I

A Critical Look at PERT Analysis

Richard M, Nicholson

UNIVERSITY OF MARYLAND COMPUTER SCIENCE CEN.TER

COLLEGE PARK, MARYLAND

https://ntrs.nasa.gov/search.jsp?R=19710023019 2020-03-21T03:43:57+00:00Z

Technica 1 Report TR-154 NGL-21-002-008

A p r i l 1971

A C r i t i c a l Look a t PERT Analys is

Richard M. Nicholson

ABSTRACT

PERT-Program Evaluat ion and Review Techn iqueana lys i s

i s descr ibed and the errors which can r e s u l t from PERT

ana lys i s are ind ica t ed . A model f o r performing the PERT

ca l cu la t ions without t hese errors r e s u l t i n g is der ived

and output from t h i s model is disp layed . A system f o r

p ro jec t management incorpora t ing t h i s model is descr ibed .

A program l i s t i n g of t he model implemented is presented .

T h i s research was supported i n par t by Nat ional Aeronautics and Space Adminis t ra t ion Grant NGL-21-002-008, t o t h e Com- pu te r Science Center of the Univers i ty of Maryland.

TABLE OF CONTENTS

ii

CHAPTER I

1 e 1 BACKGROUND

Effective ma gement of a research and development

project requlres the early establishment of a project schedule

andl+a frequent review of progress t b ensure that the schedule

is being met. This schedule provides the basis for the speci-

gication of a completion date

for the estimation of total cost as well as for the scheduling

of material deliveries, manpower and facilities allocation,

)

ad serves as a planning tool

subassembly delivery datea, etc, Act-1 progress is compared

to thiB schedule to d8tac.t; any deviation which might occur

so that a revised sc

to correct the deviation,

duls can be prodixced, or action taken

Perfoxye any such project involves the execution of IC .'

lengthy'sequences of individual tasks e Consequently, ex-

pected completion time i a a omplex function based on the

expected duratfan of amoh individual t a s k

calculated with some degree of uncertainty, PEBT - Program Evaluation and Review Technique - is a management tool which

8 for thg systemat'ic expression of the interrelation-

ships between the indkvidual tasks of a project and for the

statistical estimatian of overall completion time given

estimates of time required to perform the invidual tasks.

. . . .. .. . . . . . - -. . . . . . . .. . .. . . . ... . . .. . . .

I

Although there are many kinds of jobs

the performance of a number of interrelated

not utilized for all of these; for example,

are applied to the scheduling of a job shop

2

which involve

tasks, PERT i s

other techniques

or assembly line

activities. In the construction industry, a technique known

as the Critical Path Method (CPM) is employed; the CPM approach

is basically the same as that which will be described for

PERT, with the exception that the time required to accomplish

each task is regarded asa deterministic, rather than prob-

abiliskic, quantity. Any project composed of tasks about

which significant experience and historical data are available

is a strong candidate for CPM.

PERT is applied specifically in the management of those

projects involving tasks which are unique to the particular

project, rather than being of a routine or repetitive nature.

It may be said that the planners of a project employing

PERT are faced with the problem of estimating creativity;

however, it is assumed that the manager or engineer respon-

sible for providing the time estimates for a particular task

has sufficient experience in similar jobs to enable him t o

specify a range of times in which the task may be expected t o

be accomplished and to provide a reasonable "most likely"

time for its accgmplishrnent.

Since individual tasks, or sequences of tasks, can

frequently be performed simultaneously, there may be flexibility

in the schedule time requirements for certain of the tasks.

