a t e r than i t s L.S.T.  99 Hence, b y D e f i n i t i o n  10,  *1 * y > * 2  or and, b y D e f i n i t i o n  12,  f = t  2  - t  x  T h e r e f o r e t h e d u r a t i o n o f t h e a c t i v i t y i s g r e a t e r than i t s t o t a l float.  Theorem 3 F i n i t e upper and lower bounds f o r the maximum r e s o u r c e requirement o f a p r o j e c t may be d e r i v e d from t h e c h a r a c t e r i s t i c s o f the network representing the p r o j e c t . Proof F o r any f e a s i b l e s c h e d u l e , t h e r e s o u r c e s o f an a c t i v i t y a r e r e q u i r e d f o r t h e d u r a t i o n o f t h e a c t i v i t y , somewhere between i t s E.S.T. and its  L.F.T.  T h e r e f o r e , by D e f i n i t i o n 23,  t h e T o t a l Resource P r o f i l e i s t h e u n i o n  o f t h e r e s o u r c e p r o f i l e s r e s u l t i n g from a l l p o s s i b l e f e a s i b l e schedules. Consequently, no f e a s i b l e s c h e d u l e c o u l d have a maximum r e s o u r c e r e quirement  l a r g e r t h a n t h a t r e q u i r e d f o r t h e T o t a l Resource  Profile.  Hence, t h e minimum upper bound f o r t h e maximum r e s o u r c e requirement o f a p r o j e c t i s t h e maximum r e s o u r c e requirement o f the T o t a l  Resource  Profile.  By D e f i n i t i o n 22,  t h e Common Resource P r o f i l e shows t h e r e s o u r c e r e -  quirements t h a t a r e common t o a l l f e a s i b l e s c h e d u l e s .  100 Hence, t h e Common Resource source p r o f i l e s  P r o f i l e i s the intersection o f the r e -  r e s u l t i n g from a l l p o s s i b l e f e a s i b l e s c h e d u l e s .  I t s e x i s t e n c e and c o n s t r u c t i o n i s e s t a b l i s h e d b y Lemma 3-1 •  Con-  s e q u e n t l y , no f e a s i b l e schedule c o u l d have a maximum r e s o u r c e r e quirement  s m a l l e r than t h a t r e q u i r e d f o r t h e Common Resource  Pro-  file. Hence, a n e t w o r k - c o n s t r a i n e d lower bound f o r t h e maximum r e s o u r c e requirement  o f a p r o j e c t i s the maximum r e s o u r c e requirement  Common Resource  o f the  Profile.  The t o t a l a r e a e n c l o s e d by a r e s o u r c e p r o f i l e f o r any s c h e d u l e o f a p r o j e c t i s f i x e d , as i t r e p r e s e n t s t h e t o t a l number o f r e s o u r c e hours r e q u i r e d b y t h e p r o j e c t .  The minimum h e i g h t f o r t h e p r o f i l e  would o c c u r i f t h e r e s o u r c e requirement were c o n s t a n t i . e . a r e c t a n g u lar profile.  F o r , i f t h e demand a t some time was lower, a c o r r e s p o n d -  i n g i n c r e a s e would be r e q u i r e d elsewhere  t o maintain t h e t o t a l resource-  hour e x p e n d i t u r e . Hence, t h e r e s o u r c e - h o u r - c o n s t r a i n e d lower bound f o r the maximum r e s o u r c e requirement o f a p r o j e c t i s the r a t i o  o f t h e t o t a l number o f  resource-hours r e q u i r e d t o t h e p r o j e c t d u r a t i o n .  The n e t w o r k - c o n s t r a i n e d lower bound may be f u r t h e r r e f i n e d b y c o n s i d e r i n g each a c t i v i t y o f t h e p r o j e c t i n t u r n — r e m o v i n g b u t i o n t o t h e Common Resource its  i t s contri-  Profile, i f relevant—and placing i t i n  e n t i r e t y on t h e p r o f i l e , i n t h a t p o s i t i o n between i t s E.S.T. and  <  101 i t s L . F . T . which minimizes the resource requirements over the time span.  The maximum resource requirement for any of the modified pro-  f i l e s may increase the lower bound. resource-hour-constrained  The better of t h i s bound and the  lower bound i s found by the subroutine  SB6-B0UND. The resource-hour-constrained  lower bound i s generally the  more effective o f the two lower bounds.  This i s shown below i n  Table I I , which i s a contraction of Table XI on Page 102 .  Networks  with a high proportion of c r i t i c a l and n e a r - c r i t i c a l a c t i v i t i e s , resource requirements inversely proportional to f l o a t ,  or with  are more l i k e l y to  y i e l d a higher network-constrained bound.  TABLE I I COMPARISON OF TWO LOWER RESOURCE BOUND ESTIMATES Range of Activities  Number of Networks  1-20  75  ho  21-ljO  1*1-60  20 35  :  60+  Percentage of Best Results Res-hr.Bd. Network Bd.  10.7 50.0 85.0 9li.3  89.3 55.0 15.0 5.7  Bounds and P r o f i l e s D e f i n i t i o n 25 s  An EXCESS RESOURCE PROFILE i s a T o t a l Resource P r o f i l e or a modification of i t such that the time span, during which an a c t i v i t y i s contributing resources to the p r o f i l e , i s i n excess of the duration of the a c t i v i t y for at l e a s t one a c t i v i t y .  102 Definition  26 :  The SLACK o f an a c t i v i t y i n an Excess Resource P r o f i l e i s t h e d i f f e r e n c e between t h e time span, d u r i n g w h i c h the a c t i v i t y i s c o n t r i b u t i n g r e s o u r c e s t o t h e p r o f i l e , and t h e d u r a t i o n o f t h e a c t i v i t y .  Definition  27  A TROUGH i s a c o n t i n u o u s time p e r i o d i n a r e s o u r c e p r o f i l e d u r i n g which t h e r e s o u r c e requirements a r e n o t g r e a t e r t h a n t h e l o w e r bound.  Definition  28 :  A PEAK i s a c o n t i n u o u s time p e r i o d i n a r e s o u r c e p r o f i l e d u r i n g which t h e r e s o u r c e requirements a r e g r e a t e r t h a n t h e lower bound.  Definition  29 :  The EARLIEST CONTRIBUTING TIME (E.C.T.) o f an a c t i v i t y i s t h e e a r l i e s t time a t w h i c h t h e a c t i v i t y i s c o n t r i b u t i n g r e s o u r c e s t o an Excess Resource P r o f i l e . A s i m i l a r meaning i s g i v e n t o t h e LATEST CONTRIBUTING TIME (L.C.T.) o f a n a c t i v i t y .  Definition  30 :  The DELAY o f an e v e n t i s t h e d i f f e r e n c e between t h e E.T. o f t h e event and t h e e a r l i e s t E.C.T. o f any a c t i v i t y s t a r t i n g a t the event.  Definition  31 :  The OVERLAP o f an event i s t h e d i f f e r e n c e between t h e e a r l i e s t E.C.T. o f any a c t i v i t y s t a r t i n g a t t h e event and t h e l a t e s t L.C.T. o f any a c t i v i t y t e r m i n a t i n g a t the event. I f t h e l a t t e r o c c u r s b e f o r e t h e former, the OVERLAP i s z e r o .  Lemma U-l ~'  The s l a c k o f e v e r y a c t i v i t y i n t h e T o t a l Resource P r o f i l e project i s equal t o i t s t o t a l  of a  float.  Proof By D e f i n i t i o n s 23 and 29, t h e E.C.T. o f an a c t i v i t y i n the T o t a l Resource P r o f i l e e q u a l s t h e E.S.T. o f t h e a c t i v i t y , and t h e L.C.T. o f t h e a c t i v i t y e q u a l s i t s L.F.T.  103 Therefore, by D e f i n i t i o n 26, the slack » L . F . T . - E . S . T . - the duration. But, by D e f i n i t i o n 11,  L.F.T.  a  L . S . T . • the duration  Hence,  the slack • L . S . T . - E . S . T .  Therefore, by D e f i n i t i o n 12,  the slack = the t o t a l float of the  activity.  Lemma h-2 The overlap of every event i n the T o t a l Resource P r o f i l e of a project i s equal to i t s event s l a c k . Proof For a T o t a l Resource P r o f i l e , By Definitions 23 and 29, the E . C . T . of each a c t i v i t y » i t s E . S . T . and the L . G . T . of each a c t i v i t y = i t s  L.F.T.  The E . S . T . of an a c t i v i t y - the E . T . of i t s i n i t i a l event and the L . F . T . of an a c t i v i t y » the L . T . of i t s terminal event. Hence, by Definitions 9 and 31> the overlap of every event equals i t s event s l a c k .  Lemma k-3 For a feasible schedule r e s u l t i n g from an Excess Resource  Profile  of a project, the slack of each a c t i v i t y must be reduced to zero. Proof By D e f i n i t i o n 25, an Excess Resource P r o f i l e i s derived from the  101*  T o t a l Resource P r o f i l e ,  T h e r e f o r e , by Lemma 1*-1, and D e f i n i t i o n s  13, lh and 15, n o n - c r i t i c a l a c t i v i t i e s i n an Excess Resource may have nonzero  Profile  slack.  I n a f e a s i b l e s c h e d u l e , by D e f i n i t i o n 18, each a c t i v i t y c o n t r i b u t e s resources f o r p r e c i s e l y i t s duration.  T h e r e f o r e , by D e f i n i t i o n 26, t h e s l a c k o f e a c h a c t i v i t y must be z e r o .  Theorem h F o r a f e a s i b l e schedule r e s u l t i n g from an E x c e s s Resource file  Pro-  o f a p r o j e c t , t h e o v e r l a p o f each event must be r e d u c e d t o  zero. Proof By D e f i n i t i o n 25, an Excess Resource P r o f i l e i s d e r i v e d from t h e T o t a l Resource P r o f i l e .  T h e r e f o r e , by Lemma h-2, a l l events w i t h  nonzero e v e n t s l a c k may have nonzero o v e r l a p . I n a f e a s i b l e s c h e d u l e , b y Lemma lj-3, a l l a c t i v i t i e s have z e r o s l a c k . Therefore, by D e f i n i t i o n s  18 and 26, t h e E.C.T. and t h e L.C.T. o f  each a c t i v i t y i s t h e s t a r t and f i n i s h time o f t h e a c t i v i t y . I f the  o v e r l a p o f an event i s nonzero, t h e n , by D e f i n i t i o n 31, t h e  f i n i s h time o f an a c t i v i t y t e r m i n a t i n g a t t h e event i s l a t e r t h a n the  s t a r t time o f an a c t i v i t y b e g i n n i n g a t t h e e v e n t .  But, by  D e f i n i t i o n 2, an e v e n t i s a s i n g l e moment o f t i m e and so cannot o c c u r at d i f f e r e n t times. T h e r e f o r e t h e o v e r l a p o f each event must be z e r o .  105 Theorem 5 Consider any a c t i v i t y ( i , j ) , with slack s, E.C.T. t ^ and L.C.T. t . 2  Let  the i n i t i a l event i and the terminal event j of the a c t i v i t y  have E.T., delay and overlap t^,d^,v^ and tj,d-j,Vj respectively. I f an Excess Resource P r o f i l e has a trough from t j to (t^+s), then t  2  and s may both be reduced by (s-k),  where k * 0 i f t ^ f e t ^ + d^ + v^ and k  a  t ^ • d^ • v^ - t ^ otherwise,  without a f f e c t i n g the lower bound, provided that t  2  - ( s - k ) ^ t j • dj  I f an Excess Resource P r o f i l e has a trough from (tjj-s) to t , 2  then t ^ may be increased and s decreased by (s-m), where m • 0 i f t < t j + dj and k = t 2  2  - t j - dj otherwise, without a f f e c t i n g  the lower bound, provided that t j + ( s - k j g t j • d^ + v^ .  Proof I f an Excess Resource P r o f i l e has a trough from t ^ to (t]+s), f o r a c t i v i t y ( i , j ) , then i t i s best to reduce the slack by reducing t « 2  This has the e f f e c t o f lowering the t o t a l resource-hours contributing to peaks and so of improving the chances of lowering these peaks.  If  there i s a peak between t-^ and (t-j+s) then t h i s conclusion no longer applies, as reducing the slack by increasing t ^ may be necessary to reduce the intervening peak. However, no reduction of t  2  should occur i f t h i s a f f e c t s the  scheduling of other a c t i v i t i e s .  Theorem h establishes that, f o r  a f e a s i b l e schedule, the overlap of each event must be reduced t o  106 zero.  Consequently, r e d u c t i o n i n the s l a c k o f one a c t i v i t y  a f f e c t the s l a c k o f o t h e r a c t i v i t i e s .  Hence t h e f o l l o w i n g  could restric-  tions. I f t ^ < t ^ + d ^ • v^, some o f the a c t i v i t y s l a c k must be r e t a i n e d , so t h a t a c t i v i t i e s may  c o n t i n u e t o have t h e i r L.C.T. a t  (see f i g u r e 2 ( a ) ) .  A l s o , i f t h e reduced L.C.T. o f t h e a c t i v i t y  still  1  (t-j+di+v^). was  g r e a t e r t h a n ( t ^ + d j ) , the r e d u c t i o n would a f f e c t those a c t i v i -  t i e s whose E.C.T. a r e a t ( t j + d j ) .  (see f i g u r e  2(b)).  A s i m i l a r argument a p p l i e s i f an Excess Resource P r o f i l e has a t r o u g h from ( t - s ) t o t 2 , 2  f o r some a c t i v i t y ( i , j ) .  This  shows t h a t i t i s  b e s t t o d e c r e a s e t h e s l a c k by i n c r e a s i n g t ^ , s u b j e c t  t o t h e same r e s -  t r i c t i o n s as above.  ACTIVITY ( k . l )  ACTIVITY ( i , j )  -\  tj_  4  <  ACTIVITY ( j . p )  ACTIVITY ( i , j )  •4-  1  ts  mm  it  FIGURE 2.(a)  FIGURE 2(b)  POSSIBLE DEPENDENCIES AMONGST ACTIVITIES  -+  1 0 7  The a l g o r i t h m t h a t i s d e r i v e d from Theorem $ — s u b r o u t i n e  SB7-CH0P—  i s i t e r a t i v e i n s o f a r as an a d d i t i o n a l o r l a r g e r t r o u g h may be i n t r o duced b y a r e d u c t i o n , and t h i s i n t u r n may p e r m i t o t h e r r e d u c t i o n s t o be made.  I t s e f f e c t i v e n e s s i s dependent upon the v a l u e o f the lower  bound and upon t h e p r o x i m i t y o f t h e Excess Resource P r o f i l e t o i n d i c a t i n g a f e a s i b l e schedule.  A l t e r n a t i v e l y , t h e e f f e c t i v e n e s s i s dependent  upon t h e p r o p o r t i o n o f t h e d u r a t i o n o f the r e s o u r c e p r o f i l e t h a t i s i n troughs.  The procedure  i s t h e r e f o r e p a r t i c u l a r l y u s e f u l i n answering  the s l i g h t l y d i f f e r e n t managerial question:  "Can a f e a s i b l e  schedule  be found which uses a maximum o f x u n i t s o f r e s o u r c e ? "  Logical Implications  Theorem 6 C o n s i d e r any a c t i v i t y ( i , j ) , w i t h s l a c k s, E.C.T. t ^ and L.C.T. t « 2  L e t the i n i t i a l  event i and t h e t e r m i n a l event j  o f t h e a c t i v i t y have E.T., d e l a y and o v e r l a p t ^ , d^, v ^ and tj,  d j , Vj r e s p e c t i v e l y .  I f t ^ + s < t ^ '+ d^ + v ^ , then t h e r e i s a r e s t r a i n t on t h e L.C.T. o f a l l a c t i v i t i e s t e r m i n a t i n g a t event i . If t  2  - s>tj  + d j , t h e n t h e r e i s a r e s t r a i n t on t h e E.C.T.  o f a l l a c t i v i t i e s s t a r t i n g a t event j .  Proof I f t ^ + s < t ^ + dj_ + v ^ ( s e e f i g u r e 2 ( a ) ) , then, f o r each a c t i v i t y  108  ( k , i ) w i t h L.C.T.  T , i n order t h a t  z e r o , as p e r Theorem k,  T ^ t-^ + s . m  the s l a c k o f t h e a c t i v i t y If t  I f T > t ^ + s, then T m  ( k , i ) must be s u i t a b l y  and  m  reduced.  - s > t j • d j ( s e e f i g u r e 2 ( b ) ) , t h e n , f o r each a c t i v i t y  2  (j,p)  may be l a t e r reduced t o  m  w i t h E.C.T.  T , i n o r d e r t h a t Vj may be l a t e r reduced t o e  z e r o , as p e r Theorem k,  T  e —  ~  s  »  T  e  < - f c  2 " » s  t  h  e  n  T  e  m u s  '  b  be s u i t a b l y i n c r e a s e d , and t h e s l a c k o f t h e a c t i v i t y ( j , p ) s u i t a b l y decreased*  Theorem 7 An u n f e a s i b l e s c h e d u l e may be i m p l i e d b y an Excess Resource Profile.  Proof C o n s i d e r an event i w i t h o v e r l a p v ^ .  Let the a c t i v i t i e s  i n g a t i have L.C.T. and s l a c k s :  and S-^, T ^  T^  and S , 2  terminat...  Let, t h e a c t i v i t i e s s t a r t i n g a t i have E.C.T. and s l a c k s : T ^ and s ^ , g  T  e 2  and s , . . .  I f max  2  ( T - S p ) > m i n ( T q + S ) , then v ^ can never be reduced t o z e r o m p  e  q  and t h e r e f o r e , by Theorem k  f  an u n f e a s i b l e s c h e d u l e i s i m p l i e d .  The i m p l i c a t i o n s o f Theorems 6 and 7 a r e p r e c i s e l y those e f f e c t s on o t h e r a c t i v i t i e s t h a t were a v o i d e d i n Theorem 5.  From t h e p r o o f o f  109 Theorem 7,  • max ( T p ) - min ( T q ) . m  e  Therefore the updating o f the  overlap i s achieved i n a s i m i l a r f a s h i o n t o the f e a s i b i l i t y s u b r o u t i n e SB8-EVBAL.  This f e a s i b i l i t y  check—see  check must be made a f t e r  d e c i s i o n , and f o r e v e r y i m p l i c a t i o n o f t h e d e c i s i o n .  As w i t h Theorem 5,  a p p l i c a t i o n o f Theorem 6 i s i t e r a t i v e and may l e a d t o a c h a i n o f implications.  I t i s implemented by t h e s u b r o u t i n e  every  SB9-ESRED.  CHAPTER X I ?  SOLUTIONS  Introduction The  concept  o f t h e Excess Resource P r o f i l e i s used i n an a l g o r i t h m  w h i c h h e u r i s t i c a l l y f i n d s a minimax v a l u e .  Knowledge o f t h e p r o p e r t i e s  o f t h e p r o f i l e i s f u l l y employed i n r e d u c i n g t h e number o f d e c i s i o n s t o be made and i n m a i n t a i n i n g t h e d e r i v a t i o n o f a f e a s i b l e s c h e d u l e .  In  the d e s c r i p t i o n o f t h e a l g o r i t h m , t h e p a r e n t h e s i z e d s u b r o u t i n e s r e f e r t o the r e l e v a n t l i s t i n g i n Appendix ' D ' .  The l o c a l approach, w h i c h i s i n -  t r o d u c e d as a b a s i s f o r comparison o f r e s u l t s , i s r e p r e s e n t e d b y an adaptation o f Martino's M u l t i - r e s o u r c e A l l o c a t i o n Procedure,  which i s  d e s c r i b e d i n d e t a i l and i s implemented b y t h e s u b r o u t i n e S B I T T M A P , a l s o l i s t e d i n Appendix *D»,-  The  Global Solution  Theorem 8 The  maximum r e s o u r c e requirement  o f an Excess Resource P r o f i l e  i s t h e maximum r e s o u r c e requirement  o f any f e a s i b l e  schedule  t h a t can be d e r i v e d from t h e p r o f i l e , assuming t h a t no d e c i s i o n "that has been made t o produce t h i s Excess Resource P r o f i l e  from  the T o t a l Resource P r o f i l e i s r e v e r s e d .  Proof F o r any f e a s i b l e s c h e d u l e , t h e r e s o u r c e s o f an a c t i v i t y a r e r e q u i r e d  Ill f o r the duration of the a c t i v i t y somewhere between i t s E . S . T , and i t s L . F . T .  Therefore, by D e f i n i t i o n 23, the T o t a l Resource  P r o f i l e i s the union of a l l possible feasible schedules, and consequently gives an upper bound to the maximum resource requirement of any schedule.  The effect of any decision i n the reduction of  the T o t a l Resource P r o f i l e to a f e a s i b l e schedule i s to change the E . S . T . or the L , S , T . of an a c t i v i t y .  Consequently, the same argu-  ment may be applied to show that the Excess Resource P r o f i l e i s the union of a l l feasible schedules that are possible with a l l the previous decisions upheld, and therefore gives a current upper bound to the maximum resource requirement for these schedules.  The consequence of t h i s theorem i s that a converging i t e r a t i v e procedure may be evolved which approaches the lower resource bound— although i t may not actually reach i t .  The product of t h i s convergence  i s a minimax value, which i s found by the following algorithm. 1)  Perform the c r i t i c a l path a n a l y s i s .  (SB3-CP)  2)  Find the best lower resource bound.  (SB6-B0UTJD)  3)  Create the T o t a l Resource P r o f i l e .  k)  Make any automatic adjustments that are now p o s s i b l e .  (SBU-RES) (SB7-CH0P)  See Theorem 5 and the discussion on Page 105 • 5)  Find the maximum resource requirement.  (SB10-RSMX)  6)  I f t h i s equals the lower bound, go to step 18.  (SB18-MAIN)  112 7)  Find the amount of resource necessary to reduce the current maximum below the next maximum or to the lower bound, whichever i s higher.  8)  (SB18-MAIN)  Find those a c t i v i t i e s which are contributing resources to the current maximum.  9)  (SB11-RELACT)  Make a l i s t of the combinations of these a c t i v i t i e s  the adjustment  of whose E . C . T . or L . C . T . w i l l achieve the required reduction. (SB12-SELECT and SB13-FEAS) The a c t i v i t i e s from step 8 are considered one at a time and with e i t h e r t h e i r E . C . T . increased or t h e i r L . C . T . decreased appropriately. I f the l i s t i s empty, the same procedure applies with a c t i v i t i e s two at a time and, i f there i s s t i l l no combination, three at a time, e t c . P r a c t i c a l considerations of both time and space p r o h i b i t the f u l l implementation of t h i s procedure and only certain combinations are actually examined. 10)  I f no combination of these a c t i v i t i e s w i l l provide the required r e duction, go to step 15.  11)  (SB18-MAIN)  Find the best combination on the l i s t .  (SB29-WK2)  The p r i o r i t y system to f i n d the best combination uses the following ordered set of c r i t e r i a . The second c r i t e r i o n i s used only i f using the f i r s t results i n a t i e . A) Maximum sum of t o t a l f l o a t s remaining after the adjustments. B) Minimum sum of resource requirements. Results using other p r i o r i t y systems are examined i n Chapter XV. 12)  Make the required adjustments and the consequent implied adjustments. (SB29^WK2 and SB9-ESRED) See Theorem 6 and the discussion on Page 107.  1  113 13)  If a feasibility list,  check f a i l s , remove the combination from the  recover the o r i g i n a l conditions, and go t o step 10.  (SB8-EVBAL and SB29-WK2) See Theorem 7 and the discussion on Page 108,  Hi)  Go to step lu  15)  Reduce the required amount of resource reduction by one.  16)  I f the required reduction i s zero, go t o step 18.  17)  Repeat steps 8 - 13.  18)  Remove a l l excess resource requirements and e s t a b l i s h event times.  (SB18-MAIN) (SB18-MAIN)  (SB18-MAIN)  (SB18-MAIN)  (SB15-TTDT) 19)  Check that no simple reduction i n the maximum resource requirement may be made.  (SB15-TIDT)  In step 18, excess resource requirements were removed from the p r o f i l e from that end of each a c t i v i t y nearer to the maximum. Each a c t i v i t y contributing to the maximum i s removed from the p r o f i l e and replaced to minimize i t s e f f e c t . I f the maximum i s reduced, the procedure i s repeated for the new set of a c t i v i t i e s comprising the new maximum. 20)  Stop.  (SB18-MAIN)  A h e u r i s t i c method i s used i n step 1 1 t o find the best combination. Step 19 i s a precaution, necessitated by t h i s method, to avoid a feasible schedule with a maximum resource requirement that could be lowered by a simple adjustment.  lilt I t i s f a l s e t o consider  t h a t f o l l o w i n g a l l t h e p a t h s , determined  by t h e p o s s i b l e combinations f o r each maximum, i n a t r e e - l i k e would e s t a b l i s h a genuine optimum bound.  search  A l l t h a t i s required t o lower  a maximum b y a c e r t a i n amount i s t h a t t h e sum o f t h e r e d u c t i o n s  of each  a c t i v i t y c o n t r i b u t i n g resources  The r e -  t o t h e maximum b e t h i s amount.  s u l t a n t number o f p o s s i b i l i t i e s i s t h e r e f o r e  e x t r e m e l y l a r g e and an  enumeration would n o t be p r a c t i c a l *  The  Local Solution The  procedure i n t r o d u c e d  here i s a m o d i f i c a t i o n o f t h e M u l t i -  r e s o u r c e A l l o c a t i o n Procedure (MAP), w h i c h i s d e s c r i b e d by M a r t i n o  (10).  M a r t i n o a c t u a l l y u s e d MAP t o f i n d t h e s h o r t e s t s c h e d u l e f o r a p r o j e c t w i t h a given resource  constraint.  The o b j e c t i v e i s t o p r o v i d e  son w i t h t h e suggested g l o b a l s o l u t i o n . p a t h a n a l y s i s , t h e same l o w e r r e s o u r c e  a compari-  Consequently, a f t e r t h e c r i t i c a l  bound i s u s e d as a n i n i t i a l r e f e r -  ence p o i n t and t h e method i s a p p l i e d t o f i n d i n g a minimax v a l u e , f o r f e a s i b l e schedules only.  The a l g o r i t h m w h i c h produces t h i s v a l u e i s as  follows t 1)  Perform t h e c r i t i c a l p a t h a n a l y s i s .  2)  F i n d the best resources.  3)  S e t the c l o c k t o zero.  k)  F i n d a l l a c t i v i t i e s t h a t can s t a r t a t t h i s time.  5)  I f t h e r e a r e no such a c t i v i t i e s , go t o s t e p  lower bound.  T h i s determines t h e amount o f a v a i l a b l e  8.  115 6)  Order t h e a c t i v i t i e s , a c c o r d i n g t o the f o l l o w i n g p r i o r i t y system. A) S m a l l e s t r e m a i n i n g t o t a l f l o a t . B) L a r g e s t need o f r e s o u r c e - h o u r s . C) L a r g e s t need o f r e s o u r c e s . D) S m a l l e s t sequence code, under the J - P r i o r i t y System w h i c h assumes i - j event o r d e r i n g . Each t e s t i s used o n l y i f the preceding t e s t r e s u l t s i n a t i e .  7)  Schedule a c t i v i t i e s t o s t a r t , a c c o r d i n g t o t h e s o r t e d l i s t , w h i l e r e s o u r c e s a r e a v a i l a b l e . I f an a c t i v i t y r e q u i r e s more r e s o u r c e s than a r e a v a i l a b l e , i t s s t a r t i s d e l a y e d and t h e scan o f the l i s t of a c t i v i t i e s continues.  8)  I f t h e r e a r e no a c t i v i t i e s i n p r o g r e s s and none s t i l l t o be scheduled t o s t a r t , stop.  9)  I f t h e r e are no a c t i v i t i e s i n p r o g r e s s and some s t i l l s c h e d u l e d t o s t a r t , go t o s t e p 12.  10)  t o be  Of t h e a c t i v i t i e s i n p r o g r e s s , f i n d the e a r l i e s t time a t which ends.  i  ._  one  ..  11)  Move the c l o c k f o r w a r d t o t h i s t i m e . A d j u s t the times o f t h e a c t i v i t i e s i n p r o g r e s s , and t h e r e m a i n i n g t o t a l f l o a t s o f t h o s e a c t i v i t i e s whose s t a r t s have been d e l a y e d .  12)  I f no r e m a i n i n g t o t a l f l o a t i s n e g a t i v e , go t o s t e p  13)  Increment t h e p r e s e n t lower bound by  Ui)  Go t o s t e p  3.  one.  U.  CHAPTER XV  RESULTS, EXTENSIONS AND  CONCLUSIONS  Introduction A comparison between the r e s u l t s o b t a i n e d by the g l o b a l s o l u t i o n and b y MAP  show t h a t , by i t s e l f ,  the g l o b a l s o l u t i o n i s o n l y s l i g h t l y  b e t t e r and t a k e s c o n s i d e r a b l y l o n g e r t o c a l c u l a t e a r e s u l t .  However,  the combination o f t h e two methods produces a s i g n i f i c a n t o v e r a l l r e d u c t i o n i n resource requirements.  The l i k e l y percentage o f b o t h  critical  e v e n t s and c r i t i c a l a c t i v i t i e s appear t o have upper and lower  bounding  curves.  D i f f e r e n t p r i o r i t y systems are examined and p r e d i c t i o n s are  made about f u t u r e s o l u t i o n s .  Some p o s s i b l e developments o f t h e g l o b a l  approach a r e d e s c r i b e d .  Comparative  Results  The absence o f knowledge o f the optimum lower r e s o u r c e bound makes any a b s o l u t e judgement o f a method i m p o s s i b l e . t i o n was  compared w i t h MAP,  the comparison  f o r 7h  a l o c a l approach.  random networks,  5 a c t i v i t i e s t o 1 1 0 events and 30h  The  The  global solu-  detailed results of  r a n g i n g i n s i z e from 5> events  and  a c t i v i t i e s , are given i n Table X  on Page 1 0 0 . A c o n d e n s a t i o n o f t h e s e r e s u l t s i n T a b l e I I I shows t h a t t h e g l o b a l s o l u t i o n , p r o d u c e s b e t t e r r e s u l t s more o f t e n t h a n the l o c a l s o l u t i o n b u t i n d i c a t e s an o v e r a l l d e c r e a s e i n r e s o u r c e requirements o f o n l y 2.1$.  T h i s advantage  i s more pronounced  i n s m a l l networks w i t h up  117 to 1$ events, where the o v e r a l l decrease i s l.h%»  The decrease i n  the remaining networks i s only 0,25%. Table IV presents a summary of the comparative times for the same networks.  The global approach takes far longer to find a solution  than the l o c a l approach, and the r a t i o of the times i s seen to increase exponentially with the size of the network. t i a l i s caused by the implementing programs.  Some of the time differenAfter the preliminary  c a l c u l a t i o n s , which are the same for both methods, the l o c a l procedure finds i t s s o l u t i o n within one subroutine, while the global approach uses f i f t e e n i n t e r a c t i n g subroutines. The time f o r the g l o b a l method could be reduced by r e s t r i c t i n g the search for f e a s i b l e combinations (see w e l l as by more e f f i c i e n t coding.  step 9 i n the algorithm), as  However, except f o r small networks,  i t i s u n l i k e l y that the a d d i t i o n a l cost of time would be worth the possible s l i g h t reduction i n the maximum resource requirement. I t i s i n t e r e s t i n g to notice t h a t , comparing the better of the two r e s u l t s with that obtained from MAP, gives an o v e r a l l reduction of 8,1%, which i s evenly spread amongst the different sizes of network. This large decrease would j u s t i f y the use of both methods as a combined procedure.  Different P r i o r i t y Systems Three different p r i o r i t y systems f o r the global solution were compared.  They were:  TABLE I I I  A COMPARISON OF RESULTS  Range of Events  Range of Act.  No. of Nwk  Total Resource Requirements Global L o c a l  D i f f . o f res.req. w.r.t. L o c a l Global Better  No. o f B e s t Results Glob. Loc.  5- 5 10- 10 15- 15 20- 20 25- 30 35- 50 60-110  5- io 9 15- 32 9 21*- 56 9 35- 87 l£ 1*3-132 12 61-176 9 105-301* 8  331* 605 680 1602 1211 1032 1355  331 665 753 1561 1238 1035 1379  - 3 + 60 • 73 - 1*1 • 27 + 3 • 21*  + 5 + 60 + 85 + 111 + 117 + 67 + 118  7 9 7 9 8 1* 3  8 2 2 10 I* 5 5  5-110  5-30U 7U  6819  6962  + 11*3  • 563  1*7  36  OVERALL  TABLE I V  A COMPARISON OF TIMES  Range o f Events  Range o f Act.  5- 5 10- 10 15-15 20- 20 25- 30 35- 50 60-110  5- 10 15- 32 21*- 56 35- 87 1*3-132 61-176 105-301*  OVERALL  5-110  5-301*  dumber o f Ntwks.  T o t a l Times ( s e c . ) Global Local  Ratio o f Times  9 9 9 18 12 9 8  1*.2 25.7 66.6 518.1* 806.3 11*01.5 9768.2  1.5 5.2 8.9 3l*.5 1*8.0 1*9.5 296.3  2.8 1*.9 7.5 15.0 16.8 28.2 33.0  71*  12590.9  W*3.9  28.3  119 P I - H i g h e s t r e m a i n i n g event o v e r l a p s ; H i g h e s t event o v e r l a p s ; H i g h e s t r e m a i n i n g a c t i v i t y s l a c k s ; Lowest r e s o u r c e r e q u i r e m e n t s . P2  - H i g h e s t remaining a c t i v i t y s l a c k s ; Lowest r e s o u r c e r e q u i r e m e n t s .  