3 tions produce a sehedule listing completion

ell 8s f o p the entire project

flexibility xists, Given

s a relatively ob-

jectives standam3 hich to measure ppoogress,

as deveboped in 1947-58 in an all-out effort to

e completion of the Polaris Fleet Ballistic Project,

ith the successful

nd agencies and contractors

t e by two ye BT

inuteman and

1s Nike-Zeus, Pershing and Ha

projects @ Continued success pplications has re-

nt b snt that BERT

ojects over

million? [ I] 8

of soft

accept basic indi

ttle more t tion of the hand

leullated ~ @ t ~ ~ a , ich is descfibed in the second go

The soncept is inv riabliy implemented 'by using

ptions and b %yzing this simplified

4 data by using the concept of llcritical path". Briefly, this

critical path approach assumes that there is one critical se-

quence of events (called the critical path) and all analysis

of the project is relative to this particular sequence. Re-

cently,

Ringer [3] ha.ve criticized the commonly accepted techniques used

in project scheduling, demonstrating that some of the simplifying

assumptions of PERT tend to produce an optimistic schedule,

The purpose of this thesis is to investigate the assumptions

inherent in most PERTirhplementation~, particularly that of a

"critical" path, and to demonstrate a computer program which

accurately reflects the interdependence of sequences of simul-

taneous activities in large scale projects, accounting for the

effects of uncertainty in individixal activities in the process

of projecting overall completion times.

researchers such as MacCrimmon and Ryavec [2] and

1.2 Standard PERT Definitions

Let us first develop some of the standard notion used

in PERT,

Activities. By activity is meant an individual task

which is not further Subdivided, that is, activities are regarded

as the basic tasks which are involved in the overall project,

and become the basic building blocks of the PERT system.

Activities are denoteds a, b, c, ... . Associated with 'an activity is a length of time or duration required for its com-

pletion alone. Typically this is not a fixed quantity of time,

? I

but is more accupately expressed as a statistical distribution

which indicates the minimal, maximal, and expected (mean)

- ~

time required - for its completion, ~- _ _ -- A

The distribution associated with the activity a is

represented as a probability distpibution function (pdf), fa($)

I This function is defined I

and

The form of the pdf utilized in PERT is further defined by the

, minimal point ma and maximal point Ma, so that

Clearly, the expected time of completion of a is a function of

this distribution fa ana is denoted in the usual fashion

or more simply just pa.

The extent to which the component tasks of the project

are broken down into subtasks, and eventually into the smallest

task or activities, depends on the ability to accurately esti-

mate these distributions fa, ' fa can be estimated with reasonable accuracy is regarded as an

activity and not furthep analyzed,

Typically, any task f o r which

I

6

However, it is unrealistic in practice to expect a

project manager to.know the form of a given activity

distribution fa' It is much more likely that he can estimate

only the minimal, maximal, and usual durations, which corres-

pond to the extremal and modal points of fa, respectively.

For this reason, all PERT implementations assume a standard

form of distribution called the beta distribution and vary its

parameters to yield the extremal and modal points,, denoted

here as ma, Ma andm0dea.

The pdf of the beta distribution is

a B K (t - mal 0 elsewhere,

(Ma - t) , for ma 5 t 5 PI,

where a , B > - l e

Figure 1 illustrates a few examples of beta pdf with

specified parameters.

,t m3 mode 1 mode3 1 m3 m

..

I *2 modeZ lVI2 I

Figure 1, Examples of the Beta Distribution

?

The beta distribution has characteristics which seem

intuitively to represent the manner in which activities may

be expected to be distributed:

I) It is unimodal, Th re is a single most probable completion time,

2 ) It is continuous, "Continuity reflects the property that if an activity has a particulars probability of being completed in a small interval, the prob- ability is only slightly increase when the size of the interval is increasede" 4

3 ) The distribution has definite end points, The activity must consume a non-negative quantity of time, and there is assumed to be some maxj.rnum time which the activity will not exceed, barring an "Act of God,"

It is condition ( 3 ) that seems to make the beta a better

''standard" distribution than the better known normal distri-

bution.

There is no*attempt made to estimate the exponential

parameters andB.of 'the beta distribution, However, as

Figure 1 demonst

specify a unique beta distribution; curves 1 and 2 are iden-

tbs, map Ma arid modea are insufficient to

tical in these values and yet epesent different pdf @'s ,

etivity pdf is furtheno defined by the assumption of 2

a mean pa = (ma 9 4 mode, #,)/6 and a variance oa =

(Ma - From these two additionaLs ssumpt%ons, a

unique pdf can be assigned to activity a.

Although the existence of a beta distributed pdf is used

to justify several theoretical results in the application of

~~~~

1/ Mac Crimmon and Ryavec [ 2.1, po 7 0

8

PERT, in actual iaplementation a rather surprising simplifi-

cation is made, Only information about the mean f l and

variance oaz is retained about each activity, and only these

two quantities are employed in further anaiysi.:::.

a

One of the major aspects of this thesis will questiot?

1) the exclusive use of beta pdf's, and 2 ) aalculations based

and a2 alone, We will develop a PERT implementation

which is not bound by either of these restrictions, and use it

to point out several anomalous situations,

The network. In any large project the initiation of

certain tasks or activities must await the completion of other

activities, For example, the assembly of power supply can-

not commence until the design is completed and the components

have been procured. Representation of this interdependence

between activities is the second essential aspect of the

PERT analysis , c

We seek a convenient way to represent the time of

completion aspect of activities together with their inter-

dependence, The most rewarding structure has been a labeled

directed gpaph or network, called the PERT network. Tradi-

tionally, a directed graph G is a point set P, together

with a relation A on P denoted G = ( P t A ) .

of ordered pairs { (p,q) I p,qcP), where each ordered pair is

Thus, A is a set

commonly called an arc (or edge) of the graph. 21

2_/ Most graph theoretic terminolo y and results will be found in Busacker and Saaty [ b y , unless otherwise noted,

? I

9

In this model the arcs { a ) of espond to the

activities of the project, The points P of the graph comes-

pond to unique points.in time hich are h o r n as events in

PERT terminology, Events are denoted by integers,

1 is said to be labeled if there exist

functions on P, Q Y ~ or both, into a set of labels, In a

PERT network the arm { a) a ith their associated

pdf fa (01- some abbreviated pepre on of" it such as 2

) and the points i are labeled with ma8 E'II&, modeas 8 9

specific times in the time continuum, 2l The actual form of the graph, feeap configuration of its

res and points, reflects the interdependence of the activi-

ties, In general, an event (point) eo esponds to the point

in time (as yet unspecified hen an activity may be initiated

or when it is completed, E ts, t h e ~ ~ ~ o e, serve as initial

and terminal points of BT network, If an event is the terhinal point of sevepal arcs, it represents

the point in time hen _all those activities have been comb

pleted, In the graphical representation, an event is iden-

ti f ied by an lh 8 d an 'activity

identified by a Roman letter,

6 is initiated by event 10 and terminated by event 28 . .

For example, in Figure 2 activity

21 one of the major efinements of $his implementation will be to peplace the scalar labels on the points by distri5u- tions,

i I

10

Figure 2, PERT Bepresentation of Activity and Events

The PERT network is constructed so that all activties

terminated by an event are shown entering the event, and all

activities initiated by the event are shown leaving it, It

has already been stated that the basic PERT assumption is

that an event does not occur until all activities preceding

it have been completed, It is evident that this dependence

relation, together with the assumption that the project can

logically be completed, implies that a partial ordering is

necessarily imposed upon $he events (as well as the tasks)

of any PERT network.

any partially ordered set can be represented by an acyclic

graph, or network, and conversely, the path relation on an

acyclic graph induces a partial ordering of its points.

More sinply, the existence of a path from event i to event j

implies that j cannot precede i , and bhe PERT network itself

is an acyclic directed graph, Figure 3 shows a typical

PERT network layout, normally called the system flow plan,

This flow plan illustrates the dependency concept. Neither

activity b nor activity c can be initiated until the occur-

rence of event 10, which is in turn dependent upon the com-

pletion of activity a.

As is well known (Knuth [ 5 ] page 2 5 9 ) ,

1.1

Figure 3 9 System Flow Plan

Fnsom the examples in Figures 2 and 3 , we notice fire

distinguished svents identified as events 0 anclh, which

correspond to the initiation and completion of the entire

project, Graphs of this configuration are called two ter-