P3  - Lowest r e s o u r c e r e q u i r e m e n t s ; H i g h e s t r e m a i n i n g a c t i v i t y  slacks.  E v e n t o v e r l a p s were c o n s i d e r e d t o a v o i d u n f e a s i b l e s c h e d u l e s .  Highest  r e m a i n i n g a c t i v i t y s l a c k s and l o w e s t r e s o u r c e requirements were b o t h c o n s i d e r e d t o l e a v e the g r e a t e r f l e x i b i l i t y i n the Excess Resource file.  The d e t a i l e d r e s u l t s a r e g i v e n i n T a b l e X I I on Page 1^3  .  Pro-  These  a r e summarized i n T a b l e V, i n which s i x p o i n t s a r e awarded f o r e a c h network, w i t h f o u r p o i n t s f o r the b e s t system and two p o i n t s f o r t h e next b e s t . A l t h o u g h t h e r e was s e l e c t e d because i t was to  little  d i f f e r e n c e between P2 and P3,  P2  was  i n t u i t i v e l y more l o g i c a l and because i t appeared  t a k e l e s s time t o produce  a solution.  However, comparative  times a r e  n o t shown. The f a c t t h a t P2 a c h i e v e s as good r e s u l t s as P3 was Pascoe (5U)  s u p p o r t e d by  i n his investigation of d i f f e r e n t c r i t e r i a f o r l o c a l  solu-  t i o n s , i n w h i c h he observed t h a t most p r i o r i t y systems w i l l g i v e the same r e s u l t s f o r the minimax. problem.  T h a t any p r i o r i t y system  will  a c h i e v e s i m i l a r s u c c e s s suggests t h a t , i f a c o n s i s t e n t l y good s o l u t i o n is  t o be found, a new  c r i t e r i o n needs t o be d i s c o v e r e d .  I t i s probable  t h a t t h i s c r i t e r i o n w i l l be e s t a b l i s h e d from g l o b a l c o n s i d e r a t i o n s and a p p l i e d w i t h a l o c a l decision--making p r o c e d u r e .  An example o f t h i s  c o u l d be t h e r e s o u r c e - c o n s t r a i n e d f l o a t d e f i n e d by W i e s t  (63).  120 TABLE V  COMPARISON OF PRIORITY SYSTEMS 'Range of Events  Range•of Act.  5- 5 10-10 15-15 20-20 25-30 35-60 OVERALL 5-60  'Number of Ntwrks.  Points Given PI -P2 -P3  5 - 10 15- 32 2U- 56 35- 87 1*3-132 61-211  9 9 . 9 18 12 12  16 18 22 35 18 Hi  20 20 17 29 33* 30  18 16 15 hh 21 28  3li5 618 681 I6I1O 1267 1585  33U 605 680 1602 1211 Ui97  338 617 68I1 1550 1215 1508  5-211  69  123  Hi9  lh2  6136  5929  5912 .  I  Resource Requirements P2 PI P3  121  Percentage o f C r i t i c a l A c t i v i t i e s and Events I n any problem w i t h f i x e d c r i t i c a l p a t h l e n g t h , t h e p r o p o r t i o n o f c r i t i c a l a c t i v i t i e s and o f c r i t i c a l events a r e o f paramount importance.  As t h e s c h e d u l i n g o f c r i t i c a l a c t i v i t i e s i s determined,  the t r u e s i z e o f t h e problem i s dependent critical activities.  upon t h e number o f non-  C r i t i c a l e v e n t s r e p r e s e n t f i x e d times t h a t may  be used as r e f e r e n c e p o i n t s — f o r example, c r i t i c a l e v e n t s are o f t e n used t o s u b d i v i d e a p r o j e c t . K e l l e y (hh)  notes t h a t " i n a l l ' r e a l ' p r o j e c t s s t u d i e d t o date,  l e s s t h a n 10% o f t h e a c t i v i t i e s have been c r i t i c a l . "  Of n e c e s s i t y ,  any s t u d y o f t h e a c t i v i t i e s / c r i t i c a l a c t i v i t i e s and e v e n t s / c r i t i c a l events r e l a t i o n s h i p s must be e m p i r i c a l . lished.  However,, none has been pub-  F i g u r e s 3 and h i n d i c a t e t h e r e s u l t s f o r o v e r 150 random n e t -  .' works l i s t e d i n T a b l e s X H I and XIV on Page 155.  Points that are r e -  p e a t e d a r e o n l y shown once. Reasonable  e s t i m a t e s o f t h e range o f the l i k e l y percentage o f  b o t h c r i t i c a l a c t i v i t i e s and c r i t i c a l e v e n t s may be made by c o n s i d e r a t i o n o f t h e upper and l o w e r bounding o f p o i n t s i n t h e graphs.  curves which a r e i m p l i e d b y t h e bunching  The mean and v a r i a n c e a t d i f f e r e n t l e v e l s a r e  l i s t e d i n T a b l e s VI and V I I .  Developments I n s t e a d o f m i n i m i z i n g t h e maximum r e s o u r c e requirement, i f t h e r e s o u r c e a v a i l a b i l i t y v a r i e d throughout the d u r a t i o n o f t h e p r o j e c t ,  12it TABLE VI LIKELY PERCENTAGE OF CRITICAL ACTIVITIES  Number of Act. 20 UO 60 80 100 120 11*0  Percentage C r i t . Act. Mean Variance 1*3.0 26.1* 20.0 16.0 I3.U 11.6 10.0  Number of Act.  27.0 12.9  160 180 200 220 21*0 260 280  8.5 6.5  5.1 lul  3.5  'Percentage C r i t . A c t . Mean Variance 9.0 8.0  7.5  7.1 6.9 6.8 6.8  TABLE VII LIKELY PERCENTAGE OF CRITICAL EVENTS Number of Events 10 20 30 liO  50  60 70  Percentage C r i t . Ev. Mean Variance  67.5 53.lt  lt2.3 3iw3 28.6  2i*.5 21.3  32.5 27.1 22.7 19.3  I5.lt 12.0 8.8  3.0  2.5  2.2 2.1 1.9 1.8 1.8  12$ i t would be e a s y t o modify t h e g l o b a l p r o c e d u r e t o m i n i m i z e t h e excess r e q u i r e m e n t o v e r a minimum bound which was a f u n c t i o n o f t i m e .  A  s l i g h t adjustment t o t h e d e f i n i t i o n s o f peaks and troughs would be r e quired. I f t h e time p e r i o d over which t h e minimax r e s o u r c e requirement o c c u r r e d was m i n i m i z e d , i t would i n d i c a t e t h e s m a l l e s t e x t e n s i o n o f the c r i t i c a l p a t h l e n g t h n e c e s s a r y t o reduce t h i s peak.  Delaying the  p r o j e c t end r e s u l t s i n a l l a c t i v i t i e s b e i n g n o n - c r i t i c a l and i n i n c r e a s i n g t h e L.F.T. o f each a c t i v i t y .  The o n l y e f f e c t on t h e a l g o r i t h m  i s t o i n c r e a s e t h e magnitude o f t h e problem. The g l o b a l approach may e a s i l y be adapted t o s o l v e s u b - m i n i m i z a t i o n problems.  Having minimized t h e maximum requirement f o r one r e s o u r c e ,  a c t i v i t y s l a c k i s r e s t o r e d , w i t h o u t e x c e e d i n g t h i s maximum, a c c o r d i n g t o a p r i o r i t y determined b y the T o t a l Resource P r o f i l e o f t h e next r e source*  BIBLIOGRAPHY  PART 'A' —PRIMARY SOURCES  I. 1  2  3  h 5)  BOOKS AND BIBLIOGRAPHIES  Anon.  A p p l i c a t i o n s and Techniques o f O p e r a t i o n s R e s e a r c h — A  . DOT and NASA Guide t o PERT/Cost System D e s i g n , Washington: U n i t e d S t a t e s Department o f Defense and t h e N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n , 1962. An I n t r o d u c t i o n t o t h e Use o f RAMPS.  C.E.I.R.(UK),  1962.  NASA PERT and Companion Cost System Handbook. Washington: N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n , 1962.  •< PERT Summary Report. Phases I and I I . Washington: States Navy Special Projects O f f i c e , 1958.  United  6)  A r c h i b a l d , Russell D. and V i l l o r i a , Richard L . Network Based Management Systems (PERT/CPM). New York: John Wiley and Sons, 1967.  7)  Baar, James and Howard, William E . Bruce, and Company, I960.  8)  Baker, Bruce N. and E r i s , Rene L . An Introduction to PERT-CPM. New York: Richard D. Irwin, 19o%.  9)  Battersby, A l b e r t . Network Analysis for Planning and Scheduling. London: MacMillan, 1967.  Polaris.  New York:  Harcourt,  10)  Martino, R . L . Project Management and C o n t r o l . Volume I I I — A l l o c a t ing and Scheduling Resources. New York: American Management A s s o c i a t i o n , 1965.  11  Sobczak, Thomas V. Network P l a n n i n g — A B i b l i o g r a p h y . Journal o f I n d u s t r i a l E n g i n e e r i n g , 13 (November/December 1962).  127 Wattel, Harold L . ( e d . ) . Network Scheduling and Control Systems. Yearbook of Business Volume I I . A series of Masters theses. New York: Hofstra U n i v e r s i t y , January 1961*. Williams, Dwight L . Planning of Research and Development Work. New York: A Wallace Clark and Company Report, undated.  TECHNICAL PAPERS Beckwith, R . E . A Cost Control Extension of the PERT System. I n s t i t u t e of Radio Engineers Transactions on Engineering Management, EM9 (December 1962). Berman, E . B . Resource A l l o c a t i o n i n a PERT Network under Continuous A c t i v i t y Time-Cost Functions. Management Science, 10 (July 1962*). Burgess, A . R . and Killebrew, James B. Variation i n A c t i v i t y Level on a C y c l i c a l Arrow Diagram. Journal of I n d u s t r i a l Engineering7 13 (March/April 1962). Charnes, A . and Cooper, W.W, A Network Interpretation and a Directed Subdual Algorithm f o r C r i t i c a l Path Scheduling. Journal of I n d u s t r i a l Engineering, 13 (July/August 1962). , Cooper, W.W. and Thompson, G.L. C r i t i c a l Path Analyses v i a Chance Constrained and Stochastic Programming. Operations Research, 12 (May/June 1961*77 C l a r k , Charles E . The Optimum A l l o c a t i o n of Resources among the A c t i v i t i e s of a Network. Journal of I n d u s t r i a l Engineering, 12 (January7February 1961). • The Greatest of a F i n i t e Set of Random Variables. tions Research, 9 (March/April 196l"n  Opera-  . The PERT Model for the D i s t r i b u t i o n of an A c t i v i t y Time. A l e t t e r to the E d i t o r . Operations Research, 10 (May/June 1962). Crows ton, W. and Thompson, G.L. Decision CPM: A Method for Simultaneous Planning, Scheduling and Control of Projects. Operations Research, 15(May/June 1967).  128 23)  Dantzig, G . B . , Ford, L.R. and Fulkerson, D.R. A Primal-Dual Algorithm. Contained i n Annals of Mathematics Study No. 38, edited by H.W. Kuhn and A.W. Tucker. Princeton University Press, 1956.  21;)  Davis, Edward W. Resource A l l o c a t i o n i n Project Network Models— A Survey. Journal of I n d u s t r i a l Engineering, 17 ( A p r i l 1966).  25)  Dewitte, L . Manpower Leveling of PERT Networks. for Science/Engineering, March/April 196k.  26)  Dimsdale, Bernard. Computer Construction of Minimal Project Networks. IBM Systems Journal, 2 (March 1963).  27)  E i s n e r , H. A Generalized Network Approach to the Planning and Scheduling of a Research Program. Operations Research, 10 (January/February 1962).  28)  Elmaghraby, Salah E . On Generalized A c t i v i t y Networks. of I n d u s t r i a l Engineering, 17 (November 1966). "  29)  Data Processing  Journal  . An Algebra for the Analysis of Generalized A c t i v i t y Networks. Management Science, 10 ( A p r i l 196h).  30)  Everett H I , H. Generalized Lagrange M u l t i p l i e r Method for Solving Problems of Optimum A l l o c a t i o n of Resources. Operations Research, 11 (May/June 1963).  31)  Freeman, Raoul J . A Generalized PERT. A l e t t e r to the E d i t o r . Operations Research, 8 (March/April I960).  32)  . A Generalized Network Approach to Project A c t i v i t y Sequencing. I n s t i t u t e of Radio Engineers Transactions on Engineering Management, EM 7 (September I960).  33)  F r y , B . L . SCANS—System Description and Comparison with PERT. I n s t i t u t e of Radio Engineers Transactions on Engineering Management, EM 9 (September 1962).  3U)  Fulkerson, D.R. A Network Flow Computation f o r . P r o j e c t Curves. Management Science, 7 (January 1961).  35)  Cost  . Expected C r i t i c a l Path Lengths i n PERT Networks. Operations Research, 10 (November/December 1962)1  129 Fulkerson, D.R. Scheduling i n Project Networks. Rand Corporation Memo. RM-U137-PR. Rand Corporation, June 1961w Goldberg, Charles R. An Algorithm for the Sequential Solution of Schedule Networks. A letter to the Editor. Operations Research, 12 (May/June 1961*). Grubbs,- Frank E. Attempts to Validate Certain PERT Statistics or "Picking on PERT." A letter to the Editor. Operations Research, 10 (November/December 1962). Hartley, H.O. and Worthara, A.W. A Statistical Theory for PERT C r i t i c a l Path Analysis. Management Science, 12 (June 1966). Hayward, P. and Robinson, J.A. ESP—Business Computations— 300001—Preliminary Analysis of the Construction Scheduling Problems. Engineering Department Report. E.I. Dupont de Nemours and Company, December 1956. Healy, Thomas L. Activity Subdivision and PERT Probability Statements. Operations Research, 9 (May/June 1961). Jewell, William S. Risk-Taking i n C r i t i c a l Path Analysis. Management Science, 11 (January 1965). Kelley, James E., J r . Parametric Programming and the Primal Dual ., Algorithm. Operations Research, 7 (May/June 1959). C r i t i c a l Path Planning and Scheduling; Mathematical Basis. Operations Research, 9 (May/June 1961). and Walker, M.R. C r i t i c a l Path Planning and Scheduling. Proceedings of the Eastern Joint Computer Conference, December 1959. Lambourn, S. Resource Allocation and Multi-Project Scheduling (RAMPS): A New Tool i n Planning and Control. Computer Journal, f> (January 196JH Levy, F.K., Thompson, G.S. and Wiest, J.D. Multi-Ship Multi-Shop Workload Smoothing Program. Naval Research Logistics Quarterly, 10 (March 1963). MacCrimmon, Kenneth R. and Ryavec, Charles A. An Analytical Study of the PERT Assumptions. Operations Research, 12 (January/  February 196ITH  130 McGee, A.A. and M a r k a r i a n , M.D.. Optimum A l l o c a t i o n o f Research/ E n g i n e e r i n g Manpower w i t h i n a M u l t i - P r o j e c t O r g a n i z a t i o n a l S t r u c t u r e . I n s t i t u t e o f Radio E n g i n e e r s T r a n s a c t i o n s on E n g i n e e r i n g Management, EM 9 (September 1962). Malcolm, D.G. e t a l . A p p l i c a t i o n o f a Technique f o r Research and Development. O p e r a t i o n s R e s e a r c h , 7 (September/October1959). Meyer, W.L. and S h a f f e r , L.R. E x t e n s i o n s o f the C r i t i c a l P a t h Method t h r o u g h the A p p l i c a t i o n o f I n t e g e r Programming. U n i v e r s i t y o f I l l i n o i s Department o f C i v i l E n g i n e e r i n g , J u l y  1963.  Moshman, Jack, Johnson, Jacob and L a r s o n , Madalyn. RAMPS—A Technique f o r Resource A l l o c a t i o n and M u l t i - P r o j e c t S c h e d u l ing. S p r i n g J o i n t Computer Conference. American F e d e r a t i o n o f I n f o r m a t i o n P r o c e s s i n g S o c i e t i e s Conference P r o c e e d i n g s ,  23 (1963).  P a r i k h , S h a i l e n d r a C. and J e w e l l , W i l l i a m S. Decomposition o f P r o j e c t Networks. Management S c i e n c e , 11 (January 196577 Pascoe, T.L. An E x p e r i m e n t a l Comparison o f H e u r i s t i c Methods f o r A l l o c a t i n g Resources . A Ph.D. t h e s i s . Cambridge U n i v e r s i t y E n g i n e e r i n g Department, 1965. P e t r o v i c , R a d i v o j . O p t i m i z a t i o n o f Resource A l l o c a t i o n i n P r o j e c t Planning. O p e r a t i o n s Research, 16 ( J u l y 196b"). P r a g e r , W i l l i a m . A S t r u c t u r a l Method o f Computing P r o j e c t Cost Polygons. Management S c i e n c e , 9 TApril 1963J. P r i t s k e r , A. A l a n B. and Happ, W. W i l l i a m . GERT: G r a p h i c a l E v a l u a t i o n and Review Technique. P a r t I — F u n d a m e n t a l s . J o u r n a l o f I n d u s t r i a l E n g i n e e r i n g , "T7 (May 1966). , and Whitehouse, Gary E. GERT: G r a p h i c a l E v a l u a t i o n and Review Technique. P a r t I I — P r o b a b i l i s t i c and I n d u s t r i a l Engineering Applications. Journal of I n d u s t r i a l Engineering, 17 (June 19667. ' " Roy,  B. The Method o f P o t e n t i a l s . Reference C.R. Academie de S c i e n c e s , 1959.  T.2lt8,2li37.  Paris:  Sobczak, Thomas V. A,Look a t Network P l a n n i n g . I n s t i t u t e o f Radio E n g i n e e r s T r a n s a c t i o n s on E n g i n e e r i n g Management, EM 9 (September 1962).  131 61)  Thompson, Robert E . Adjusting Network Plans with "PERT Slack Bonus." Journal of I n d u s t r i a l Engineering, 17 (March 1966).  62)  Van Slyke, Richard M. Monte Carlo Methods and the PERT Problem. Operations Research, 11 (September/October 1963).  63) Wiest, Jerome D. Some Properties of Schedules f o r Large Projects with Limited Resources. Operations Research, 12 (May/June  1961).  61*)  Wilson, R . C . Assembly Line Balancing and Resource Leveling. Summer Conference on Production and Inventory C o n t r o l . University of Michigan, 1961*.  PART 'B'—SECONDARY SOURCES I . BOOKS 65)  Anon.  66)  .  67)  .  IBM 701*0/701*1* PERT COST H Program. IBM #70l*0-CP-03X. International Business Machines Corporation, 1961*. Management Planning and Control Techniques. OTS Selective Bibliography SB-510. WashingtonJ United States Department of Commerce, 1963. Minuteman PERT System Indoctrination.  Courses I and I I .  United States A i r Force B a l l i s t i c s System Division,~l96l7 68)  __. PERT & CPMj Proven Tools for Management Planning and ControlT Management Science Series Presentation #102361*5. Burroughs Corporation, 1965.  69)  PERT/Time for Burroughs B200/B300 Series Systems. Management Science Series MSS-007. Burroughs Corporation, 1965.  70)  •  Program Evaluation Procedure: Technicians Handbook. Dayton, Ohio: Wright-Patterson A i r Force Base Dictorate of Systems Management, undated.  71)  .  SCANS—Network Construction Procedure. ment Corporation, 1961.  Systems Develop-  132  k  72  Anon.  73  •  7k  . WSPACS MOD Z e r o . A Demonstration Model. Dayton, Ohio: W r i g h t - P a t t e r s o n A i r F o r c e Base D i c t o r a t e o f Systems Management, I960.  SPECTROL P r o g r e s s R e p o r t Phase 2. C o r p o r a t i o n , I960. TOPS:  19537"  Systems  The O p e r a t i o n a l PERT System.  Development  Aerospace C o r p o r a t i o n ,  75  Dean, B.V. ( e d . ) . O p e r a t i o n s R e s e a r c h i n R e s e a r c h and Development. Chapter~S* New Y o r k : John W i l e y and Sons, 1963.  76  E v a r t s , Harry F.  77  Getz, CW. PERT: A New Management P l a n n i n g and C o n t r o l Technique. New York: American Management A s s o c i a t i o n , 1962.  78  Goode, H. and Machol, R. H i l l , 1957.  79  L e v i n , R i c h a r d I . and K i r k p a t r i c k , C h a r l e s A. P l a n n i n g and C o n t r o l w i t h PERT/CPM. New Y o r k : M c G r a w - H i l l , 19W,  80  M a r t i n o , R.L. P r o j e c t Management and C o n t r o l . Volume I — F i n d i n g t h e C r i t i c a l P a t h . New Y o r k : American Management A s s o c i a t i o n , 196U.  81  U.  I n t r o d u c t i o n t o PERT.  Boston:  A l l y n and  Bacon, 196k*  System E n g i n e e r i n g .  New York:  McGraw-  P r o j e c t Management and C o n t r o l . Volume I I — A p p l i e d O p e r a t i o n a l P l a n n i n g . New Y o r k : American Management A s s o c i a t i o n , 1965.  TECHNICAL PAPERS  82) Anon. Expo '67 F o l l o w s ' C r i t i c a l P a t h ' . March 19VJ.  B u s i n e s s Automation,  83)  . P r o g r e s s t o PERT a t Idthgows. (March 196777  The Computer B u l l e t i n , 10  8!*)  . R e p o r t on C r i t i c a l Path S c h e d u l i n g o f R e p a i r s and Resources f o r Brush D i e s e l E l e c t r i c Locomotive No. 901. C o o r d i n a t i o n and P l a n n i n g Department R e p o r t JER/EE7CPM. ,702l*/55. Steel Company o f Wales, 1961*.  133 A l p e r t , Lewis and Orkand, D.S. A Time Resource Trade-Off Model f o r Aiding Management Decisions. Technical Paper No. 12. S i l v e r Spring, Maryland: Operations Research, 1962. Bildson, R.A. and G i l l e s p i e , J . R . C r i t i c a l Path Planning— PERT Integration. A l e t t e r to the E d i t o r . Operations Research, 10 (November/December 1962). Black, O . J . An Algorithm for Resource Leveling i n Project Networks. An unpublished paper. Yale U n i v e r s i t y , Department of I n d u s t r i a l Administration, May 1965. Blanning, Robert W. and Rao, Ambar G. A Note on "Decomposition of Project Networks." A l e t t e r to~the E d i t o r . Management Science, 12 (September 1$65). Boyan, E . A . Target Commitment Scheduling. Lecture notes f o r Course 15:71. Massachusetts Institute of Technology,  19U6.  Briggs, William A . Minimum Excess Cost Curve. Algorithm #217. Communications of the ACM, 6 (December 1963). Clayton, Ross and Glenn, Robert. Analysts look at PERT through Eyes of the S c i e n t i s t and Engineer. Naval Management  Review, (5ctober 1962.  Clingen, C T . A Modification of Fulkerson's PERT Algorithm. A l e t t e r to the E d i t o r . Operations Research, 12 (July/August  19610.  Constantine, Larry L . and Donnelly, James F . PERGO: A Project Management T o o l . Datamation, October "~ Digman, Lester A . PERT/LOB: Life-Cycle Technique. I n d u s t r i a l Engineering, lo" (February 1967). Dimsdale, Bernard. On Project Networks. ing Centre, 1962.  Journal of  UCLA Western Data Process-  Eisenman, Burton and Shapiro, Martin. Evaluation of a PERT Network. Algorithm #119. Communications of the ACM, 5~TAugust 1962). Fey, C P . A p p l i c a t i o n of Least Cost Estimating and Scheduling. IBM Management Science Report MS-1. Bethesda, Maryland: International Business Machines Corporation, 1962.  •131* 98)  F i r t h , A.N.O. Optimization Problems; Solution by an Analogue Computer, Computer Journal, 1* (1961/1962).  99)  F i s h e r , A . C . , Liebman, J.S. and Nemhauser, G.L, Computer Construct i o n of Project Networks. Communications of the ACM, 11 TJuILy  19SW.  100)  Fondahl, J.W. A Noncomputer Approach to the C r i t i c a l Path Method f o r the Construction Industry. Stanford U n i v e r s i t y , 1962.  101)  Ford, L.R. and Fulkerson, D.R. A Simple Algorithm for Finding Maximal Network Flows and an Application to the Hitchcock Problem. Canadian Journal of Mathematics, 9 (19573.  102)  Frishberg, M.C. Least Cost Estimating and Scheduling—Scheduling Phase Only (LESS). IBM 650 ProgramTibrary F i l e No. 10.3.005. International Business Machines Corporation, undated.  103)  Gass, S. and Saaty, T . The Computational Algorithm f o r the Parametric Objective Function. Naval Research L o g i s t i c s  Quarterly, 2 (19557.  101*)  Ghare, P.M. Optimal Resource A l l o c a t i o n i n A c t i v i t y Networks. paper to the 2ttth National Meeting. Houston, Texas; Operations Research Society of America, November 1965.  A  105)  Handa, V.K. Project Cost Curve Equivalent Linear Graphs. A paper to the 27th National Meeting. Boston: .Operations Research Society of America, May 1965.  106)  H i l l , Laurence S.  Some Cost Accounting Problems i n PERT Cost.  Journal of I n d u s t r i a l Engineering,  17 (February 196671  107)  Howden, W.E. A Program for the Construction of PERT Flow Charts. The Computer Journal, 10 (November 1967).  108)  Hu, T . C . P a r a l l e l Sequencing and Assembly Line Problems. tions Research, 9 (November/December 1961).  109)  Hudson, James P. Program Evaluation Procedure. A Description of the WWDC PEP Program. Dayton, Ohio: Wright-Patterson A i r Force Base Aeronautical System D i v i s i o n , undated.  110)  Kahn, A . B . Topological Sorting of Large Networks. tions of the ACM, 5 (November 1962).  Ul)  K e l l e y , James E . , J r . Scheduling A c t i v i t i e s to Satisfy Constraints on Resources. Contained i n I n d u s t r i a l Scheduling, edited by J . F . Muth and P . L . Thompson. New York: P r e n t i c e - H a l l , 1963.  Opera-  Communica-  135  112)  K i r k p a t r i c k , T . I . and C l a r k , N.R. PERT as an A i d to Logic Design. IBM Journal of Research and Development, 10 (March 1966).  113)  K l e i n , Mark M.  Scheduling Project Networks.  the ACM, 10 ( A p r i l  Communications of  196717  111*)  Lass, S . E . PERT Time Calculations without Topological Ordering. Communications of the ACM, 8 (March 1965).  115)  Lasser, D . J . Topological Ordering of a L i s t of Randomly Numbered Elements of a Network. Cc>mmunications of the ACM, I* ( A p r i l 19oT)T  116)  Leavenworth, B. C r i t i c a l Path Scheduling. cations of the ACM, 1* (March 1961).  117)  Malcolm, D . G . R e l i a b i l i t y Maturity Index (R.M.I.)—An Extension of PERT i n t o R e l i a b i l i t y Management. Journal of I n d u s t r i a l Engineering, 11* (January/February 1963).  118)  Mauchly, John W. A p r i l 1962.  119)  Moder, J . J . PERT.  120)  Newnham, Donald E . , et a l . A PERT Control Centre for Management of Major B a l l i s t i c Modification Programs. Journal of I n d u s t r i a l Engineering, 16 (July/August 1965).  121)  N o e t t l , John N. and Brumbaugh, P h i l i p , information Concepts i n Network Planning. Journal of I n d u s t r i a l Engineering, 18 (July 196*717  122)  Odom, Ralph Q. and Blystone, Eugene E . A Case-Study of CPM i n a Manufacturing S i t u a t i o n . Journal of I n d u s t r i a l Engineering, 15 (November/December 1961*).  123)  P h i l l i p s , C e c i l R. Fifteen Key Features of Computer Programs for CPM and PERT. Journal of I n d u s t r i a l Engineering, 15 (January/ February 1961*).  121*)  Pocock, J.W. PERT as an A n a l y t i c a l A i d f o r Program Planning— Its Payoffs and Problems. Operations Research, 10 (November/ December 1962).  C r i t i c a l Path Scheduling.  Algorithm #1*0.  Communi-  Chemical Engineering,  and P h i l l i p s , C.R. Project Management with CPM and New York: Reinhold Corporation, 1961*.  136 125) P r o s t i c k , Joel M. Loop Tracing i n PEP-PERT Networks.  Paper presented at 16th National Conference. Los Angeles: Association for Computing Machinery, 1961.  126)  Roper, Don E . C r i t i c a l Path Scheduling. Engineering, 15 (MarchT&pril 1961*).  127)  Roseribloom, R.S. Notes on the Development of Network Models f o r Resource A l l o c a t i o n i n R. and D. P r o j e c t s . I n s t i t u t e o f Radio Engineers Transactions on Engineering Management, EM 11 (June 1961*).  128)  Roy, B. and Dibon, M. 1966.  129)  Salveson, Melvin E . P r i n c i p l e s of Dynamic Weapon Systems Programming. Naval L o g i s t i c s Quarterly, 8 (1961^.  130)  Tate, A . E . Introduction to the Resource A l l o c a t i o n Problem. Paper presented at 2nd C r i t i c a l Path Analysis Symposium. London: Operational Research Society, June 1961*.  131)  Tonge, F . M . The Use of Heuristic Programming i n Management Science. Management Science, 7 (October I960).  132)  Wagner, Harvey M.,. G i g l i o , Richard L . and Glaser, R. George.  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The Gamma and Beta functions are related by: B(p*l,q+1) - T ( p + 1 ) r ( q + l ) / r(p+q+2)  = pT(p) r(q*D/(p+q+l)r(p q+D +  Hence  , by (3)  B(p+l,q+l) - pB(p,q+l)/(p+q+l)  The Beta d i s t r i b u t i o n i s defined as: b(x) » xP(l-x) /B(p+l,q+l) q  for  0 4t x ^ 1. db(x)/dx = 0  i f pxP~ .(l-jc) 1  q  - xP.qU-x)* " = 0 1  1  Let X be the mode of the Beta d i s t r i b u t i o n , then, as X / 0 and X / 1,  p(l-X) - qX i.e.  X = p/(p+q)  I f m i s the mean of x, then m = Substituting (5) i n (7),  m -  xb(x)dx J  X 0  x  p+1  •  (l-x) dx/B(p+l,q+l) q  139 From (1), Applying  ra  (U),  * B(p+2,q+l)/B(p+l,q+l) m = (p*l)/(p+q+2)  (8)  L e t V ( x ) be t h e v a r i a n c e o f x, and E ( x ) be t h e expected then  V(x) - E ( x - m )  i.e.  V(x) = E ( x ) -  2  ra  2  =  x b(x)dx  (10)  2  S u b s t i t u t i n g (5) i n (10) g i v e s E ( x ) -  x  2  From (1), Applying  (9)  2  2  E(x )  v a l u e o f x,  -  (ii),  E(x )= 2  A p p l y i n g (U) a g a i n ,  E(x )» 2  p + 2  (l-x) dx/B(p+l,q+l) q  B(p+3,q+D/B(p+l,q+l) (p+2)B(p+2,q+l)/(p+q+3)B(p+l,q*l) (p+l)(p+2)/(p+q+2)(p+q+3) ...  (11)  S u b s t i t u t i n g (11) and (8) i n (9), V(x) = (p+l)(p+2)/(p+q+2)(p+q+3) - (p*l) /(p+q+2) 2  2  S i m p l i f y i n g , V(x) - (p+l)(q+l)/(p+q+2) (p+q+3) 2  (12)  Hence t h e mode, mean and v a r i a n c e o f the B e t a d i s t r i b u t i o n a r e g i v e n by  (6), (8) and (12).  PERT For PERT, t h e axes and t h e s c a l e o f t h e Beta, d i s t r i b u t i o n have t o be altered. Let T , T Q  m  pessimistic  and Tp r e p r e s e n t t h e most o p t i m i s t i c , most l i k e l y and most times,  then t h e t r a n s f o r m a t i o n f o r t h e d i s t r i b u t i o n o f t h e time t i s : t = T  0  + (T -T ).x p  (13)  0  Note t h a t when x=0, t = T  0  and when x = l , t = T . p  HiO From (6) and (13), the mode of the d i s t r i b u t i o n i s T m - T Q + (Tp-T ).p/(p+q) 0  i.e.  T m - (qTo+pT )/(p+q)  (Hi)  p  From (8) and (13), the expected time,  T e * T Q + (T p -T 0 ).(p+l)/(p+q+2) - (T +qVpTp+Tp)/(p+q+2) 0  Hence, from (Iii), T e «= (T Q + (p q)T m • Tp)/(p*q+2)  (15)  +  The variance,  V(t) = ( T p - T Q ) 2 . V ( x )  Hence  V(t) - (Tp-T0)2.(p+l)(q+l)/(p+q+2)2(p+q+3)  ....... (16)  The PERT assumptions are that T e • (T Q + liT m + T )/6 p  and that V(t)= ( T p - T ) /36 2  0  (17) (18)  I f the most l i k e l y time i s the mid-point of the extreme times, from equation (15), any value of (p+q) w i l l correctly equate the expected time with the most l i k e l y time. For the symmetric d i s t r i b u t i o n , p = q Therefore, from (16) and (18),  :^6(p+l) 2 » U(p+1)2 (2p+3)  As p i s nonnegative,  9 • 2p • 3 i.e.  p • 3  Consequently, p=sq 3 represents the unique symmetric Beta d i s t r i b u t i o n =:  for which the variance i s given by equation (18).  Uil Grubbs, i n reference (38), has shown that the only Beta d i s t r i b u t i o n s that s a t i s f y  the PERT assumptions are the following three  p a i r s of values f o r p and q : 1) 2) 3)  p = 2 + p » 2 P * 3  2 and q - 2 2 and q » 2 + and q = 3  2 2  Further Analysis For s i m p l i c i t y , consider the basic range ( 0 , 1 ) .  From (17), the 'PERT1 mean mp a  (liX+l)/6  where X i s the mode.  Consider the general case. (  For some s,  mp » (sX+l)/(s+2)  ...  ,.  Let k = p + q  (20)  Then, from (6),  p - kX  And, from  m = (kX+l)/(k+2)  (8),  (19)  ,  (21) (22)  mp - m - (sX+l)/(s+2) - (kX+l)/(k+2) Hence  nip - m = (s-k)(2X-l)/(k+2)(s+2)  (23)  From equation (23), there i s no e r r o r i n the PERT assumption for the expected time, i f s - k  or i f X = l / 2 .  From (18), the 'PERT' variance  ¥ p » 1/36  Consider the general case. For some r ,  VD « r  (2h)  12.2 From (20) and (21), q - k(l-X)  ....  (25)  From (12), 7 = (kX+l)(k-kX+l)/(k+2)2(k+3)  (26)  V p - V - r - (-k 2 X 2 +k 2 X+k+l)/(k+2) 2 (k+3) Hence, V p - V = ( k 2 X ( X - l ) + r(k+2)2(k+3) - (k+l))/(k+2)2(k+3) As the function  ...  (27)  f(X) - X(X-l) i s symmetric about X - l / 2 , for the  range (0,1), a t which value i t has a maximum value of l / l i , from equation (27), V p would have a minimax e r r o r i f r(k+2)2(k+3) - (k+1) - k 2 / 8 i.e. i f  r - (k +8k+8)/8(k+2)2(k+3)  (28)  2  For reasons described i n the PERT Summary Report (5), as quoted on Page 28, the value chosen i s r = 1/36. From (28),  8(k+2)2(k+3) - 36(k +8k+8) 2  2k + 5k - I»0k - 1*8 • 0 3  2  (k-l*)(2k +13k+12)  -0  2  (29)  2 Consequently, as 2k + 13k + 12 >0 f o r  kSO, k  a  I*, as assumed  i n PERT, i s the only fixed value of k which w i l l give the minimax error f o r the variance r » l/36. As the mode X i s given for each a c t i v i t y , i f the weight k i s f i x e d , the Beta d i s t r i b u t i o n , and consequently i t s variance, i s determined by equations (21) and (25).  Conversely, the variance r may be  f i x e d , i n which case an appropriate value of k should be chosen.  Hi3 Let r - 1/36 From (26),  (k+2)2(k+3) - 36(kX+l)(k-kX+l) k? + k2(36X2-36X+7) - 20k - 2h - 0  Let  ..  (30)  n = 36X2 - 36X + 7  Then, as O ^ X S l ,  -2=gn«7  (31)  The consequence of equation (31) i s to give upper and lower bounds to k .  Thus,  2.8<k<6  (32)  Some solutions of the cubic equation (30) are given below i n Table VTII.  TABLE VIII SAMPLE PAIRINGS OF X AND k k  0.02 0.15 0.27 0.50 %  3  h  5 6  - X  k  0.73 5 0.85 h 0.98 3  An empirical study of the difference between expected times,  derived both with various fixed values of k and with fixed variance, and the corresponding actual times would be i n t e r e s t i n g .  I t seems  l i k e l y , however, that the r e s u l t s would question the v a l i d i t y of the Beta d i s t r i b u t i o n assumption.  APPENDIX 'B'  THE ACTIVITY TIME/COST RELATIONSHIP  C o n v e n t i o n a l l y , assumptions about t h e c o n t i n u o u s a c t i v i t y t i m e / c o s t r e l a t i o n s h i p have been made w i t h o u t c o n s i d e r a t i o n o f t h e components o f the c o s t .  There a r e r e s o u r c e s , s u c h a s manpower, which a r e r e q u i r e d  i n amounts determined b y t h e a c t i v i t y d u r a t i o n , and which have a c o s t p r o p o r t i o n a l t o b o t h the amount and t h e d u r a t i o n .  It is,  then, reason-  a b l e t o assume t h a t the t o t a l c o s t o f an a c t i v i t y i s the sum o f a cons t a n t term, a time-dependent term and a time-and-resource-deperident term.  The a n a l y s i s o f the t i m e / c o s t f u n c t i o n may be extended b y c o n -  s i d e r i n g t h e e f f e c t o f t h e t i m e / r e s o u r c e r e l a t i o n s h i p on t h i s  function.  I t w i l l be shown t h a t r e a s o n a b l e assumptions about t h e t i m e / r e s o u r c e f u n c t i o n can r e s u l t i n an u n r e a s o n a b l e form f o r the t i m e / c o s t f u n c t i o n .  L e t C and t r e p r e s e n t the T o t a l Cost and D u r a t i o n o f an a c t i v i t y . L e t m r e p r e s e n t t h e amount o f time-dependent r e s o u r c e s r e q u i r e d . Let t i ,  m^, c ^ and t , m , c 2  2  2  r e p r e s e n t the ' c r a s h ' and 'normal' t i m e s ,  and r e s o u r c e r e q u i r e m e n t s and c o s t s r e s p e c t i v e l y . Let ' k j , 3=0,1,2,...'  Hence  t^t^=t . 2  be n o n n e g a t i v e c o n s t a n t s .  Then C = ICQ + k ^ t + k m t  (i)  As m i s time-dependent, l e t m • f ( t )  (2)  2  From (1), %  C - k  or  C - ko + ( k + k f ( t ) ) . t  0  * k t + k t.f(t) x  2  1  2  (3)  (U)  Consider the a c t i v i t y time/resource f u n c t i o n , f ( t ) .  The o b j e c t  i n expending a d d i t i o n a l r e s o u r c e s i s t o reduce t h e a c t i v i t y d u r a t i o n . However, i t i s t o be e x p e c t e d t h a t t h e r e s h o u l d be a minimum r e s o u r c e requirement and a p o i n t o f r e s o u r c e s a t u r a t i o n .  Although d i f f e r e n t i n  c o n c e p t , these p o i n t s a r e e q u i v a l e n t t o t h e • 'normal' and ' c r a s h ' c o n d i tions.  I t i s the curve between these extremes t h a t needs t o be c o n s i -  dered.  A s i m p l e form o f t h i s curve might be  L i , L2 and a .  + L t , a  2  f o r constants  T h i s i n c l u d e s t h e l i n e a r c a s e , when a =» 1.  L i and L2  a r e determined b y t h e extreme p o i n t s . f(t) • L! • • L t  ...  a  2  f ^ t ) - Lgat " 3  From ( 3 ) ,  C -  Hence  C = k  (6)  1  • k^t + k t.(L +L t ) a  2  0  1  2  + (k *k L )t • k ^ t * * 1  2  2  2  (8)  a  2  - k2L2a(a+l)t ~  2  (7)  1  1  dC/dt = k i + k L ^ •* k L 2 ( a + l ) t d C/dt  a  (9)  1  I n o r d e r t h a t r e s o u r c e requirements s h o u l d decrease as time i n c r e a s e s , from e q u a t i o n (6), p r e c i s e l y one o f e i t h e r s h o u l d be n e g a t i v e .  t h e t i m e / c o s t curve has a maximum, when  tju  - (ki+k Li)/k2L2(a l).  01  +  2  or a  I f LJJ i s n e g a t i v e , then, from e q u a t i o n s (8) and  (9), a  (5)  If  l i e s between t i ,  and t , which 2  w i t h a p p r o p r i a t e v a l u e s i s c l e a r l y p o s s i b l e , t h e n t h e t i m e / c o s t curve s h o u l d be m o d i f i e d t o c o n s i s t o f a d i s c r e t e p o i n t a t ( t ^ , c i ) and a d i s c o n t i n u i t y u n t i l t h a t p o i n t where t h e l i n e C = c ^ i n t e r s e c t s t h e  11*6 o r i g i n a l curve f o r t h e second time (see F i g u r e 5 ( a ) ) . curve i s u n a l t e r e d . than  '1*,  The r e s t o f the  I f a i s n e g a t i v e and i t s a b s o l u t e v a l u e i s g r e a t e r  s i m i l a r a n a l y s i s shows t h e p o s s i b i l i t y o f a minimum between  ti and t . 2  I n t h i s c a s e , t h e r e can be no advantage i n e x t e n d i n g t h e  d u r a t i o n beyond tm, and the curve s h o u l d be m o d i f i e d by d e c r e a s i n g %2  t  o  t ^ (see F i g u r e 5 ( b ) ) .  i s l e s s than ' 1 ' ,  I f a i s n e g a t i v e and i t s a b s o l u t e v a l u e  then i n o r d e r f o r c o s t t o decrease w i t h time a t  from e q u a t i o n (7) i t i s n e c e s s a r y t h a t ( k ^ + k L ^ )  be n e g a t i v e ,  2  i.e.  Lj<-  k^/k . 2  all,  T h i s c o u l d r e s u l t i n a maximum between the extreme  p o i n t s , r e q u i r i n g t h e same m o d i f i c a t i o n as above. I n o r d e r t h a t the t i m e / c o s t r e l a t i o n s h i p be o f the expected c r e a s i n g form, b o t h the ' r e s o u r c e - h o u r s decrease w i t h t i m e .  1  de-  and the amount o f r e s o u r c e s must  This i s i n d i c a t e d by equations  (3) and (1*).  The  1.  t i m e / c o s t r e l a t i o n s h i p i s l i n e a r i f and o n l y i f a =» -  I t i s q u i t e p o s s i b l e t h a t deeper a n a l y s i s o f c o s t components may  y i e l d o t h e r problems.  The o b j e c t o f t h i s a n a l y s i s i s t o i l l u s t r a t e  the e x t e n t t o which the u s u a l assumptions may  c  be i n e r r o r .  l  Cost  9-2 *1  V  *2  Time  FIGURE  5(a)  POSSIBLE TIME/COST CURVES  FIGURE  5(b)  APPENDIX «C« RESULTS The results given below i n Tables IX to XIV were obtained using the program listed i n Appendix 'D . 1  In order to simplify the column  headings, the following abbreviations are used.  BD  -  Lower Resource Bound.  BD1  -  Resource-Hour-Constrained Lower Resource Bound.  BD2  -•  Network - Constrained Lower Resource Bound.  MAP  -  Minimax Solution obtained by MAP.  MIN  -  Minimax Solution obtained by the Algorithm.  NA  -  Number of Activities  NE  -  Number of Events i n the Network.  NW  -  Number of Networks Considered.  PCA  -  Percentage of C r i t i c a l Activities i n the Network.  PCE  -  Percentage of C r i t i c a l Events i n the Network.  TM1  -  Time taken to obtain Minimax Solution by the Algorithm, i n seconds.  TM2  -  Time taken to obtain Minimax Solution by MAP, i n seconds.  i n the Network.  TABLE IX  EVENT/ACTIVITY RELATIONSHIP WITH VARIABLE CONTROL ' C  98*  NW  To tal NA  5 10 15 20 25 30 35 1*0 1*5 50 55 60 65 70  100 100 100 100 100 100 100 100 100 100 100 100 100 100  63U 1596 2550 3535 1*1*61 5i*U5 6288 7308 8209 9215 10151 11136 12021* ' 12996  100 686 100 1581* 100 2566 100 3531 100 1*1*73 100 5U26 100 6396 100 7321* 100 8270 100 9230 100 10166 100 11118 100 12096 100 13037  7 16 26 35 U5 51* 63 73 82 92 102 111 121 130  -2 + 2 - 3 + 2 - 1* + 3 - 3 + 1* -6 + 5 - 8 + 6 - 5 + 7 -6 + 7 - 8 +10 -8 + 9 -7 + 8 -7 + 8 - 9 +11 - 9 * 9  5 10 15 20 25 30 35 1*0 us 50 55 60 65 70  100 100 100 100 100 100 100 100 100 100 100 100 100 100  811* 2302 3663 1*91*0 6298 7682 9238 101*65 12056 1331*9 1 787 16203 17650 1901*8  100 100 100 100 100 100 100 100 100 100 100 100 100 100  8 23 36 50 61* 78 93 105 120 131* 11*7 162 177 190  - 3 -7 -6 -10 -11 -11 -16 -16 -16 -16 -19 -12 -18 -20  t  C  NE  2  1*  NW  Total NA ?  856 2252 3636 U997 61*11 7838 9326 101*53 11863 13U03 11*606 16251* 17658 18986  Rounded Average  Variance  + 3 + 5 + 8 + 10 + 13 + 10 + 12 + 1U + 13 + 13 + 13 + 16 + 13 + 18  U*9 TABLE IX (Continued)  c  NE  NW  Total NA  6  5 10 15 20 25 30 35 1*0 1*5 50 55 60 65 70  100 100 100 100 100 100 100 100 100  975 2611* 1*1*66 61*1*9 8221* 991*0 11985 11*031* 15581*  NW  Total NA  Rounded Average  100 100 100 100 100 100 100 100 100 100 100 100 100 100  925 2768 1*613 61*09 8285 10158 1191*8 13979 15885 171*23 191*1*3 21262 23061* 25058  9 27 1*5 61* 83 100 120 11*0 157 171* 191* 213 231 251  9S%  Variance - 2 - 6 -12 -13 -15 -11* -19 -22 -20 -22 -22 -27 -27 -31  • 2 +•• 8 +11 •11 +18 •16 •17 +21 •26 •31 +29 •25 +26 +26  150 TABLE X SOLUTIONS TO THE RESOURCE MINIMAX PROBLEM  NE  NA  BD  MIN  TM1  MAP  TM2  5  5 6  38 23  38  0.1 0.1 0.2 O.li  38 23 33  0.1 0.1  7  9 9  9  10 10 10 10  15  16 20 20 20 27 28 29 32  15  33 33  1*1*  1*1* 1*7 1*5  28  Uli  1*0 31  36 50 1*5 58  102 1*9 67  57 51  1*0  25  50  35 35 37 1*1* hh 20  33 33  2l* 30  35  36 37 37 1*2  1*6 1*6 1*7 1*8  58 58 60  23  62 112  51* 63 51*  33  38  1*5 50 52 85  102  53 81* 70  1*5 52  0.9  66 87 102 70 101  2.3 2.8  o.u 3.6 1*.0  5.9  71*  1*3  111* 78  75 73  71*  81  i.i  1.9 2.5 l*.3  1*9 59  73  1*5 35  1.0  1*0  57 75  81i  70 71  1*7  68  63  1*7  32  1*1*  lt.7  88  60 1*9 77  0.5 0.5 0.9 o,5  61*  72  70  31*  90 72 86 70  95 62 93 90 88  103  62 72 136 102 79 70 92  2.7 6.9  6.9  11.2 6.2  10.6 5.0 8.2 10.9 10.1  21**9  Hi.6  15.5  27.2  29.6: 35.1* 33.9  .  72 91 76 72 73 79 101 69 79 91 83 102  0.3  0.1 0.1  0.3 0.3 0.1 0.1  0.3  0.1  0.1* 0.8  0.1 0.6 1.0 0.9 1.0 0.2 0.6  0.5 0.8  1.1* 0.5  i.i 1.6  0.7  0.6 1.1  0.7  1.1 1.7 1.8 1.3  1.5 2.2  1.1* 3.7  151 TABLE X (Continued) NE  NA  BD  20  65 69 71 77 78 87  61 99 86  25 30  35 1*0  50  71*  76 77  1*3 1*3 68  1*7 63 68  1*9 50 52 81 87 95 112 118 132 61 61*  1*9 86 72 101*  71* 71*  93 127 91 137 176 105  53 92  95 90 121 87 91 55 67 75 120  KEN  71*  133 99 91 107 91 69 78 96 68 100 107 126 80 106 121 107 153 125 108 67 85 102 162  TML  MAP  1*6.0 29.8 37.0 51*.7 50.9  79 132 103 87 88 81* 71 98 98 66 101 86 111*  7l*.l  12.1 20.8 1*1.5 21.6 17.5 33.2 56.5 82.5 135.1 128.5 10l*.7 152.3  1*0.1*  39.9 68.1 56.2 80.7 11*6.5 126.2 1*09.2 l*3l*.3 11*5.3 1*76.9 1038.2  81*  132 111 122 155 111 107 81*  81*  112 166  TM2 2.3  l*.l  2.1 2.7 3.3 2.2 1.2  1.1*  1.5  1.1* 1.6 1.2 3.2  3.1*  7.7 1*.2 7.5 13.7 1*.0 1.3 2.6 2.7  3.1* 7.1*  93 138 125 130 11*9 186 112 135 186  307.3 976.1 1801*. 6  136 11*2 185 11*1 171 168  100  11*2 211 131 192 260 277  69 113 176 100 130 156 87 106 152 161*  221*  1860.7  211*  3.3 23.3 33.3 11*2.5  110-\  301*  203  233  3159.1  222  56.3  60  70  97 130  U*l*  3.0 7.2 17.9 3.8 7.7 26.1  TABLE XI DETAILED COMPARISON OF TWO LOWER RESOURCE BOUND ESTIMATES  ACTIVITY RANGE  TOTAL NW  BEST RESULTS BD2 .BOTH  - BD1  1-  10  50  h  U6  0  11-  20  25  k  21  0  21-  30  25  8  15  2  31- UO  15  10  5  0  h i - 50  10  8  2  0  50-100  30  27  3  0  100+  15  15  0  0  TABLE XII  COMPARISON OF TWO PRIORITY SYSTEMS WITH THE CHOSEN SYSTEM P2  NE  5  NA  PI  P2  P3  5  38 23 33 33 1*1* 33 56 1*7 38  38 23 33 33 1*1* 33 1*5 1*7 38  38 23 33 33 1*1* 33 1*9 1*7 38  1*5 51* 53 83 102 61* 83  U5 50 52 85 102 53 81* 70 61*  U5 50 57 79 102 51 81* 75 71*  1*0 57 75 111* 78 75 73 88  1*0 63 85 111* 70 65 79 90 78  81* 90 71*  75 88 62 75 79 102 67 68  6 7 9 9  99  10 10 10 10  15  16  20 20 20 27 28 29 32 15  20.  2l* 25 30 35 35 37 1*2* 1*1* 56  35 36 37 37 1*2 1*6 1*6 1*7  71  63  51 56 78 111* 69 75  71 88 79  81*  106 70 71* 11*0 93 65 68  80  72  86 95 70 62  TABLE XII (Continued) NE  NA  PI  P2  P3  20  1*8  92 98  93 90 88 103  87 85 91 107  58 58  60  65 69 71 77 78 87 25  U3  1*3  68 30  1*9  50 52 81 87 95 112 118 137 35 1*0  61 61* 71* 71*  93  127  50  60  91  88  93 76 129 96 82 89 97  133  99 91 107 91  123 95 93 91 88  92 69 105  78 69 96  81*  79 102 107  68 100 107 126 80 106 121 107 153  121*  70 110 128 112 169  71*  71*  73 99 62 103 107 127 71 113 128  103 11*5  118 112  125  125 109  72  67 85 102 162  68 103 162  93 138 130  98 133 150  130 11*9  150  91  111* 178 91*  137 176  170 128  105 11*2 211  153 207  128  108  186  81*  11*7  179  155 TABLE XIII THE RELATIONSHIP BETWEEN THE NUMBER OF ACTIVITIES AND THE NUMBER OF CRITICAL ACTIVITIES  NA  5 5  6 7 7 7 7 7 8 8 8 9 9 9 9 9/ 9 9  10 10 10 10 10 10 10 10 10 11 11  PCA  NA  PCA  60.0 60.0 66.7 57.1 1*2.9 57.1 57.1 57.1 37.5 37.5 37.5  11  18.2 33.3 31.3 50.0 25.0 37.5 35.3 35.3 31.6 31.6 25.0 35.0 35.0 20.0 37.5 20.0 23.1 22.2 25.9 29.6 21.1* 32.1 20.7 31.0 20.0 21.9 21.9 21.2 21.2  kk.k  l*l*.l* l*l*.lt 33.3 22.2 22.2 22.2 30.0 30.0 1*0.0 1*6.0 20.0 30.0 30.0 30.0 1*0.0 27.3 27.3  15  16 16 16 16 17 17 19  19 20 20 20 20 21* 25  26  27 27  27 28 28 29  29 30 32 32 33 33  35 35 35 35 36 36 37 37 37 38 1*2 1*2 1*3 1*3 M 1*1* 1*6 1*6 1*7 1*8 ii9 50 52  56 57 58 58  60  61  PCA  NA  PCA  22.9 20.0 22.9 20.0 19.1* 19.1* 2l*.3 2l*.3 21.6 23.7 21.1* 23.8 25.6 23.3 20.5 18.2 17.1* 19.6 21.3 18.8 21*. 5 16.0 17.3 16.1 12.3 12.1 15.5 15.0 16.1*  6b 65  U*.l 15.1* 16.9 16.2 H*.5 12*.l 17.6 17.6 13.3 11.7 H*.l 12.1*. 10.8 15.7 15.5 16.1 12.6 12.2 13.3 lit.3 9.8 H*.0 17.9 12.6 13.5 11.5 12.1* 10.9 9.8  65 68 69 71 71* 71* 75  77 78 81 83 83 81* 87  87 90 90 91 92 93  95 95  96 96 97 101 102  NA  102 102 103  101*  105  112 111* 111* 116 118 118 119 121 123 126 126 127 127 127 129 129 131 132 137 11*0 11*0 11*2 H*3 11*6  PCA  NA  PCA  9.8 12.8 9.7 11.5 8.6 10.7 11.1* 8.8 13.8 12.7 11.9 10.9 13.2 10.6 12.7 8.7 7.9 11.8 11.0 10.9 10.1 13.7 9.9 8.8 8.6 9.3 7.8 9.8 8.9  150 150  9.3 9.3 9.7 5.8 ll.l* 9.1* 11.3 8.1 9.3 8.0 6.6 9.9 7.0 8.0 8.5 9.7 7.1* 8.5 8.9 6.7 7.1 6.9 6.5 6.3  151* 156 158  159  160 160 161 163 168 172 172 175 176 176 176 189 192 193 211 260 277 301*  156 TABLE XIV THE RELATIONSHIP BETWEEN THE NUMBER OF EVENTS AND THE NUMBER OF CRITICAL EVENTS PCE  NE  NW  5 10 15  35  30 100.0 30 90.0 30 86.7 53.3 35 65.0 l*o.o 20 61*. 