minal graphs, and from the problem constraints it is obvious

that all PERT networks must be two terminal networks,

The typical procedure in implementing PERT is to draw

the system flaw plan forthe project and then associate time

estimates with each activity, Once these quantities have been

added to the network, the entire graphic presentation is fre-

quently referred to as a PERT chart, In the PERT chart

in Figure 'I, and subsequent1

W, 3 presented beneath the arc representing a will represent the extreme and modal points of the activity pdf, while

the three numbers [ma, modea,

) are placed above the a c , These five scalars become

labels for the arc, This representation is standard in texts

describing PERT d will be used for the prsesent, even,

though it will need refinement in later sections, d In all

d Note that 41) the letters associated with each arc (activity) and numbers associated w i t h each point (event) are not labels, but merely arbitrary identifiers; ( 2 ) labels need not be scalars, and we could use the p&f*s themselves (in either closed or approximate form) as labels,

i

f I

12

calculations, the activity pdf8s will be assumed independent, .

though clearly there are cases in which they may not be; .if

the duration of an activity is dependent upon its starting

date (e.g., if it is dependent upon the weather), it is not

truly independent of the activities preceding.

It is evident that the structure of the PERT network,

together with the labels on the activities { a must be

given by the project description. The whole aim of PERT

analysis is simply t o then assign a consistent set of labels

to the events. In the most simple representation these event

labels are a single scalar which indicate the time at which

the event occurs, In a more sophisticated representation

these labels might be distributions indicatingthe range of pos-

sible event times.

In order to assign a consistent set of labels to the

events, and in particular to event -(and thus estimate the

total duration of the project), it is necessary to obtain a

time estimate f o r each of the paths through the PERT chart.

The definition of path found in Busacker and Saaty will be

employed;

A path is a set of arcs which, if properly

ordered, form a path progression, The no-

(i,j) represents a path from event i

t o event j,

The concept o f path length used in determining project duration

differs from the normal graph theoretic definition (which

is usually simply a count of the arcs involved), Mean path

I I

--- %engt& 1s t h e SLBYR sf t h e expected times of the activities

hich csmppisse the path;

fop all ens denoted L

Variance of is the sum of the vapiances of the

activities whieh eornjxxis~ the path; -----!-

0 for all %E denoted L ( P I , a (52

--- Mi nilrim h. is .the sum of the minimum estimates of the

activities which ecPrrmpPfse the path;

denoted Lm ( P ) ,

e maximum time estimates

hick ssmprise the path;

f o p all a € , denoted L 1. disregard the activity pdf

once it has been usecP to e t e the mean and variance o f t h e

activity dumt ion , only the rYlean path length L and its variance

E 2 arPe used in t h e typical implementation, Using the (5

mean path length ial event 0 to

renee of event i, or expected e

defined;

ed EETis can be

EETi = max e It is ~~~~~~ in sis to call this value the early

_LI=__B_ event ___n__ 'time of event ii% me slight Qustffiaation of this

~ ~ ~ ~ i ~ o ~ o ~ y will be given later, but in the authorRs opinion

eensislten@y with litePa$ure* In igum 4, each activity

i I

14

(edge of the graph) has been labeled with its corresponding

distribution f , (For simplicity in this arid the following a examples only the parameters (pa, oa2) and [ma, modea, 1 are given,)

days ( a l l estimates will be stated in days, although clearly

any unit of time may be used),

The early event time f o r 10, EETIO is e jght

EET, = 4

Figure 4, Calculation of Early Event Time (EET)

Although the expected time for the occurrence of event 10

is eight days after project initiation, the time estimates

allow for its occurrence in six days (activities a and b

requiring three days apiece, and c requiring not more than

six days),

maximum times estimated, event 10 ill not occur until ten

days have elapsed. The terms earliest possible time EFT

and latest time LPT will be introduced to repre-

sent the extreme values, The earliest possible time for

Occurrence of event i is defined,

On the other hand, if all activities require the

Emi = max [Lm ( P ) for all.

The latest possible time for the occurrence of event i is

similarly defined

LFT~ = max C L ~ ( p ) for all

15

s indicated, Emi and L are not defined in existing PERT

procedures; rathey, St is assumed

a normal distmaf

t EETp is the mean of

equal to Lt,2 ( p C ) ) where

€? - i s the same path that de%ermined €SETi (agy path P ( 0 , i ) for

is less than L ) will be denoted m the initial to the terminal

e function ins it maximum (isee9

EET:,) is called the critical s before, the overall

variance is assumed to be L,

event in this model is the expected time for the occurrence

J' Thus, EETi for any

of event i, and in particular EET, is the expected time for

the completion of the entire project, Thus it is-reasonable f o r

the project manager to set EET, as overall project goal.

ith this in mind, e may ask how late any event can

d still meet t e overall goal of EET, So late event

f is defined as the difference between the overall -

project comple't;ion date ET, and the expected length of time

Poequired to corn ities subse uent to event i;

LETi = EET,- max 1

T - I Y ~ ~ X {L for all P(i,m)] 0)

so that

(i, m )]= EET,

9 If mope than one path is found mum, the path with the largest variance La2 is taken to be the critical path,

equal to its maxi-

for all P ( 0 , m ))

and event i is found to be on the critical path, Other-

wise LET > EET s meaning there is a grace period between the

expected time and the deadline (i.e., time which still assures

reasonable probability of meeting the overal progect goal

EET m for the event, This period is called slack,and all

events not on the critical path have positive slack, It is now

evident what is "critical" about the critical path, and why

PERT analysts pay particular attention to the critical path.

The slack of an event represents a buffer which will absorb

some slippage in the completion of activities on paths pass-

ing through the event, and there is no slack on the critical

path; consequently, a delay in the completion of an activity

on the critical pathuuill'be expected to result in ppoject

overrun.

i i

The term early event time for EETi appears to have

osen to oontrast with late event time LET' in this i

determination of slack. Further, it is apparent that in

this implementation of PERT two consistent labels for each

event (point) of the network are determined and that their dlf-

ference is the slack,

The late event times of the PERT network present

themselves as appropriate project milestones, If the late

event time LET for any event i is not realized, there is

a high likelihood of a delay in project completion, It is

also apparent that one of the

i

I L-

1 :

17

Iscritical" path, be several paths *on which the

events contain negative slack !LETi < EET. 1; if the original

schedule EET is to be met, all such paths must be shortened

by the acceleration of one or mope activities of each, The

PERT chart ill indicate hich of the paths, if any, still

contain slack, so that the project manager may determine

hetber any pDesources m e available for shifting from activities

There may no

1

W

on these paths to activities on the no critical path.(.s) e I

l , 3 Analysis by the Critical Path Method

Sihce the entire goal of the crPitical path analysis

in PERT is to assign EETi and LETi in each event, let us

illustrate the procedure by an example,

Figure 5 shows a subnetwork of a PERT chart of a major research and development project, Labels beneath the arcs

are consistent with the notation introduced eaylier; for each

activity, extrema1 and modal estimates of completion time

are listed, Using the expkessions for mean = (ma 4 mode, + a,)/6 and variance3 o a

culated f o r each activity in Figure 6,

2 = (Ma - ma)/36, these values are cal- 2

Thenex% step in a critical path analysis is _the deter-

mination of the early event time EETi fop each event; this

operation is called the forward iteration or forward passe

above event i enclosed in a square, Thus,

the early event time for event IO is shown as 161 in Figure This quantity is found by adding the 6 days required to 1

---

I

18

Figure $. Network fiw Calculation of Ear ly Event Times_ and Late Event T i m e s

Figure 6.: Calculation of e t i v i t y Expected Times and Vam?fances

0 e

20

complete activity a to the EETO which is always taken to

be 0, Activity b is expected to consume 2 days beginning

6 days after project initiation, so its completion (event 20)

should occur at 6 4 2 = 8 days. Similar calculations may be