0 l*l*.0 2U 53.3 36.7 13 51.1*  ko  13  20 25 30  U5 50 55  60 70  h 7 2 5 5  31.1*  1*7.5  30.0  1*2.2 32.0 25.5 26.7  25.7  NW  PCE  NW  WE  11 2 1 7  80.0 80.0 80.0  11*  60.0 5 70.0 13 1 73.3 1 Uo.o 55.0 8  1* 5 1 3 2 3 3 2 1 1 1 1 1 1 3  1*6.7  60.0 35.0 60.0 l*o.o 5o.o 33.3  1*2.9  28.6 l*o.o 27.5 37.8 28.0 20.0 25.0  21.1*  5 1 U 3 1 1 1 3 3 1 2 2 3 1  1 * 1 1 1  56.0 32.0  1*6. 7 30.0  1*0.0  NW  1 2 5  l l l  37.5 22.5 22.2 26.0  1 1 2  23.3 20.0  1 1  PCE  NW  PCE  NW  60.0  66.7  6 8  5o.o 60.0  h  50.0  8  U5.0  7 6  52.0  2  1*8.0  9  1*3.3  26.7 37.1  U 1 3  1*0.0 2 3l*.3  1  35.0 20.0 20.0  2 1 1  32.5  1  20.0  1  16.7  1  APPENDIX ' D «  THE PROGRAM  The f o l l o w i n g i s t h e l i s t i n g o f a program w h i c h implements the p r o c e d u r e s  that have been d e s c r i b e d i n P a r t 'B'. The program  i s w r i t t e n i n F o r t r a n IV.  I t was compiled b y t h e IBM F o r t r a n Com-  p i l e r on t h e IBM 701*1* a t t h e U.B.C. Computer Centre d u r i n g August,  1968.  N.R.ARDEN I SN  ..  :_  [  _„  FORTRAN SOURCE LIST  11/17/68  PAGE  1  SOURCE STATEMENT  $I8FTC 0 INTEGER MXN,NFA,FA(400),B,I,J,C,MX,BB 1 * INTEGER NA,A,AT, AM, AS,BS ,NE, ES ,ATA8,CPL ,RS,RTAB,RLST, FE,NFE 2 COMMON /CO/NA,A{400,2),AT(400) ,AMI 400),AS(400), BS(400)»NE, 3... 1 ES(150,2),ATAB(700,2),CPL,RS(400),RTAB(150,3), * 2 RLST(2300),FE(150),NFE INTEGER POINT 4 COMMON /C6/P0INT 5 * REAL PCA,PCE,T1,T2,T3,T4,T5,T6,T7,T8,T9 6 7 * KM = .400 . DO 1 NE=110.150.10 10 * DO 1 J=4,6,2 11 * 12 C =J T l = CLCCK(O.O) 13 * CALL PERT(C,20,25,HXN) 14 IF(NA.GT.MXN) GO T O 1 15 CALL TAB 20 * Tl = CLOCK ( TD/60.0 21 * C * C SET BACKWARD SLACK TO ZERO $ C 00 5. I = l....NA 22 A. BS ( I )= 0 23* 5 C C DO CRITICAL PATH ANALYSIS AND OUTPUT RESULTS C T2 = CLOCK(0.0) 25 * CALL CP(FA,NFA) 2.6 T2 = CLOCK(T2)/60.0 27 IF(NA.EQ.O) GO TO 1 30 CPL = ES(NE,1) 33 PCA = 100.0*FLOAT(NFA)/FLOAT(NA) 34 * 35 * B = MX(RStl»CPL) 36 WRITE (6,103) B . 103 FORMAT(/IX,16HINITIAL SOUND = ,14) 37 CALL STORE(1,0) 40 * POINT = 4 41 T3 = CLOCK(O.O) 42 CALL TIDY 43 T3 = CL0CK(T3)/60.0 44 * 45 CALL STORE(ltl) 46 T4 = CLOCK(O.O) CALL BOUND(B) 47 T4 = CLOCK(T4)/60.0 50 * * BB =B 51 T5 = CLOCK(0.0) 52 * CALL NAP(88) 53 * WRITE(!6,104) BB 54 F0RMATI/1X.12HMAP BOUND = ,14) 55 * 104 T5 = CL0CK(T5)/60.0 56 CALL ST0RE(1,1) 57 T6 = CLOCK(0.0) 60 CALL RES 61 * IF(NA.EQ.O) GO TO 1 62 * CALL CH0P(1,NA,B) • 65 *  * t * .* * * * * *  +  *  * * * *' t  t  * *. * * * * * *  .  '  .,  FORTRAN SOURCE LIST  N.R.ARDEN  ISN  „  66 67 70 71 72 73 74 75 76 77 102 103  SOURCE STATEMENT  * * * * * * 107 * * 1  T6 = CL0CMT6)/60.0 PCE = 100.0*FL0AT(NFE)/FL0AT(NE) POINT = 2 T8 = CLOCK(0.0) CALL MAIN(B) T8 = CLQCK(T8)/60.0 WRITE(6,107) NE J.NA.NFE NFA,PCA.PCE,Tl.T2,T3,T4,T5,T6,T7,T8,T9 FORMAT(IX,13,15,316,2F9.2,9F9. 3) STOP CONTINUE STOP END t  USER FUNCTION SUBPROGRAM REFERENCES C L O C K M X  NO MESSAGES FOR ABOVE IME 14HRS L7MIN 5C.0SEC  ASSEMBLY  t  11/17/68 * * * * t *  PAGE  2  FORTRAN SOURCE LIST  N.R.ARDEN I SN 0 1 2 3 4 5 6  7 10 11 13 14 15 16 17 20  SOURCE STATEMENT  * * * * * * * * *  21 * 22 23 * 26 27 * 30 31 * 32 * 35 * 36 37 * 40 *  41 42  * * * *  $I8FTC SB1 SUBROUTINE PERT(C,ATC,AMC,MXN) INTEGER C,ATC,AMC,MXN,N,NO,JNT,I,J,K INTEGER NA,A,AT,AM,AS,BS, NE,ES , ATAB,CPL,RS,RTAB,RLST,FE,NFE COMMON /CO/NA,A(400,2),AT(400) ,AM.( 400) , AS( 400) ,B S(400),NE » 1 ES(150,2),ATA 8(700, 2),CPL,RS(400),RTA8(150,3), 2 RLST(2300),FE(150),NFE COMMON /CI/ NM50) REAL X,Y,YY,YYY,Z,ZZ  c  11/17/68  PAGE  3  #  * * •* * *  *  * C INITIALIZE C * NA = 0 * DO 2C0 1 = 1,NE 200 N(I) = 0 X =CLOCK(0.0) X = RAND(X) * Y =C YY = A TC YYY = AMC NO = NE - 1 C C SET UP A RANDOM NUMBER OF RANDOM ACTIVITIES LEAVING ALL EVENTS - EXCEP * T c FOR THE FINAL EVENT.  c  c c c c  DO 201 1=1,NO J = 1.0 + Y*RAND(0.0) IF(J.GT.(NE-I)) J = NE - I Z = I +1 ZZ = NE - I DO 201 K=1,J NA = NA + 1 IF(NA.GT.MXN) RETURN A(NA,1) = I A{NA,2) = Z + ZZ*RAND(0.0) AM(NA) = 1.0 + YYY*RAND(0.0) AT ('NA). = 1.0 + YY*RAND(0.0)  * *  _  * * * *  _  _  _  *  DUMMY EVENTS COULD BE INTRODUCED BY SUBSTITUTING AT(NA) = YY*RAND(0. * 0) AND IF(AT(NA).EQ.O) AM(NA) = 0. * JNT = A(NA,2) N.IJNT), = N(JNT) + 1  201 C C ADD A RANDOM ACTIVITY INTO ANY EVENT - EXCEPT FOR THE INITIAL EVENT C NOT THE TERMINAL EVENT OF ANY RANDOM ACTIVITY ASSIGNED SO FAR. * C 45 * DO 202 1=2,NO 46 * IF(N( I ) .GT.O) GO TO 202 NA = NA + 1 51 * 52 * IF(NA.GT.MXN) RETURN 55 A{NA , 2) = I 56 A(NA,1) = 1.0 + FLOAT(I-1)*RAND(0.0) 57 AM(NA) = 1.0 + YYY*RAND(0.0) 60 AT(NA) = 1.0 + YY*RAND(0.0)  * *  *  * WHICH IS  *  *  *  *  *  .  .  .  ,  FORTRAN SOURCE LIST SB1  N.R.ARDEN  SOURCE STATEMENT  I SN 61 * 202 63 * 64 *  CONTINUE RETURN END  USER FUNCTION SUBPROGRAM REFERENCES CLOCK  RAND  NO MESSAGES FOR ABOVE ASSEMBLY I ME 14HRS 18MIN 08.6SEC  11/17/68  PAGE  4  N.R.ARDEN ISN  _  .  0 1 2 3 4  FORTRAN SOURCE LIST SOURCE  11/17/68  STATEMENT  * SIBFTC SB3 SUBROUTINE CP(FA,NFA) * INTEGER NFA,FA(NFA),WL,N,T,SM,NIN,F, I , J , K, X , Y, Z INTEGER NA,A,ATfAMfAS BS,NE,ES,ATAB,CPL,RS,RTAB,RLST,FE,NFE * * COMMON /C0/NA,A(400,2),AT(400),AM(400),AS(400),BS(400),NE, 1 ES(150,2),ATAB(700,2),CPL,RS(400),RTAB( 150,3), * 2 RLST(2300) ,FE(150),NFE CONMON /CIV NIN(150) 5 6 COMMON /C2/ WL(400,2) 7., COf/MQN /C3/ F(400,2) * C C INITIALIZE C 10 T = 0 11 * N = 0 12 FS(1,1) = 0 13 DO 300 I=1,NA 14 * F(1,1) = BS(I) 15 * F( I , 2 )= 0 16 IF(8S(1I ) .GT.O) F(I,2) = 1 21 * 300 CONTINUE 23 * DO 310 1=1,400 24 * 310 R S( I ) = 0 * C C ADD THOSE ACTIVITIES LEAVING THE INITIAL EVENT TO CURRENT LIST. C 26 DO 302 J=1,NE X = ATAB(J,2) 27 NIN(J) = ATAB(X,2) 30 * 302 32 Y = ATAB(1.1) X = ATAB(Y , 1 ) 33 34 DO 303 J=1,X 35 N = N +1 36 I = Y +J 37 W L ( N , 1 ) = ATAB(I,1) 40 Z = WL(N,1) 41 IF(F(Z,2).EQ.0) RS(1) = RS(1) + AM(Z ) 44 AS(Z) = T 45 * 303 WL(N,2) = AT(Z) + BS(Z) C C FIND THE SCONEST TIME AT WHICH EITHER AN ACTIVITY ENDS OR THE FORWARD C OF AN ACTIVITY ON THE CURRENT LIST BECOMES ZERO. * C 47 * 314 SM = WL(1,2) 50 Z = WL ( 1 , 1 ) IF(F(Z,2).EQ.l) SM = F(Z,1) 51 54 * DO 304 J=1,N 55 Z = WL(J,1) IF(F(Z,2).EQ.1.AND.F(Z,1).LT.SM) SM = F(Z,1) 56 IF(F(Z,2).EQ.0.AND.WL(J,2).LT.SM) SM = WL(J,2) 61 * 64 304 CONTINUE C * C ADJUST ALL TIMES AND ADD NEW ACTIVITIES WHERE POSSIBLE C 66 J = T +1 f  * * * *  *  * *  * * * * *  * * * * * *  * * * * * *  * * * * * * *  SLACK  PAGE  6  FORTRAN SOURCE LIST SB3  N.R.ARDEN ISN 67 70 73 74 76 77 ICO 1C1 102 105 1C6 111 113 114 115 120 121 122 125 126 131 1 32 133 134 135 136 137 14 0 141 143 144 147 150 151 153 154 157 162 163 166  167 170 171 172 173 175 176 177 200 201 2C2  11/17/68  PAGE  7  SOURCE STATEMENT 4  * * * 311 * * *  * * 4  *  306 313  * * * * * 4  f  * * *  *  309 308  307 * * 305 4  - J + SM - 1 IF(K.GT.400) GO TO 326 DO 311 I=J,K R S{I) = RS(J) T = T + SM DO 306 J=1,N WL(J,2) = WL(J,2) - SM Z = WL(J.l) IF(F(Z,2).EQ.0) GO TO 306 F(Z,1) = F(Z,1) - SM IF(F(Z,1).EC.0) F(Z,2) = 0 CONTINUE J = 1 Z = WL(J.l) IF(WL(.Jt2),.GT.O) GO TO 305 X = A( Z,2) NIN(X) = NIN(X) - 1 IF(NIN(X).GT.O) GO TO 308 ES(X,1 ) = T IF(X.EG.NE) GO TO 301 Y = ATAB(X » 1 ) I = ATAB(Y , 1) DO 309 K = l I N = N + 1 Z = Y + K WL(N,1) = ATAB{Z 11) Z = WL(Ntl') AS(Z) = T WL(N,2) = AT(Z) + BS(Z) N =, N - 1 IF(J.GT.N) GC TO 314 DO 307 K=J,N WL(K,1) = WL(K+1,1) WL(K,2) = WL(K+li2) GO TO 313 IF(T.GE.400) GO TO 326 IF(F(Z,2).EQ.O) RS(T+1) = RS(T+1) + AM(Z) J = J + 1 IF(J.LE.N) GO TO 313 GO TO 314 K  4 4 4  *  4 4  * 4 4 4  ^  * *  *  :  4  *  * * 4 4  * * 4  *  * * * * * 4 * C * C START RETURN SCAN BY ADDING THOSE ACTIVITIES ENTERING THE FINAL EVENT * * * C CURRENT LIST. * C * * 301 N = 0 * ES(NE,2) = 0 * DO 315 J=1,NE * X = ATAB(J,1) * * 315 NIN(J) = ATAB('Xtl) 4 * Y = ATAB(NE,2) 4 4 X = ATAB(Y » 2) * DO 316 J=l ,X * 4 N = N + 1 4 Z = Y +J WLlN.l) = ATAB(Z,2)  TO  N.R.ARDEN I SN  :  SOURCE STATEMENT  FORTRAN SOURCE LIST SB3  203 * Z = WL(Ntl) 204 * AS(Z) = T - AS(Z) - AT(Z) - 8S(Z) 205 * 316 WLIN,2) = Mil) + BS(Z) c C FIND LATEST TIME AT WHICH AN ACTIVITY STARTS * C 20 7 322 SM = WL(lr2) 210 DO 317 J=1,N 211 * IF(W L ( J »2). L T. S M) SM = WL(J,2) 214.. 31.7 CONTINUE C C ADJUST TIMES AND ADD NEW ACTIVITIES WHERE POSSIBLE. C 216 T - T - SM 217 * DO 312 J=1,N 220 312 WLU,2) = WL(J,2) - SM 22 2 * J = 1 223 324 IF(WL<J,2).GT.O) CO TO 318 226 Z = WLIJtl) 227 X = A(Z,1) 230 NIN(X) = N1N(X) - 1 231 IF(NIN(X).GT.O) GO TO 319 234 E S ( X , 2 )= T - ES ( X , 1 ) 235 * IF fX. E.Q.I). GC TO 320 240 Y = ATAB(X,2) 241 I = ATA8(Y,2) 242 DO 321 K=1,I 243 N = N + 1 244 * Z = Y +K 245 WL(N,1) = ATAB(Z t 2) 246 * Z = WL(N,1) 247 AS(Z) = T - AS(Z) - AT(Z) - BStZ) 250 321 WLi'N,2) = AT(Z) + BS(Z) 2 52 319 N = N - 1 253 * IF(J.GT.N) GC TO 322 256 DO 323 K=,J,N 257 * WUK,1) = WL(K+1,1) 260 * 323 WL(K,2) = WL(K+1,2) 2 62 * GO TO 324 318 J = J + 1 263 264 IF(J.LE.N) GC TO 324 267 GO TO 322 * C C ADD THOSE ACTIVITIES WITH ZERO SLACK TO A LIST. C 320 NFA = C 270 271 * a DO 325 1=1,NA 272 * IF (ASM ) .GT.O) GC TO 325 NFA = NFA + 1 275 276 * FA(NFA) = I 277 325 CONTINUE 3C1 RETURN 302 326 NA = 0 303 * RETURN 3C4 END  11/17/68  t * * * * * * * * * * T-  * *  * * * *  * * * * * * * * +  *  PAGE  8  FORTRAN  N.R.ARDEN ISN  SOURCE LIST  11/17/68  SOURCE STATEMENT 0 1 2 3 4  4 ' SIBFTC SB4 * SUBROUTINE RES * INTEGER WL,F{150,3),CRN,NF,I,J,K,X,Y,Z * INTEGER NA,A,AT,AM,AS,BS,NE,ES,ATA8,CPL,RS,RTAB,RLST,FE,NFE * COMMON /C0/NA,A(400,2),AT(400),AM{400),A S(400),BS(400),NE, * 1 ES(150,2),ATAB(700,2),CPL,RS(400),RTAB(150,3), 4 2 RLST ( 2300 ) , FE ( 150) , NFE 5 * COMMON /C2/ WL(400,2) 6 * NFE = 0 7 * DO 400 1=1,NE ' 10 * IF(ES( 1,2).GT.O) GO TO 400 13 4 NFE = NFE + 1 14 4 FE(NFE) = I . 15 4 400 CONTINUE ' 17 4 DO 401 1=2,NFE 20 4 K = NFE - I + 1 21 4 DO 401 J=1,K 22 4 X = FEU) 23 4 Y = FEU+i) 24 4 IF(ES(Y,1).GE.ES(X,1)) GO TO 401 27 4 F E U + 1) = X 30 4 FEU) = Y 31 4 401 CONTINUE 34 4 NF = NFE - 1 35 4 DO 402 I = 1,NF 36 4 DO 402 J= 1 , 3 37 4 402 F(I,J) = 0 42 4 DO 403 I=1,NA 43 4 X = A( I ,1) 44 4 X = ES(X,1) 45 4 WL(I,1) = CRN(X,1) 46 4 X = WL(I,1) 47 4 F ( X , 1) = F ( X , 1 ) + 1 50 4 Y = AM,2) 514 Y=ES(Y,1)+ES(Y,2) 52 4 WL(I ,2 ) = CRN(Y,2) 53 4 Y = WL(I ,2) 54 4 IF(Y.NE.X) F(Y,2) = F(Y,2) + 1 57 4 IF(Y.LE.X+1) GO TO 403 62 4 X = X + 1 63 * Y = Y - 1 64 4 DO 405 J=X,Y 65 4 40 5 F(J,3)=F(J,3) + 1 67 4 403 CONTINUE 714 J = 1 72 4 DO 406 1 = 1,NF 73 4 DO 406 K=l,3 74 4 IF(J.GT.2300) GO TO 412 77 4 RLST( J ) = F ( I ,K) ICO 4 RTAB(I ,K) = J 101 4 406 J = J + RLSTU) + 1 104 4 DO 408 1=1,NA 10 5 4 X = WL ( I , 1 ) 1C6 4 I = RTAB(X,1 ) 107 4 Z = Z + RLST(Z) - F(X,1) + 1 :  4 * 4 4 * *  1  4 4 * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 +_ 4 4 4 4 4 4_ 4 * 4 4 4_ 4 4 4 4 4 _4_ 4 4 4 4 4 4 4 4 4 *  4  4_  4  PAGE  10  N.R.ARDEN ISN 110 111 112 * 113 t. 116 117 120 121 * 122 * 12 5. 126 127 130 * 131 132 133 135 * 137 140 142 * 143 * 144 ±. 145 t 146 14 7 * 150 * 153 154 155 156 * 157 160 *  FORTRAN SOURCE LIST SB4  11/17/68  SOURCE STATEMENT  409 408 410  411 412 490  RLST(Z) = I F(X,1) = F(X,1) - 1 Y = WL(I ,2 ) IF(Y.EG.X) GO TO 408 7. = R T A B ( Y , 2 ) Z = Z + RLST(Z) - F (Y , 2) + 1 RLST(Z) = I F.(Y,2) = F(Y,2) - 1 IF(Y.LE.X+1) GO TO 408 X = X + 1 Y = Y - 1 DO 409 J=X,Y Z = RTAB(J,3) Z = Z + RLST(Z) - F(J,3) + 1 RLST(Z) = I F(J,3) = F(J,3) - 1 CONTINUE DO 410 1=1,CPL RS( I ) = 0 DO 411 I=1,NA X = A(1,1) X = E S ( X , 1 )+ 1 Y = A(I,2) Y = ES(Y,1) + ES(Y,2) DO 411 J=X,Y R'S(J) = RS(J) +AM ( I ) RETURN NA = 0 WRITE(6,490) FORMAT(//IX,25HARRAY - RLST RETURN END  *  * * *  *  * * * *  * * t  * *  OVERFLOWED)  USER FUNCTION SUBPROGRAM REFERENCES I  CRN  •I  NO MESSAGES FOR ABOVE ASSEMBLY ME 14HRS 19MIN 15.6SEC  i  PAGE  11  N.R.ARDEN I SN  FORTRAN SOURCE LIST  1 1/17/68  SOURCE STATEMENT  0 * $IBFTC SB5 INTEGER FUNCTION MX(G,J,K) 1 * 2 * INTEGER J » KtG'( K ) ,1 MX = 0 3 * DO 700 I-J,K IF{G(I).GT.MX) MX = G(I) 5 * CONTINUE 10 * 700 RETURN 12 * END 13  NO MESSAGES FOR ABOVE ASSEMBLY ME 1AHRS 1.9MIN 30.1SEC  * ; = N 4  4  PAGE  12  N.R.ARDEN ISN  FORTRAN SOURCE LIST 0 1 2 3 4  i  '  * * *  *  i  5 * 6 * 7 * 1  11/17/68  PAGE  13  SOURCE STATEMENT  0  12 13 14 15 20 21 22 23 24 26 30 31 34 35 36 37 40 41 44 45 46 47 50 51 54 55 60 63 64 65 70 73 75 76 10.1 104 1C5 1C6 107  *  * * * * * * * * * *  * *  * *  *  *  SIBFTC SB6 SUBROUTINE BOUND(B) INTEGER B,M,RT,FB,RMAX,BN,R,MX,X,Y,Z,I*J INTEGER NA,A,AT,AM,AS,BS ,NE,ES,ATAB,CPL,RS,RTAB,RL ST, FE,NFE COMMON /CO/NA,A(400,2),AT(400),AMI 400),AS(400),BS(400). NE, 1 ES( 150,2 ) ,ATAB( 700, 2) , CPL , RS ( 400 ) , RT A B ( 1.50,3) , 2 RLST(2300),FE(150),NFE COMMON /C4/ RT(400) M =0 DO 600 1=1,CPL RT(I) = 0 600 DO 601 1 = 1 , NA M = M + AM(I)*AT(t) X = ATU) - A S ( I ) IF(X.LE.O) GC TO 601 Y = A( I , 1 ) Y = ES(Y,1) + A S U ) + 1 X = Y + X -1 DO 602 J=Y,X RT ( J ) = RT ( J ) + AM ( I ) 602 601 CONTINUE FB = M/CPL IF(FL0AT(F8).LT.FL0AT(M)/FLOAT(CPL)) FB = FB + 1 J =1 RMAX =. MX(RT,J,CPL) R =0 DO 603 I=1,NA X = AT(I) - AS(I) IF(X.GT.O) GC TO 603 Y = A(1,1) Y = ES ( Y , 1 )+1 X = Y + AS(I) BN = RMAX + AMU) 606 Z = MX(RT, Y,Y + A T U ) - l ) + AMU) IF(Z.LT.BN) BN = Z 604 Y = Y +1 IF(Y.EQ.X) GO TO 605 IF(RT(Y).EQ.RT(Y-1)) GO TO: 604 GO TO 6G6 Z = MX(RT,X,X + A T U ) - l ) + AM(I) 605 IF(Z.LT.BN) BN = Z IF(BN.GT.R) R = BN 603 CONTINUE B = FB IF(RMAX.GT.B) B = RMAX IF(R.GT.B) B = R WRITE(i6,610) M, CPL, FB, RMAX, R ,B F0RMAT(1X,6I6) 610 RETURN EMC  USER FUNCTION SUBPROGRAM REFERENCES  * * * * * * * * * * * * * * * * * *  * *  * *  '  „,  N.R.ARDEN ISN  FORTRAN SOURCE LIST  11/17/63  PAGE  15  SOURCE STATEMENT  0 $IBFTC SB7 SUBROUTINE CHOP(J,K,B) 1 * INTEGER 8,L,N,II,C ,X,Y,,Z,XX,YY,ZZ,I 2 3 INTEGER NA,A,AT,AM,AS,BS,NE,ES ,ATAB, CPL,RS,RTAB,RLST,FE,NFE COMMON /CO/NA,A(400,2),AT(400),AM(400),AS(400),BS(400),NE, 4 1 ES(150,2),ATAB(700,2),CPL ,RS(400),RTAB(150,3), 2 RLST(2300),FE(150),NFE 5 * 806 N = 0 6 * DO 800 I=J.K IF{AS(I).EQ.O) GC TO 800 7 * Z = A(1,1) 12 ZZ = A(I,2 ) 13 X = ES(Z,1) + BS(I ) + 1 14 15 * Y = X + AS ( I) - 1 .XX = X + AT ( I ) 16 YY = Y + AT(I) 17 IF(A S(I)-E S(Z,2).LE.0) GO TO 802 20 * 23 * DO 801 L=X,Y 24 * IF(RS(L).GT.B) GC TO 802 27 801 CONTINUE IF(AS(I)-ES(Z,.2).LT.ES(ZZ,2) ) GO TO 802 31 * 34 * N = 1 35 AS( I ) = ES(Z ,2) 36 * C = 1 37 * CALL EVBAL(ZZ,C) 40 * ES(ZZ,2) = C 41 * IF(C.LE.O). ES(ZZ,2) = 0 XX = XX + ES(Z,2) 44 * 00 803 L=XX,YY 45 # 46 t 803 RS(L) ~ RS(L) - AM(I) GO TO 800 50 IF(AS(I)-ES(ZZ,2).LE.O) GO TO 800 802 51 54 DO 804 L=XX,YY 55 IF(R.SIL).GT.B). GO TO 800 CONTINUE 60 * 804 62 * IF(AS(I)-ES(ZZ,2).LT.ES(Z,2)) GO TO 800 65 * N = 1 66 * BS(I) = AS( I ) - ES(ZZ,2) 67 AS(I) = ES(ZZ,2) 70 * C = 1 CALL EVBAL(Z,C) 71 * 72 * ES(Z,2) = C IF(C.LE.O) ES(Z,2) = 0 73 * Y = Y - ES(ZZ,2) 76 DO 805 L=X,Y 77 RS(L) = RS(L) - AM(I) 100 * 805 800 CONTINUE 1G2 104 * IF(N.EC.l) GO TQ 806 107 RETURN 110 i END  NO MESSAGES FOR ABOVE ASSEMBLY IE 14HRS 20MIN 07.0SEC  *  * * * * *  *  * *  * * *  * * * t •. *• * * *  J  N.R.ARDEN I SN  _  FORTRAN SOURCE LIST SOURCE  11/17/68  STATEMENT  0 $IBFTC SB8 * SUBROUTINE EVBALMtO 1 * INTEGER I ,C,X,.Y,Z, J,K,L, MIN,MAX 2 INTEGER NA,A,AT,AM,AS,BS,NE,ES,ATA8,CPL,RS,RTAB,RLST,FE,NFE 3 * COMMON /C0/NA,A(400,2),AT(400),AM{400),AS(400),BSI 400),NE, 4 * * I ES(150,2),ATAB(700,2),CPL,RS(400),RTAB(150,3), * * 2 RLST(2300),FE(150),NFE * * C * * C INPUT C CASE 1..C IS SET TO ZERO IF CHECKING FEAS IBLE ACTIV ITY SLACKS ABOUT TH * E EVENT. * * C CASE 2..C IS SET TG ONE IF UPDATING EVENT SLACK. * C C OUTPUT * C CASE 1..IF C POSITIVE,EVENT SLACK CANNOT BE REDUCED TO ZERO. * C CASE 2..C INDICATES UPDATED VALUE OF EVENT SLACK. * * C IF,( I .EQ. l.OR. I .EQ.NE ) GO TO 1002 5 * X = ATAB(1,1) 10 * * Y = ATAB(X,1) 11 * MIN = CPL 12 * DO 10C0 J=1,Y 13 * K = X +J 14 . 