used to determine the early event times f o r events 3 0 , 40, 5 0 , 60, 80, 90 and 100 because each has only one path leading

to it from event 0. These times are shown in Figure 7,

\

Figure 7. E a r l y Event Times with No Pkrallel Paths Considered

2 1

Given EETgo and EETlO0, w e can c a l c u l a t e EET110.

C (0,110) is the path pass ing through event 100 - i t s length

i s 2 9 days, as compared t o a l eng th of 28 days for t h e path

passing through event 9 0 , s o E E T ~ ~ o is 29 days, Continuing

through the network, EETi is c a l c u l a t e d f o r each event i ,

a f t e r e a r l y event times on a l l t he paths ( 0 , i ) have been

determined, F igure 8 shows e a r l y event times for each event

i n t h e network, EET, is 32 days, wi th t he c r i t i c a l path

c o n s i s t i n g of a c t i v i t i e s a, b, c, d, and e,

Figure -8 , PERT Net om?k wi&h.,Alp Early Event Times Calculated

i

22

If the overall project target date is set at 32 days,

LET is 32, The expected time to perform activity j is

4 days, so LETj = 32 - 4 = 28.

event is shown in Figure 9,enclosed in the circle above the

early event time. Calculation of the late event times in

an operation known as the backward pass is analogous to the

CO

The late event time for an

ard pass, with LET, serving as the initial time for

calculations as EET 0

culation of LET80 is

100; late event time

after calculation of

did for the forward pass. Thus, the cal-

dependent upon the path through event

LETi for any event i can be found only

late event times for all events on all

(i, m ) * Figure 9 shows the early and late event times

for all events in the PERT chart; slack may be determined for

any event by subtracting the early event time from the late

event time,

0 , 10, 20, 30, 40,w) was found to be critical

path with a length L ) of 32 days. As will be shown in

Chapter 111, this quantity accmately reflects the expected

time to complete all of the activities on this path in se-

quencer assuming accurate time estimates f o r the individual

activities, Thus, if the project were to be executed a number

of times or @ttPialsn, with values of the individual activities

n randomly from their pdf8ss, the mean value that would

be obtained for the length of this path is 32 days, with the

values obtained on various tPials ra ing over the interval

of 25 to 39 days,

ork Showing E a r l y and Late Event Times

i

24

However, if we examine the path consisting of

activities a, bp,c, k, n, 0, and p, whose expected value is

31 days but whose extreme values a,re 25 and 3 7 , w e see t h a t

there is considerable overlap in its range of values with

that of the critical path, In the performance of the series

of trials of the project, it is reasonable to expect that for

certain trials a larger value would be obtained on this path

than on the ftcriticaltl path. If, then, for each trial the

larger of the two path values is chosen to represent the

completion of the entire project for this trial, we are in

effect simulating the performance of a project containing these

two paths over a number of trials, Because for each trial

we have assigned to project completion time the larger of

the values representing completion of the two paths, the

mean value for project completion may be expected to be greater

than the mean value found for the "criticalf1 path. As more

paths nearly equal in length to the llcriticallt path are

added to the project, the probability that the overall ppoject

completion time exweds the length of any of the paths con-

tinue to increase, Thus it appears that if the individual

path lengths are treated as distributions (which is consistent

with the assumption that activity times are distributions)

rather than deterministic valves, the project duration cal-

culated may in some cases be different from that obtained

in ordinary PERT analysis,

Monte Carlo simulation techniques be utilized to account for

Ringer [ 3 ] has proposed that

the effects of iylcilivi ual activity distributions; however, we

intend to present accomplishing this without

resorting to laepe ted tf.Pals,

ve seen here ho the typical PERT analytic tech-

niques may be applied to p sject scheduling, but we have also

seen that the ~ ~ ~ ~ ~ ~ t ~ o ~ date predicted by the use of these

echniques might fa rliela than the time pleedicted by

o the r statistic 1 methods, In Chapter 11, an example wherein

the PERT-calculated completion time is clearly less than that

calculated by analytical means will be ?resented, and addi-

tional questions ill be raised c0ncernin.g the individual

activities,

6

CHAPTER I1

ANOMALIES ARISING FROM CRITTCAL PATV \SSUMPT?n',T!T'

2 . i Effects of Parallel Path Distributions

As we noted in the description of the PERT network,

' an event i is considered to occur only when all activities on

- a l l paths p(0,i) have been completed,

occurrence of i is the maximum time L ( p ) obtained on all

paths p ( O , i ) ,

That is, the time of

The expected time of occurrence of i is then

E(i) E [max (L ( P I for all O(O,il]].

c ( O , i ) is clearly longer than anyp,' (O,i), that is, if

L,(p,)>max EL, ( p , ' ) ] , then E [ max { L o ) ] = E [L(D~) I = LcL (aC). pdfas of the individual paths p ( O , i ) ,

made in PERT analysis is that the paths p e t

contribute to the determination of p i .

ample in the previous chapter suggests, the addition of other

paths whose pdf*s overlap that of pc can cause the expected-

time of event i to f a l l later than the time calculated using

In this case, it is not necessary to consider the

The simplification

(0, i) never

However, as the ex-

lo' The following example demonstrates quanti-

ltatively the effect on caused by the inclusion of a second

( 0 , i ) whose pdf is identical to that of pC(O,i).

The network dshown in Figure 10 contains two critical

paths pc(0,30); the mean path lengths LM(pC) and variances

6J This example is from MacCrimmon and Ryavec [2 ] . 26

2 7

c) are identical, While each path is shown to be

composed of only two activities, any of the activities a, b,

C, d may be thought of as paths whose durationsare represented

by the pdfDs shown; In addition, there may be other paths

C 8 ( 0 , 3 0 ) , but it will be assumed that max [LM (pct)I<L, ( 0 ~ 1 ~ To simplify the calc)lllations, discrete probabilities are used

to represent the activity pdf's; these are shown in Figure

10,

Distribution of a & d

1 2 3

t

Distribution of b ?& c

2 4 6 t

Figure 10, PERT Network with Discrete Activity Distributions

28

The pdf of the path p ( 0 , 10, 3 0 ) is determined as

follows. The pdf of event 10 is that of activity a, s o the

path pdf isa composition of fa and fb in series. Table 1.

is used to perform this calculation. In each row of i of

table la, the value ai is given f o r activity a

activity bo

f (ai and bi) = f(ai) 0 f(bi) is listed in column 6, The

pdf of activity a in series with activity b is then found

using

and bi for

The sum of ai and bi is listed in column 5;

fa. b (t). = E, f(aibi), ai+bi= t

and listed in Table lb. The mean of this path is found by

and is equal to 6

methods. Because

Calculation

2

days, the value found using standard PERT

this path is "criticalt1, Lu (0,) = 6.

of the fjO is demonstrated in Table 2. The

pdf@s of the two paths p c ( 0 , 3 0 ) are listed across the top

and down the sides of Table 2a. In the table, position

(i,j) is the maximum of ti and t

individual combinations of ti and tj (not shown) are found by

taking the products of the individual probabilities f(ti) and

f(tj), A s before, the probability for each ti in the pdf is

found by summing the probabilities of all entries with value

The probabilities of the je

ti in Table 2a.

event 30 are shown in Table 2b, and the expected time t o

complete event 3 0 is found to be about 14.8% greater than

Probabilities of the individual values f o r

2 9

a, Sum

be Distribution of sum,

mean = 6 , O

Table 1, Distribution of Activities in Series.

a. Max

be Distribution of maximum.

mean = 6,89

Table 2. Distribution of Paths in Parallel,

f

This rathap simple example, showing that the expected

time of an event i (EETS) need not be a simple function on

( 0 ) f o r all paths , O ( O , P ) , suggests an alternate implemen-

tation of the PERT concept, In fact, we know that there is

really a pmbabflity distribution function associated with

each event of a P

it is calculated, serves at best as a crude approximation of

network, The EETi for any event, however

this pdf, It would be more meaningful to store the pdf itself

at this point,

reasonable approach to the labeling of events is to

the pr-oc6dure used for the activities. We could

assume a standard form f o r the pdf and define each by providing

the [min, mode, max and ( p , ~ 2 ) ~

ing appearance :

So a PERT-chart might have

'R

i I

Certain of these labeling parameters are easily calculated,

e.g. , mi = EPTi

Mi = LPTi

B u t calculation of the really essential parameters remains

elusive even if all of the activity pdf's are given in /