1. 15 * L = ATA8 ( K» 1 ) 16 Z = BS(L) + AS(L)*(l-C) 17 IF(Z.LT.MIN) MIN = Z 22 1000 CONTINUE * 24 * X = ATAB(I,2) 25 * Y = ATAB(X,2) * MAX = C 26 * 27 * DO 1001 J=1,Y K = X +J 30 * * L = ATA8(K,2) 31 * Z = A ( L , 1) 32 Z = ES(Z,1) + BS(L) + AT(L) + AS(L)*C 33 * IF(Z.GT.MAX) MAX = Z 34 *j CONTINUE 37 * 1001 C = MAX - ES(I,1) - MIN 41 * 42 * RETURN C = 0 43 * 1002 44 * RETURN 45 END  NO MESSAGES FOR ABOVE ASSEMBLY :ME 14HRS 20MIN 23.6SEC V  PAGE  16  N.R.ARDEN ISN 0 * 1 * 2 * 3 * 4* * * 5 * 6 * .7 * 10 * 13 * 14 * 17 * 20 * 2 1 * 22 * 23 * 24 * 25 + 30 * 31 * 32 * 33 * 36 * 37 * 40 * 43 * 44 * 45 * 50 * 51 # 52 * 53 * 55 * 56 * 57 * 61 * 62 * 63 * 64 * 67 * 70 * 73 * 74 * 75 * 76 * 77 * ICO * 1C1 * 102 * 103 * 1C6 * 107 * 110 *  FORTRAN SOURCE SIBFTC  SOURCE  LIST  11/17/68  PAGE  STATEMENT  SB9 SUBROUTINE ESRED(I,NAC,ACT,PT) INTEGER I ,.NAC , AC T ( 400 ) , P T , X , Y, 7., XX , Y Y , Z Z., J , K, L, KK , I I X I N T E G E R N A , A » A T , A M , A S , B S , N E , E S , A T A 8 , C P L , RS, R T A B , R L S T , FE_,.NFE COMMON / C O / N A , A ( 4 0 0 , 2 ) , A T ( 4 0 0 ) , A M l 4 0 0 ) , A S ( 4 0 0 ) , B S ( 4 0 0 ) , N E , 1 ESU50,2),ATAB(700,2),CPL,RS(400),RTAB<150,3), 2 RLST(2300),FEU50),NFE PT = 0 X = A(1,1) Y - A ( I , 2) ' I F ( E S ( Y , 2 ) . E Q . O ) GO TO 1108 Z = ES(X,1) + BS(I) + A T U ) I F ( Z . L E . E S ( Y , 1 ) ) GO TO 1 1 0 0 XX = A T A B ( Y , 1 ) ' YY = A T A B ( X X , 1 ) Z = Z - ES(Y,1) DO 1101 J=1,YY K = XX + J L = ATAB ( K , 1 ) I F ( B S ( D . G E . Z ) GO TO 1101 K = BS(L) + ES(Y,1) + 1 KK = E S ( Y , 1 ) + I AS(L) - AS(L) + BS(L) - Z I F ( A S ( D . L T . O ) GO TO 1109 BS(L) = Z ZZ = A ( L , 2 ) I F ( E S ( Z Z , 2 ) . E Q . 0 ) GO TO 1102 C = 0 CALL EVBAL(ZZ,C) I F ( C . L E . O ) GC TO 1102 1109 PT = 1 RETURN 1102 DO 1103 II=K,KK 1-1.03 R S ( I I ) = R S ( I I ) - AM(L) NAC = NAC + 1 ACT(NAC) = L 1101 CONTINUE 1100 C = 1 CALL EVBAL(Y,C) ES(Y,2) = C IF(C.LE.O) ES(Y,2) = 0 1108 Z = BSU) + ASU ) I F ( Z.GE.ES (X,2 ) ) GO TO 1 1 0 4 XX = A T A B ( X , 2 ) YY = A T A B ( X X , 2 ) Z = Z + ES(X,1) DO 1105 vl = l , Y Y K = XX + J L = AT AB ( K , 2 ) ZZ = A ( L , 1 ) KK = E S ( Z Z , 1 ) + B S ( L ) + A T ( L ) + A S ( L ) I F ( K K . L E . Z ) GO TO 1105 K = Z + 1 A S ( L ) = A S ( L ) + Z — KK I F ( A S ( L ) . L T . O ) GO TO 1109  * *  * * * * * * * * * t t * * * * * £ * * * * * * * * * * * * * * * * * 4_ t * * * +  f_  * * * * # ±_ * * * * 4  *  17  FORTRAN SOURCE LIST SH9  N.R.ARDEN I S'N 113 116 117 120 123 124 125 126 130 131 132 134 137 140 141 H2  11/17/68  SOURCE STATEMENT * * * * 1106  * 1107 *  * 1105  * 1104 * * * *  146 *  IF(ES(ZZ,2).EQ.O) GO TO 1106 C = 0 CALL EVBAL(ZZ,C) I F ( C L E . O ) GO TO 1106 PT = 1 RETURN DO 1107 II=K«.KK RS( II) = RSI II) " AM{L) NAC = ftAC + 1 ACT(MAC) = L CONTINUE IF(ES(X,2).EG.O) RETURN C = 1 CALL EVBAL(X,C) ESIX,2) = C lF{C.LE,p) ES(X,2) = 0 RETURN END  NO MESSAGES FOR ABOVE ASSEMBLY ME 14HRS 20M-IN 43.7SEC  * * * * *  *  PAGE  18  N.R.ARDEN ISN  11/17/68  FORTRAN SOURCE LIST  PAGE  19  SOURCE STATEMENT  $IBFTC SB 10 0 SUBROUTINE RSMX{MX,RB,RE,MXS) 1 * INTEGER MX,RB,RE,MXS,I 2 * INTEGE R N A,A,AT,AM,AS, B S , NE» E S i AT AB, C PL , RS , RT AB , RL ST , FE , NF E. 3 COMMON /CO/NA,A(400,2),AT(400),AM(400),AS(400),BS(400),NE, 4 1 ES(150,2),ATAB(700,2),CPL,RS(400),RTAB(150,3), * 2 RLST(2300),FE(150),NFE 5 MX = -1 MXS = - 1 6 7 A. 1 = 0 1200 10 1 = 1 + 1 IF(I.GT.CPL) RETURN 11 14 IF(RS(I).LE.MXS) GO TO 1200 1203 IF(RS( I ).LE.MX) GO TO 1201 17 22 * RB = I - 1 MXS = MX 23 24 MX = RS(I) 25 1202 RE = I 26 * 1 = 1 + 1 IF(I.GT.CPL) RETURN 27 * IF (RSM ) .EQ.MX) GO TO 1202 32 35 .. GO. TO 1203 1201 MXS = RS(I) 36 37 * GO TO 1200 END 40  * * * *  * * *  * *  NO MESSAGES FOR ABOVE ASSEMBLY  00 QH280  V  B END  14HRS 21MIN 01.3SEC  660 CARDS READ  0 OUTPUT CARDS  748 LINES PRINTED  0 OBJ. DECK PUNCHED  N.R.ARDEN ISN  ,  !  ! ;  SOURCE STATEMENT  FORTRAN SOURCE LIST  11/17/68  0 * $IBFTC S B U SUBROUTINE RELACT(RB,RE,N,F) 1 * INTEGER RB ,RE, N, CRN,X,Y,Z,XX,YY,ZZ,I,J,K,F{400,2) 2 INTEGER NA , At AT,AM , AS, BS ,NE, ES , ATAB , CPL,, RS ,RTAB , RL ST , FE , NFE .3 COMMON /CO/NA,A(400,2),AT1400), AM(400), AS(400),BS(400),NE, 4 1 ES(150,2),ATA 8(700, 2),CPL,RS(400),RTAB(150,3), * 2 RLST(2300),FE(150),NFE 5 X = CRN(RBfl) 6 Y = CRN(RE,2) 7 N = 0 10 DO 1400 I=X,Y DO 1400 J=l,3 11 IF(I.GT.X.AND.J.GT.1) GO TO 1400 12 XX = RTAB(X,J) 15 YY = RLST(XX) 16 IF(YY.EQ.O) GO TO 1400 17 DO 1402 K=l,YY 22 Z = XX + K 23 * 24 * Z = RLST(Z) IF(AS(Z).EQ.O) GO TO 1402 25 30 N = N +; 1 31 F ( N , 1) = Z 32 * 1402 CONTINUE 34 * 1400 CONTINUE IF(N.EQ.O) RETURN 37 * DO 1401 1=1,N 42 X = F(1,1) 43 Y = A(X,U 44 * Z = ES(Y,1) + BS(X) 45 * 46 * ZZ = Z + AT(X) + AS(X) C C CODE=O...NO SLACK AVAILABLE IN RANGE. C C0DE=1...SLACK ONLY AVAILABLE AT TERMINAL END. C C0DE=2...SLACK ONLY AVAILABLE AT INITIAL END. * C C0DE=3...SLACK AVAILABLE AT BOTH ENDS. C F( I ,2)= 0 47 IF(Z.GE.RE.OR.ZZ.LE.RB) GO; TO 1401 50 IF(Z+AS(X).GT.RB) F(I,2) = 2 53 * IF(ZZ-AS(X).LT.RE) F(I,2) = F(I,2) + 1 56 61 * 1401 CONTINUE J = 0 63 * DO 1403 1=1,N 64 I F ( F ( I ,2).EQ.O) GO TO 1403 65 J = J +1 70 F ( J , l ) = F( I ,1) 71 * 72 F(J,2) = F(I , 2) 73 * 1403 CONTINUE 75 M =J RETURN 76 END * 77  1  PAGE  2  < * t  *  * * * * * *  *  _  * * *  *  *  * * *  >  N.R.ARDEN ISN 0 1 * 2 3 4 5 * 6 7 * 10 # 11 . . . ± 12 * 15 16 17 21 22 23 24 * 27 30 32 * 33 34 35 40 42 * 4 3* 44 4 5* 46 * 47 52 53  FORTRAN SOURCE LIST  11/17/68  SOURCE STATEMENT $IBFTC SB 12 SUBROUTINE SELECT(N, P ,CH) ,NCH,X INTEGER N,.F,P,CH(1000,3) ,I,J,K,M COMMON /C2/ F(400,2) COMMON /C3/ M(800) CH(1,2) = P NCH = 1 M ( 1 )= 1 M(2) = 2 K =2 1500 IF(K.GE.P) GO TO 1501 K = K +1 DO 1502 I=K,P 1502 M(I) = M(I-l) + 1 1501 DO 1503 1=1,P X = M(I) NCH = NCH + 1 IF(NCH.GT.1000) GO TO 1505 CH(NCH,1) = F(X,1) 1503 CHINCH, 2) = F(X,2) DO 1504 1 = 1 , P K = P - I +1 M(K) = M(K) + 1 IF(M(K).LE.N-P+K) GO TO 1500 1504 CONTINUE CH(1,1) = (NCH-D/P RETURN 1505 WRITE(6,1590) P,N 1590 FORMAT(//IX,36HARRAY CH OVERFLOWED TRYING TO SELECT,1 3,4HFR0M,I 5) CH( 1,1) = (NCH - 2)/P - 1 IF(CH(:1,1 ) .LT.O) CH(1,1)= 0 RETURN END - •-  NO MESSAGES FOR ABOVE ASSEMBLY ME 14HRS 13MIN 52.3SEC  * * * * * * * * * * * *  *  * * * * *  *  PAGE  4  N.R.ARDEN ISN  FORTRAN SOURCE LIST SOURCE  11/17/68  PAGE  5  STATEMENT  * 0 * $IBFTC SB13 SUBROUTINE FEAS(CH,R6,RE,RED) 1 INTEGER R8,RE,RED,N, C N C H , VAR ,M,NN ,X ,Y,Z,XX,YY,ZZ,I,J,K 2 * * 1 iCH(1000,3) * , ATAB ,RS ,RT AB ,RLST,FE,NFE , CPL INTEGER NA,A,AT,AM,AS,BS,NE,ES 3 * BS(400),NE, AS(400), ,AM(400), COMMON /CO/NA,A(400,2),AT(400) 4 * 1 ES(150,2),ATAB(700, 2) ,CPL,RS(400) , RTAB ( 1.5 0,3) , * 2 RLST(2300),FE(150), NFE * COMMON /C3/ M(800) 5 COMMON /C4/ VAR(400) ...... 6 CH(1,3 ) = 0 7 N = CH(1,1) 10 C = CH(1,2) 11 * X = RE - RB 12 * NCH = N 13 * IFJN.EQ.l ) GO TO, 1620 14 DO 1600 I=1,N 17 * DO 1601 J=1,X 20 VAR(J) = 0 21 * 1601 * DO 1602 J=1,C 23 * 1602 MU) = 2 24 .* . CALL TEST(C,X,I,CH,RB,M) 26 * DO 1603 J=1,C 27 * 1603 M(J) = 1 30 * CALL TEST(C,X,I,CH,RB,M) 32 * * DO 1604 J=1,X 33 * * IF(VAR(J).LT.RED) GO TO 1615 34 „, „ ....... 1604 CONTINUE 37 * DO 1606 J=1,C 41 * 1613 * Y = (I-l)*C + J + 1 42 * IF(CH(Y,2).EQ.M(J)) GO TO 1610 43 * * CONTINUE 46 * 1606 * DO 1607 J=1,X 50 * * 51 * 1607 VAR(J) = 0 CALL TEST(C,X,I,CH,RB,M) 53 * ± * DO 1608 J=1,X 54 * IF{VAR(J).LT.RED) GO TO 1610 55 : 1608 CONTINUE 60 * NCH = NCH + 1 62 * * DO 1609 J=1,C .6 3 64 * Y = (I-1)*C + J + 1 * Z = (NCH-1)*C + J + 1 65 * XX = CH(Y,1) 66 IF(Z.GT.1000) GO TO 1605 67 * i i CH(Z,1) = CH(Y,1) 72 73 CH(Z,2) = CH(Y,2) CH(Z,3) = (MU)*2-3)*AS( XX) 74 4 * Z Z = A ( X X,. 1 ) 75 K = RE — ES(ZZ,1) - BS(XX) 76 * , = - K IF(M(J ) .EQ.l.AND.K.LT.AS(XX))CH ( Z3) 77 + AS(XX) K = ES(ZZ,1) + BS(XX) + AT(XX) - RB 102 * * CH(Z, K IF(M(J).EQ.2.AND.K.LT.AS(XX)) 3) = 103 * * 1609 CONTINUE 106 * 110 * 1610 DO 1611 J=1,C 11 1 * K = C - J +1  \  ;..;  „... .  . ....  .  .... .—  .. .  ....  _„  J  N.R.ARDEN ISN  :  112 * 115 117 * 120. * 121 124 * 125 126 130 4 131 4 132 4 133 4 135 137 * 140 141. A 142 4 145 * 146 * 147 * 150 151 * 152 4 153 * 154 156 * 157 4 160 * 161 4 162 163 4 164 * 165 4 166 171 * 172 * 173 * 176 177 200 4 202 * 203 205 2C6 207 4 212 214 * 215 * 216 * 217 220 221 * 224 22 5 *  FORTRAN SOURCE  SOURCE  I F ( M ( K ) . . E Q . l ) GO TO 1612 CONTINUE GO TO 1615 1612 .. M ( K ) = 2 IF ( K . E Q . O GC TO 1 6 1 3 K = K + 1 DO 1 6 1 4 J=K,C M{J) = 1 1614 GO TO 1613 DO 1 6 1 6 J = 1,C ' 1615 Y = (1-1)*C + J + 1 1616 CH(Y , 1 ) = 0 1600 CONTINUE 1617 J = N*C + 2 N = 1 NCH = NCH*C + 1 . I F ( N C H . G E . J ) GO TO 1619 CH(1, 1 ) = 0 RETURN 1619 DO 1 6 1 8 I=J,NCH N = N + 1 CH(N,1) = CH(I,1) CH(N,2) = CH(I,2) CH(N,3) = CH(I,3) 1618 CONTINUE C H ( 1 ,1> = ( N - 1 ) / C RETURN 1605 W R I T E ( 6 , 1 6 9 0 ) NCH,C FORMAT (//IX,32HARRAY CH OVERFLOWED 1690 I F , I 3) NCH = NCH - 1 GO TO 1617 1620 NN = 0 DO 1621 1=1,2 I F I N N . E Q . C ) GO TO 1621 DO 1622 J=1,C M( J) = I I F ( C H ( J + l , 2 ) . E Q . 3 ) GO TO 1:622 M(J) = 3 - CH(J+l,2) NN = NN + 1 1622 CONTINUE DO 1 6 2 3 J=1,X VAR(J) = 0 1623 CALL TEST(C,X,1,CH,RB,M) DO 1 6 2 4 J=l,X I F I V A R ( J ) . L T . R E D ) GO TO 1621 1624 CONTINUE NCH = NCH + 1 DO 1625 J=1,C Y = J + 1 Z = (NCH-l)*C + J + 1 XX = C H ( Y , 1 ) I F ( Z . G T . 1 0 0 0 ) GO TQ 1605 C H ( Z , l ) = CH(Y,1) C H ( Z , 2 ) = CH{Y,2) 1611  LIST  SB13  11/17/68  PAGE  6  STATEMENT 4  * * *  —>  * 4 4  * * * t 4 4 4 4  * *  * 4  *  4 4 4 4 4 4 ENTERING NO.,I3,13H0F ENTRIES 0 4 4 4 4 4 4 4 4 4 4 4 4 ' 4 4 , 4 4 4 4 4 4 4 4 4 4 4 4 4  J  N.R.ARDEN, I SN  _  226 227 230 231 234 235 240 242 244 24 5  SOURCE STATEMENT * *  * *  1625 1621  FORTRAN SOURCE LIST S813  CH(Z,3) = (M(J)*2-3)*AS(XX) ZZ = A(XX,1) K = RE - ES(ZZ,1) - 8S(XX) IF(M(J).EQ.l.AND.K.LT.AS(XX)) CH(Z,3) = - K K = ES(ZZ,1) + BS(XX) + AT(XX) + AS(XX) - RB IF(M(J).EQ.2.AND.K.LT.AS(XX)) CH(Z,3) = K CONTINUE CONTINUE GO TO 1617 END  | NO MESSAGES FOR ABOVE ASSEMBLY LIM.. I* H R S 1 M IN 17.1 SEC  11/17/68  * * *  * * *  PAGE  7  N.R.ARDEN I SN  SOURCE STATEMENT  FORTRAN SOURCE LIST  11/17/68 ,  0 * $ I 8FTC S814 SUBROUTINE TEST(C,X,I,CH,RB,M ) 1 INTEGER C » X » I,CH(1000,3),RB,M(C),VAR,J,Y,XX,YY,Z,ZZ,K 2 INTEGER NA , A . AT., AM , AS» BS »NE,ES,ATAB,CPL » RS,RTAB,RL ST, FE , NFE COMMON /CO/NA,A(400,2),AT(400),AM(400),AS(400),BS(400),NE, 4 t 1 ES(150,2),ATA8{700,2),CPL,RS(400),RTAB(150,3), 2 RLST(2300),FE(150),NFE COMMON /C4/ VAR(400) 5 * 6 DO 1700 J=1,C Y =(I-l)*C + J +1 * 10 XX = CH(Y,.l) YY = A(XX,.l) 11 IF(CH(Y,2).EG.M(J)) GO TO 1700 12 Z = ES(YY,.i) + BS(XX) + 1 - RB 15 * IF(M(J).E0.2) Z = Z + AT(XX) 16 ZZ = Z + AS(XX) - 1 21 ± 22 IF(Z.LE.O) Z = 1 IF(ZZ.GT.X) ZZ = X 25 * DO 1701 K=Z,ZZ 30 1701 VAR(K) = VAR(K) + AM(XX) 31 1700 CONTINUE 33 * RETURN 3.5 END 36 1 1  „ NO MESSAGES FOR ABOVE ASSEMBLY IME 14HRS 14MIN 33.2SEC  * * * * * * ± * * * * *  PAGE  8  N.R.ARDEN ISN  FORTRAN SOURCE LIST SOURCE  11/17/68  PAGE  9  STATEMENT  0 $I8FTC S815 * SUBROUTINE TIDY 1 2 INTEGER X,Y,Z,.XX,YY,ZZ,I ,J,K,I I, B,, N, F , MIN , MAX ,KB , MX I NT EGER NA , A , AT , AM , AS, BS , N E , ES , AT A 8, C PL , RS , R T AB , RL S T , F E, NF E .3 * COMMON /CO/NA,A(400,2),AT(400),AM( 400),AS( 400),BSI 400)i NE, 4 * t 1 ES (-15.0,2) , ATA 8(700, 2) , C PL , RS ( 400 ) , RT A B ( 150 , 3 ) , 2 RLST(23 00),FE(150),NFE * 5 * COMMON /C3/ F(4O0,2) INTEGER POINT 6 * COMMON /C6/P0INT' . . . 7 * IF(P0INT.BQ.4) GO TO 1817 10 t 13 * CALL RSMX(B,RB,Y,X) X = NE - 1 14 * DO 1800 1 = 1 ,X 15 * 16 Z = 0 17 * IF( ES( 1,2) .ECO) GO TO 180.1 t * 22 * Y = ATAB(I,2) 23 * YY = A TAB(Y,2) * 24 DO 1302 J=1,YY * XX = Y + J 25 * 26 * XX = ATAB(XX,2) * ZZ = A(XX,1) 27 * 30 * ZZ = ES(ZZ,1) + BS(XX) + AT(XX) IF(ZZ.GT.Z) Z = ZZ 31 * * 34 * 1802 CONTINUE 36 Z = Z - ES( I ,1) 37 1801 Y = ATAB(1,1) YY = ATAB(Y,1) 40 * DO 1803 J=1,YY 41 * * XX = Y + J 42 * XX = ATAB(XX,1) 43 IF(ASIXX).EQ.O) GO TC 1803 44 * 47 IF(BS(XX).GE.Z.AND.ESlI,1).LT.RB) GO TO 1804 ZZ = ES(I,.l) + BS(XX) + 1 52 * * 53 * K = ESII,1) + Z * 54 * IF(ES(I,1).LT.RB) GO TO 1819 57 K = ZZ + AS(XX) - 1 60 BS(XX) = BS(XX) + AS(XX) : 61 AS(XX) = 0 GO TO 1820 62 * * 63 * 1819 AS(XX) = AS(XX) + BS(XX) - Z + BS(XX) = Z 64 * 65 DO 1805 II=ZZ,K 1820 * 66 * 1805 RSI I I ) = RS(I I) - AM(XX) * IF(AS(XX).EQ.O) GO TO 1803 70 73 1804 ZZ = ES(I,1) + BS(XX) + AT(XX) + 1 K = ZZ + AS(XX) - 1 74 * DO 1806 II=ZZ,K 75 76 * 1806 RS( II) = RS(II) - AM(XX) * 100 1803 CONTINUE * 1C2 * 1800 CONTINUE * 1C4 * 1817 CALL RSMX(B,X,Y,XX) 105 N = 0 106 DO 1807 1=1,NA Z = A(I ,1) 107 t  )  J  N.R.ARDEN ISN 110 111 114 115 120 121 122 124 127 1.3.0 131 132 133 134 135 140 141 142 143 144 145 150 152 153 154 155 156 161 162 163 164 165 170 172 173 176 177 2C0 201 20 2 204 205 206 207 210 211 212 215 216 217 221 222 223 225 226  FORTRAN SOURCE  * * *  * 1807  *  * * *  * *  * 1809 1812  * * * * * * 1810  * *  1813  *  * * *  1814  *  * * *  1815  *  * *  1816  SOURCE  LIST  SB15  11/17/68  PAGE  10  STATEMENT  Z = ES(Z,1) + BS(I) I F ( Z . G E . Y ) GO TO 1 8 0 7 Z = Z + AT(I ) I F ( Z . L E . X ) GC TO 1 8 0 7 N = N +, 1 F.(Ntl) = I CONTINUE I F ( N . E G . O ) . GO TO 1 8 1 8 DO 1 8 0 8 1=1,N X = F ( 1,1) Y = A(X,1) Z = 0 XX = A T A B ( Y , 2 ) YY = A T A B ( X X , 2 ) I F ( Y Y . E Q . O ) GO TO 1812 DO 1 8 0 9 J = 1,YY ; K = XX + J K = ATAB(K , 2 ) II = A ( K , 1 ) K = E S ( I I „ 1 ) + A T ( K ) + B S { K ) IF(K.GT.Z) Z = K CONTINUE ZZ = C P L Y = A(X,2) XX = A T A B ( Y , 1 ) YY = A T A B ( X X , 1 ) I F ( Y Y . E Q . O ) GO TO 1813 DO 1 8 1 0 J=1,YY K = XX + J K = ATAB ( K, 1 ) K = BS(K) I F ( K . L T . Z Z ) ZZ = K CONTINUE ZZ = E S ( Y , 1 ) + ZZ - A T ( X ) I F ( Y Y . E Q . O ) ZZ = C P L - A T ( X ) Y = A(X,1) K = ES(Y,1) + BS(X) + 1 II = K + A T ( X ) - 1 DO 1814 J=K,II R S ( J ) = R S ( J ) - AM(X) MIN = B - AM(X) 11= K Z = Z + 1 ZZ = ZZ + 1 DO 1 8 1 5 J=Z,ZZ MAX = M X I R S , J , J + A T ( X ) - 1 ) I F ( M A X . G E . M I N ) GO TO 1815 MIN = MAX II = J CONTINUE K = II + AT(X) - 1 DO 1816 J=II,K R S ( J ) = R S ( J ) + AM(X) Y = A( X, 1 ) BS(X) = 1 1 - 1 ES(Y,1)  * *  J  * * * *  * +  *  * * t +  *  *  * * * * . * *  * * * * * * * 4  *  -  N.R.ARDEN ISN 227 232 234 2 35 236 237  FORTRAN SOURCE LIST S815  SOURCE STATEMENT  11/17/68  * IF(MIN.LT.8-AM(X)) GC TO 1817 * 1808 CONTINUE * 1818 WRITE(6,1890) B * 1890 FORMAT ( ..IX,16HMINI MUM BOUND = ,14) * RETURN * END  ,  MX  NO MESSAGES FOR ABOVE ASSEMBLY [ME 14HRS 14MIN 57.0SEC  •-—  -  •  -  *  *  USER FUNCTION SUBPROGRAM REFERENCES  -  *  •  i  I  PAGE  11  >/ ! ]  FORTRAN  N.R.ARDEN ISN  SOURCE 0 1 2 3 4  5 6 7 12 13 14 16 17 21 22 24 25 26 27 31 32 34 35 37 40  TIME  14HRS  * * * * * *  *  1901  *  1903  *  1900 1904  *  1 i  LIST  11/17/68  STATEMENT  $1BFTC SB16 SUBROUTINE STORE(N,C) INTEGER N,C,I INTEGER N A , A , A T , A M , A S , B S , N E , E S , A T A B , C P L , R S , R T A B , R L S T , F E , N F E COMMON /C0/NA,,A{400,2) , A T ( 4 0 0 ) , AM ( 4 0 0 ) , AS ( 4 0 0 ) , B S i ,NE, 400 ) 1 E S I 1 5 0 , 2 ) , A T A B ( 7 0 0 , 2 ) , C P L , R S < 4 0 0 ) , R T A B ( 1 5 0 ,3) , 2 RLST(230O),FE(150),NFE ' INTEGER S T A S , S T B S , S T R S , S T E S COMMON / C 5 / S T A S ( 4 0 0 , 0 2 ) , S T B S ( 4 0 0 , 0 2 ) , S T R S ( 4 0 0 , 0 2 ) , S T E S ( 1 5 0 , 0 2 ) I F ( C . E Q . l ) : GC TO 1 9 0 0 DO 1 9 0 1 1 = 1 , N A I STAS(I,N) = AS(I)  1902  * *  SOURCE  1905 1906  STBS(I,N) = BS(I) DO 1902 1 = 1 ,NE S T E S ( I ,N) = E S ( I , 2 ) DO 1 9 0 3 1=1,CPL S T R S ( I ,N) = R S ( I ) RETURN DO 1 9 0 4 1=1,NA A S(I) = STAS(I,N) BS(I) = STBS(I,N) DO 1 9 0 5 I=1,NE ES(I,2) = STES(I,N) DO 1 9 0 6 1 = 1 ,CPL RS(I) = STRS(I,N) RETURN END  NO MESSAGES FOR ABOVE 15MIN 1 3 . 1 S E C  ASSEMBLY  1  •!  * * *  *  i  * * * * *  * *  PAGE  12  11/17/68  FORTRAN SOURCE L I S T  N.R.ARDEN ISN  SOURCE 0 1 2 *  *  3 4  * * 5 *  6 7 10 11 12 14 15 16 17 20 21 22 23 25 26 27 30 31 33 36 37 40 43 44 45 46 47 52 53 54 57 60 61 62 63 64 65 67 70 71 72 73 75 76 101 102  * * * * * *  STATEMENT  * * 4  *  * * * * *  * * * * 4  *  * * *  13  l  4 $IBFTC S B 1 7 * SUBROUTINE MAP(B) INTEGER 0 V R I 4 0 0 ) , S E Q I 4 0 0 ) , I , J , K , X , Y , Z , X X , Z I , T , N , N N , R P ,$M7B,NIN,WL, * 1. .4 1 ACT INTEGER NA, A,AT,AM, AS,BS , NE , ES , AT AB , CPL , RS , RT A8 , RL ST , F£ , NFE * 4 COMMON /CO/NA,-A{ 40 0, 2) , AT ( 4 0 0 ) ,AM( 4 0 0 ) ,AS( 4 0 0 ) ,BS( 4 0 0 , ) NE, 4 1 E S ( 1 5 0 , 2 ) , A T A B ( 7 0 0 , 2 ) , C P L , R S ( 4 0 0 ) , R T A B ( 1 5 0 ,3) , * 2 RLST12300),FE(150),NFE COMMON / C I / N I N U 5 0 ) * COM MON /C2 / . WL ( 4'QO » 2 ) * COMMON / C 3 / A C T ( 4 0 0 , 2 ) DO 6 0 0 1=1,NA * OVR( I ) = AMI I ) * A T ( I ) 4 SEQ(I); = 1 0 0 0 * A ( I , 1) + A ( I , 2) 600 * T = 0 601 .* X ,=. ATAB ( 1 , 1 ) , * N = ATAB(X,1) * DO 6 0 2 1=1,N J = X + I Y = ATAB(J,1) * WL( I ,1 ) = Y 4 WL( 1,2) = A S ( Y ) 602 NN = 0 RP = 0 4 DO 6 0 3 I=1,NE X = ATAB(I,2) * NIN( I ) = ATAB(X,2) 603 4 I F ( K N . E Q . O ) GO TO 6 0 8 "'" — 4 604 J = 0 * DO 6 0 7 1=1,NN 4 I F ( A C T ( I , 2 ) . G T . O ) GO TO 6 0 6 X = ACT(1,1) RP = RP - A M ( X ) * X = A(X,2) ! . 4 NIN(X) = NIN(X) - 1 j * I F ( N I N ( X ) . G T . O ) GO TO 6 0 7 * X = ATAB(X,1) 4 Y = A T A B ( X , 1) r * I F ( Y . E G . O ) RETURN 4 DO 6 0 5 K=1,Y Z = X + K * Z = ATAB(Z,1) 4 N = N + 1 * WL(N,1) = Z * XX = A ( Z , 1 ) .4. WL(N,2) = A S ( Z ) - T + E S ( X X , 1) .605 * GO TO 6 0 7 606 J = J + 1 4= A C T ( J , 1) = A C T ( I , 1 ) * ACT(J,2) = ACT(1,2) 607 CONTINUE * NN = J 4 I F ( N . L E . 1 ) GO TO 6 1 5 608 4 DO 6 1 4 1=2,N 4= J = N - I + 1 .  *  PAGE  .1  ....  „„  „.  J \  _  •  „ ' „,  ..  ....  ,  „„„.  _  ™„_ .  ., .. . _ .. „  -  -  .  -  .  ~ .... .  -  m  -  -„  —  ~  N  J  \  i  N.R.ARDEN ISN 103 -t1C4 105 * ...1.0.6. 107 110 * 111 112 * 113 114 115 116 117 121 * 123 * 1'26 127 * 130 131 134 * 135 * 136 * 137 140 * 141 142 * 14 3 144 * 146 * 147 * 150 * 151 * 154 * 156 * 157 * 162 * 163 * 164 * 167 * 171 172 174 175 * 176 177  TIME  SOURCE STATEMENT •DG  609 610 611 612  613 614 615  616 617 618 619  613  623  IF(WL(I,2).LT.O) GO TO 623 CONTINUE DO 6 22 I=1,NN ACT( 1,2) = ACT(1,2) - SM GO TO 604 B = B + 1 GO TO 601 END  NO MESSAGES FOR ABOVE ASSEMBLY 14HRS 15MIN 35.0SEC  11/17/68  ! 1  K=ltJ  Z =WL(K,1) ZZ = WL(K+1,1): IF(WL(K+1» ?l-WL(K,2)) 612,609,613 IF(OVR(ZZ)-OVR(Z)) 613,610,612 IR(AM(ZZ)-AM(Z)) 613,611,612 IF(SEQ(ZZ)-SEQ(Z)) 612,613,613 WL(K,1) = ZZ WL(K+1 ,1) = Z ......Z = WL (K,2 ) WL(K,2) = WL(K+1,2) WL(K+1,2) = Z CONTINUE CONTINUE IF(N.BQ.O) GO TO 618 J =. 0_ DO 617 1=1,N X = WL(I , 1 ) IF (RP+AMU) .GT.B) GO TO 616 RP = RP + AM(X) NN = NN + 1 ACT(NN.l) = X ACT(NN,2) = AT(X) •GO TO 617 J = J + 1 WL(J,1) = WL(I,1) WL(J,2); = WL(I,2) . CONTINUE N = J SM = CPL DO 619 I=1,NN I F U C T U ,2).LT.SM) SM= ACT(I,2) CONTINUE T = T + SM IF (N. ECO); GO TO 621 DO 620 1=1,N WL( I ,2) = WLCI,2) - SM  620 621 622  FORTRAN SOURCE LIST S817  *  i  * 4  * *  *  i  •  *  *  * *  *  !  I  * *  *  * * * * 4 +-  • * •. .  PAGE  14  N.R.ARDEN ISN  SOURCE STATEMENT  FORTRAN SOURCE LIST  1 1/17/68  * $IBFTC SB18 * SUBROUTINE MA IN IB) PT,RED, N,F * INTEGER B, MX»RB»RE ,MXS,I,J ,CH(1000, 3) , , FE » INTEGER NA,A,AT,AM,AS, BS ,NE,ES ,ATAB,CPL , R S , R T A B » R L S TNFE * COMMON /C0/NA,A(400,2) ,AT(400) ,AM(400), AS(400),BS(400),N E, * 1 ES(150,2>,ATAB(700, 2),CPL,RS(400),RTAB(150,?) » 2 RLSTI2300), FE(150), NFE * COMMON /C2/ F(400,2) 5 RSMX(MX,RB,RE,MXS) 6 * 1300 CALL IFIMX.GT.B ) GO TO 1301 7 1302 CALL TIDY 12 RETURN 13 * IF(MXS.LT.B) MXS = 8 14 1301 17 * RED = MX - MXS + 1 CALL RELACT(RB,RE,N,F) 20 * IF(N.EQ.O) GC TO 1302 21 24 DO 1305 1=1,3 25 * IF( I.NE.3): GC TO 1306 30 * CALL SELECT(N, N,CH) 31 * IF(CH(1,1).EQ.O) GO TO 1 304 CALL FE AS ( CH ,.RB , RE , RED) 34 * 35 * IF(CH(1,1).EQ.O) GC TO 1 304 40 * GO TO 1310 SELECT(N , I,CH) 41 * 1306 CALL IF(CH(1,1).EQ.O) GO TO 1305 42 * CALL FEAS(CH,RB,RE,RED) 45 jf 46 IF(CH(il, 1 ) .EQ.O) GO TO 1 305 PT,B) 51 1310. CALL WORKICH,, IF(PT.EQ.O) GO TO 1300 52 55 CONTINUE 1305 DO 1307 1=2,RED 57 1304 J = RED + 1 - 1 60 61 * DO 1308 K=l,3 IFIK.NE.3) GO TO 1309 . i . . .62. 65 * CALL SELECT(N, N,CH) 66 IF(CH(l,l).EQ.O) GO TO 1307 1 • CALL FEAS(CH,RB,RE,J) 71 I IF(CH(l,l)..EQ.O) GO TO 1 307 72 * 75 * GO TO 1311 76 * 1309 CALL SELECT(N, K,CH) ' i IF(CHI 1,1).EQ.O) GO TO 1308 77 CALL FEAS(CH,RB,RE,J) 102 * i IF(CH(1,1).EQ.O) GO TO 1308 103 1 CALL WORK(CH, PT,B) 106 * 1311 IF(PT.EQ.O) GO TO 1302 107 * CONTINUE 112 * 1308 114 * 1307 CONTINUE 116 * GO TO 1302 117 * END 0 1 2 3 4  1  NO MESSAGES FOR ABOVE ASSEMBLY TIME 14HRS 15MIN 53.3SEC  * * * * * * * * *  *  * *  t * * * * *  *  *  PAGE  15  N.R.ARDEN ISN  FORTRAN SOURCE LIST SOURCE STATEMENT  11/17/68  PAGE  16  1  $IBFTC SB19 0 SUBROUTINE WK1(CH,PT,B) 1 PT,B,I,J,K,L,M,KK, LL,MM,X,Y,Z, INTEGER CH(1000,3), 2 * N,NN,S,SS,ACT 1 XX,YY,ZZ,MAX,II, * INTEGER NA,A,AT,AM,AS,BS,NE,ES ,ATAB,CPL,RS,RTAB,RLST,FE,NFE 3 * 4 COMMON /C0/NA,AL400,2),AT(400) ,AM(400),AS(400),BS(400) ? NE , 1 ' ES(150,2),ATAB(700, 2),CPL,RS(400),RTAB(150, 3) . 2 RLST(.2300),FE(150), NFE * COMMON /C3/ AC1"(.800) 5 6 2113 J= C H( 1,1) II = CH(1„2) 7 * MAX = I 10 IF(J.EQ.l) GO TO 2100 11 * 14 KK = 0 15 4 LL = 0 16 . MM = B 17 * NN = 0 X =1 20 * DO 2101 1=1,J 21 K =0 22 * L =0 23 24 * M =0 25 * N =0 26 4 DO 2102 Y=1,II X = X +1 27 Z = CH(X,1) 30 * 31 * YY = A(Z,1) ZZ = A(Z,2) 32 4 XX = 1 33 4 34 * S = AS(Z) AS(Z) = AS(Z) - IABS(CH(X,3)) 35 I IF(CH(X,3).LT.O) GO TO 2114 36 * CALL EVBAL(ZZ,XX) 41 * IF(XX.LE.O) XX = 0 42 * . i K = K + ES(ZZ,2) - XX 45 * i N = N + ES(ZZ,2) 46 4 GO TO 2115 47 4 SS = BS(Z) 50 4 2114 BS(Z) = BS(Z) - CH(X,3) 51 CALL EVBAL(YY,XX) 52 * IF(XX.LE.O) XX = 0 53 * K = K + ES(YY,2) - XX 56 57 * N = N + ES(YY,2) 60 4 BS(Z) = SS 2115 61 AS(Z ) = S L = L +, AS ( Z ) - IABS(CH( X, 3) ) 62 63 4 2102 M = M + AM(Z) IF(K-KK) 2101,2120,2119 65 4 a IF(N-NN) 2101,2117,2119 66 2120 67 4 2117 IF(L-LL) 2101,2118,2119 IF(M-MM) 2119,2101,2101 70 * 2118 2119 KK =K 71 72 * NN = N 73 LL = L ( 74 MM = M  4  j  *  1  * * 4 4  * t 4  4 4  *  4 4 4  *  * 4  * *  4 4 '4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4  J  i  I  N.R.ARDEN ISN  :  „  _„..  SOURCE STATEMENT  FORTRAN SOURCE LIST SB 19  MAX = I 75 * 76 * 2101 CONTINUE 100 * 2100 K = CH(1»2) CALL STORE (2,0) „ 1C1 * X' = 2 + (MAX-1)*K 1C2 * XX = X + K - 1 103 * DO 2103 I=X,XX 104 * 1G5 * Y = CHI 1,1) 106 * YY = A(Y,1) . IF(CH{Ur31.GE^ 2 104 _ ._ _Ji_7____ 112 * Z = ES(YY,,1) + BS(Y) + 1 ZZ =. Z - CH(I,3) - 1 113 * ASIY) = AS(Y) + CH(1,3) 114 * 115 * BS(Y) = BS(Y) - CH{1,3) 116 * GO TO 2105 ZZ = ES(YY,1) + BS(Y) + A T ( Y ) + AS(Y) 117 * 2104 Z = ZZ - CH(I,3) + 1 120 * AS(Y) = AS(Y) - CH(I,3) 121 * 122 * 2105 LL = 1 MM = 1 123 * ACT(1) = Y 124 * _ 125 *..2116 M = ACT(MM) CALL ESRED(M,LL,ACT,PT) 126 * 127 * IF(PT.EQ.l) GO TO 2106 MM = MM + 1 132 * 133 * IF(MM.LE.LL) GO TO 2116 136 * DO 2107 J=Z,ZZ 137 .*_.2107 RS(J) = RS(J) - AM(Y) 141 * 2103 CONTINUE 143 * CALL CHOP(1,NA,B) 144 * RETURN 145 * 2106 I F ( C H ( l , l ) . E G . l ) RETURN XX = C H (1»1 ) * K - K + 1 150 * 15,1... * IF(XX.LT.X) GO TO 2109 DO 2110 J=X,XX 154 * 155 * L = K + J 156 * CH(J,l) = CH(L,1) 157 t CH(J,2) = CH(L,2) 160 * 2110 CH(J,3) = CH(L,3) 162 * 2109 CH(1,1) = CH(1,1) - 1 163 * CALL STORE(2,1) 164 * GO TO 2113 165 * END  NO MESSAGES FOR ABOVE ASSEMBLY TIME 14HRS 16MIN 15.1SEC  11/17/68  PAGE  17  * * .  ,  #  •  * -  *  •  ..  ..  *  ..  *  " -i- •  *  ;  '  *  j  * *  *  * *  *  j  ,  :  ..  .  * *  . ...  t f ''' !  •  *  *  ••;  .  „  N.R.ARDEN ISN  I _  _  FORTRAN SOURCE  LIST  11/17/68  SOURCE STATEMENT  $IBFTC SB20 0 INTEGER FUNCTION CRN(T,K) 1 * INTEGER T , I , J , K 2 * INTEGER N A , A , A T , A M , A S , B S , N E , E S , A T A B , C P L , R S , R T A B , R L S T , F E i N F E 3 COMMON / C O / N A , A ( 4 0 0 , 2 ) , A T ( 4 0 0 ) , A M I 4 0 0 ) , A S ( 4 0 0 ) , B S ( 4 0 0 ) , N E , 4 1 ES(150,2),ATAB(700,2),CPL,RS(400),RTAB(150,3), 2 RLST(2300),FE(150),NFE DO 900 1=2,NFE 5 * J = FE(I) 6 I F ( T . L E . E S ( J , 1 ) ) GC TO 9 01 ; 1. 12 900 CONTINUE CRN = 1 - 1 14 4= 901 I F ( T . E Q . E S ( J , l ) . A N D . K . E Q . l ) CRN = 1 15 RETURN 20 END 21  NO MESSAGES FOR A80VE ASSEMBLY TIME 14HRS 16MIN 30.5SEC  i  1  I  ii  7 1 •' I  * * * 4=  * 4=  * *  * 4 4  PAGE  18  N.R.ARDEN I SN  FORTRAN SOURCE LIST  11/17/68  SOURCE STATEMENT  0 * SIBFTC SB29 1 SUBROUTINE WK2(CH,PT,B) INTEGER CH(1000,3), PT,B,I,J,K,L,M,KK,LL , MM , X , YZ, , 2 1 X X » Y Y » Z Z »M A X , I I , ;. N,NN,S,SS,ACT 3 INTEGER NA,A,AT,AM,AS,6S , N E, ES , AT AB, CPL , RS , R T AB , RL ST , Fg, NFE 4 COMMON /CO/NA, A (400, 2) ,AT(4Q0) , AM ( 400 ) , AS ( 400 ) , B S ( 40$ )',NE , 1 ES(150,2),ATAB(700,2) , CPL , R S ( 400 ) , RT AB ( 15 6 , 3 ) , * 2 2 RLST{2300),FE(150),NFE 5 * COMMON /C3/ ACT^.800) 6 * 2113 J= CH(1,1) 7 * II = CH(1,.2) 10 * MAX = 1 11 * I F ( J . E C . l ) GC TO 2100 14 LL = 0 MM = B 15 X = I 16 1=1 ,J DO 2101 17 L =0 20 M = 0 21 DO 2102 Y=1,II 22 X = X +1 23 Z = CH(X,1) 24 L = L + AS(Z) - IABS(CH(X,3)) 25 M = M + AM(Z) 2102 26 IF(L-LL) 2101,2118,2119 30 IF(M-MM) 2119,2101,2101 31 2118 LL = L 2119 32 MM = M 33 i i MAX = I 34 CONTINUE 2101 35 K = CH(12 ) 2100 37 CALL STORE(2,0) 40 X = 2 + (MAX-1)*K 41 XX = X + K -1 42 ± DO 2103 I=X,XX 43 Y = CH(I,1) 44 YY = A(Y,1) 45 IF(CH( I ,3) .GE.O) GO TO 21.04 46 Z = ES(YY,1) + BS(Y) + 1 51 ZZ = Z - CH(I,3) - 1 52 AS(Y) = AS(Y) + CH(I,3) 53 54 BS(Y) = BS(Y) - CH(I,3) 55 GO TO 2105 56 * 2104 ZZ = ES(YY,1) + BS(Y) + AT(Y) + AS(Y) 57 * Z = ZZ - CH(1,3) + 1 60 * A S(Y ) = AS(Y) — CH(I,3) 61 * 2105 LL = 1 62 * MM = 1 63 * ACT(1) = Y 64 * 2116 M = ACT(MM) 65 * CALL ESRED(M,LL,ACT,PT) 66 * IF(PT.EQ.l) GO TO 2106 71 * MM = MM + 1 72 * IF(MM.LE.LL) GO TO 2116 75 * DO 2107 J = Z,ZZ !  PAGE  19  N.R.ARDEN ISN  i  FORTRAN SOURCE LIST SB29 ,  /  11/17/68  SOURCE STATEMENT  RS(J) = RS(J) - AM(Y) 76 2107 CONTINUE 100 4=' 2103 CALL CHOP(ltNA,B) 1C2 4 ,RETURN „. 103 IF(CH{ 1, 1 ),.EQ. 1 ) RETURN 104 4 2106 XX = CH(1,1)*K - K + 1 4 107 IF(XX.LT.X) GO TO 2109 4 no DO 2110 J=X,XX 113 * L = K +J 114 4 CHIJ.l) = CH(L.l) 4 ' _.. _._ ...1.15 CH(J,2) = CH(L,2) 116 4 117 4 2110 CH(J,3) = CH(L,3) 121 4 2109 CH(1,1): = CH(1,1) - 1 CALL ST0RE(2,1) 122 4 GO TO 2113 4 123 END 124 4 - •-  > ...  .  , ~  '  4 4 : I ;  !  j . i i 1  NO MESSAGES FOR ABOVE ASSEMBLY TIME 14HRS 16MIN 50..3 SEC...  4 4 4 4 4 4 4 4 4 4  * •  4 4 4  PAGE  20  N.R.ARDEN ISN  SOURCE STATEMENT  FORTRAN SOURCE LIST  11/17/68  0 SI.8FTC SB39 4 SUBROUTINE WK3<CH,PT,B) 1 * 4 INTEGER CHI 1000,3), PT,B,I,J,K,L,M,KK, LL,MM, X,Y,Z, 4 2 * * 1 XX »YY,ZZ »MAX, 11, N,NN, S, SS • ACT \ . .4 . . INTEGER NA,A,AT,AM,AS,BS,NE,ES,ATAB,CPL,RS,RTAB,RLST,F E,NFE 3 4 COMMON /CO/NA, A (400, 2) , AT (400) ,AM.< 400) , AS ( 400 ) , B S ( 400 ) 4 * ,NE, 4 1 ES(150,2 ) ,ATAB(700, 2),CPL,RS(400),RTAB(150, 3) , 4 2 RLST(2300),FE(150),NFE 4 5 * COMMON /C3/ ACT (,80 0) 4 4 6 2113 J = CH(1,1) II = CH(1,2) 4 7 * MAX = 1 4 10 4 IF(J.EQ.l) GO TO ,2100 11 * 4 14 * LL = 0 MM = B 15 * 4 16 . * . X . = ...1 • _ 4 . ; 17 4 DO 2101 1=1,J 4 L =0 20 * 4 M =0 21 4 22 00 2102 Y = l , l l 4 X = X +1 23 * 4 4__ 24 Z = CH(X,1) r •:'" ' ""j . 25 L = L + AS(Z) - IABS(CH(X,3)) j 26 * 2102 M = M + A M ( Z ) 4 4 30 IF(M-MM) 2119,2118,2101 j 1 * 4 31 2118 IF(L-LL) 2101,2101,2119 ! ! 32 * 2119 LL = L 4 1 MM = M 4 33 MAX = I 4 34 j 35 4 2101 CONTINUE J 4 4 37 * 2100 K = CH(1,2) 4 CALL ST0RE(2,0) 40 X = 2 + (MAX-1)*K 41 * 4 XX = X + K - I 4 , 42 * DO 2103 I=X,XX 43 * 4 44 * Y = CH(I,1) 4 YY = A(Y,1) 45 * 4 46 IF(CHCI,3)..GE.O) GC TO 2 104 4 Z = ES(YY,1) + BS(Y) + 1 51 * 4 ZZ = Z - CH(I,3) - 1 52 4 ; . AS(Y)=AS(Y)+CH(I,3) 53 * 4 BS(Y) = BS(Y) - CH{ 1,3) 4 54 * 55 4 GO TO 2105 ! 9 4 56 4 2104 ZZ = ES(YY,1) + BS(Y) + AT (Y) + AS(Y) 4 j 4 57 4 Z = ZZ - CH ( I ,.3) + 1 I 4 AS(Y) = AS(Y) - CH(1,3) 60 4 4 61 4 2105 LL = 1 ' MM = 1 | * 62 4 4 ACT(l) = Y ! 63 4 4 1 64 4 2116 M = ACT(MM) CALL ESRED(M,LL,ACT,PT) 4 65 4 IF(PT.EQ.l) GO TO 2106 ...4 66 4 71 4 MM = MM + 1 ' 4 72 4 IF(MM.LE.LL) GO TO 2116 4 75 4 DO 2107 J=Z,ZZ 4 i  PAGE  21  -~ -  •  - - ~"  -- —  1  •~  "  -  -  ~ " - -  A  N.R.ARDEN ISN  „  TIME  14HRS  _  76 * ICO * 102 * 103 J . 104 107 110 * 113 + 114 * 115 * 116 * 117 * 121 122 123 124  FORTRAN SOURCE 2107 2103  2106  2110 2109  NO MESSAGES FOR 17MIN 1 C . 0 S E C  SOURCE  LIST  S839  11/17/68  STATEMENT  R S ( J ) = R S ( J ) - AM(Y) CONTINUE CALL CH0P(1,NA,B1 RETURN I F ( C H ( i , l ) . E C . l ) RETURN XX = C H ( 1 , 1 ) * K - K + 1 I F ( X X . L T . X ) GO TO 2 1 0 9 DO 2 1 1 0 J=X,XX L = K + J C H ( J , 1 ) = C H ( L »1 ) CH(J,2) = CHIL,2) CH(J,3) = CH(L,3) C H ( 1 , 1 ) = CHI 1,1) - I CALL STORE(2,1) GO TO 2 1 1 3 END  ABOVE  ASSEMBLY  * *  * *  * t  * t-  t  *  PAC  FORTRAN SOURCE LIST  N.R.ARDEN I SN  _  0 1 2 __3_ 4 5 6 7 10 11 12 13 14  11/17/68  PAGE  23  SOURCE STATEMENT  * $IBFTC SB 21  * *  * * 2200 2201  *  2202  *  * *  SUBROUTINE WORK(CH,PT,B) INTEGER CHI 1C00,3) , P T , B .INTEGER. .POINT COMMON / C 6 / P 0 I N T GO TO ( 2 2 0 0 , 2 2 0 1 , 2 2 0 2 ) , P O I N T CALL WK1(CH,PT,B) RETURN CALL WK2(CH,PT,B) RETURN CALL WK3(CH,PT,B) RETURN END  *  *  * T-  *  NO MESSAGES FOR ABOVE ASSEMBLY  00 KT63'0,  JOB  END  14HRS 17MIN  26.2SEC  800 CARDS READ  0 OUTPUT CARDS . . . .  902 J  LINES PRINTED  0 O B J . DECK PUNCHED  N.R.ARDEN ISN  FORTRAN SOURCE LIST  11/17/68  SOURCE STATEMENT  0 $IBFTC SB2 SUBROUTINE TAB 1 2 * INTEGER S(2) ,N,I,J,K,X,Y 3 * INTEGER ATAB 4 4 COMMON /CO/NA,A(400,2),AT(400),AM(400),AS(400),BS( 400),NE, 4 1 ES(150,2),ATA8(700,,2),CPL,RS(400),RTAB( 150,3), * 2 RLST(2300),FE(150),NFE 5 COMMON /C2/ N(400,2) ! 6 DO 500 1=1,NE • 7 N( 1,1) = 0 10 * 500 N(1,2) = 0 12 * DO 501 I=,1,NA J = A( 1,1) 13 14 K = A(1,2) 4 15 N( J , l ) ; = N( J , l ) + 1 16 * 501 N(K,2) = N(K,2) + 1 20 4 DO 502 K=l,2 S(K) = NE + 1 21 4 DO 502 I=1,NE 22 4 ATA8(I,K) = S(K) 23 4 24 4 J = S( K ) ATA 8(J , K) = N( I ,K) 25 4 S(K) = S(K) + N(I,K) + 1 26 4 502 DO 503 I=1,NA 31 4 32 4 DO 503 K=l,2 33 4 J = A{ I,K) 34 4 X = ATAB ( J , K ) 35 4 Y = X + ATAB(X,K) - N(J,K) + 1 36 4 ATAB(Y,K) = I 37 4 503 N.(J,K) = N(J,K) - 1 42 4 RETURN 43 4 END  NO MESSAGES FOR ABOVE ASSEMBLY  00 KV550  PAGE  2  4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 • 4 4 4 4 4 4 4 4 4 4  . . . . . . . _  

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