closed form as in the following example.

A simple network can be constructed given two activities

a, b y both with uniform pdf's

If the activities are linked in series as in the following

diagram,

then the probability of event 10 occurring at time t is pre-

cisely fa(t)e The probability of event 20 occurring at time

t is evidently

P r Z o ( t ) = pr(ta) and pr(tb) for all ta,tb such that

t, tb = t, o r

ma<t,<t - - mb

And for this example of f, and fb rectangular distributions,

t 'mb

33

Or pictorially,

e - -

ma "b b f

~n accordance with standard PERT assumptions E(fa b) =

E(fa) E(fb) a n d a 2 ( f , ~ b) = @z(f&) 0 2 ( f b ) $ so that the Gal-

culation of these quantities by s u m m i n g over the specified

paths is evidently eonsistenk RT approach, -mlated in parallel 2/

as in the following net

ork is seldom constructed with a single pair of events joined by be regarded as a -composed of' more than one activity with no effect on the results p esented here,

activities, buz either a or b'may

34

a

t h e n t he pwbability that event, 10 will occur precisely at

time t is evidently

Again, when Pa and fb are uniform distributions, we may let

X = max(ma,mb), Y = min(M,, Mb), 2 = max(Ma, Mb),

, t < X f 0

I o , t>Z

or pictorially,

- - - x Y Z

ma Ma mb Mb

r* I +

35

(assuming ma< MbS mb'Mae If mb->Na9 then evidently

fa+b =-fb; if r n a ~ F l b p f,+b = fa),

These results help to illustrate, perhaps, why the de-

signers of PERT despaired of any attempt to derive the event

pdf*s as a function individual activity pdf*s and chase in-

stead to define the mean of fa. b

and the mean of fa b as the maximum of ua and pbs

this extremely simple case of uniform activity pdf#s the re-

as the sum of

Even in

sulting event pdf s are hopelessly ieldy, And any calcu-

lations based on these derived distributions could be expected

to yield even more cumbemome pdf Os,

further demonstration of the different event pdf's

that may be de ived in solution of the network are presented

in Figure 11 through 14,

when two input pdf's (identified as Distribution A and

Distribution B ) are composed in parallel o r in series.

eral&,@ computer routines which were used t o .perform the parallel

and series compositions are those used in the implementation

hich are p l o t s of the pdfls resulting

The gen-

described i p n Ch ; they are derived in Ch

In Figure 11, two uniform distributions are combined in para-

llel, as in the above .exBmple, Figure 12 demonstrates the

o roughly triangular distributions in parallel;

f is shifted noticeably t o the right, indicating

pdted time, The Rekt two plots, Figure? 23

e 8 situation in ieh the output distri-

ore modes than the input, In Figure 13,

the series e bimodal distribtions produces a

trimodal resultant, and in Figure 14, a bimodal distribution

results when two unimodal distributions are composed in I .

parallel.

These plots demonstrate that even if all the activity

pdf's are of some standard form (e.g., normal o r beta distri-

bution), there is no assurance that the resultant event pdf's

need be of the same form, or even that they be unimodal, and

there is no simple analytic technique f o r obtaining them.

If, however, the algorithms necessary to numerically derive

resultant distributions from some arbitrary input distributions

are developed, the distributions themselves may be stored as

labels in the PERT network. The network would then have the

appearance,

37

devtates from

9-00

0

9.00 10.60

12.20 1s. PO 19.0a

TI

e

1 12.20 19.80 15. PO 17.0t

TI

0

9.00 10- 60 12.20 13.80 15.40 17.01

TIME' PflRRLLEL UN I FUR

Figure 11, Parallel Combination of Two Uniformly-Distributed Activities

.312

* .a50 I-- w

w .187

P- I--

1 H

H

DISTRIBUTIUN A

i

0

9.00 10.20 11.40 12.60 13.80 15.00 TIME

DISTRIBUTIUN B .312

.250

.1$7

e 125

.062

0

9, DO 10.20 11.4~ 11.60 13.80 15.00 TIME

RESULTANT

- 296

222

.148

* 074

0

9.00 10.20 11.40 12- 60 13.60 15.00 TI

PRRALLEL TRIRNGLE

.358

DISTAIBUTIUN R 39

?- ! I -

-J w

W

0

9-00 11.B0 12-60 13.80 TIM

DISTRIBUTIBN B e 358

.286

.215

1U3

.071

0

9. 00 10.20 11.40 12.60 13- 80 15-00 TIME

I RESULTPJNT I .190

?- .I52 cl

.111

'E -076

w a3

0 ar Q -838

0

18.00 20. 40 22.80 25.20 27.60 30.00 TIME

SER I ES B 1 -MC1OFIL

Figure 13* Series Combination of Two Activities Having Bimodal Distributions

,952

+- .762 I-

-I .571 w

w

DISTRIBUTIUN FI

[I)

Q CT a, .1BO

.381

0

9-00 10.20 11.u0 12. 60 13.80 15.00

TIME

DISTRIBUTIBN B .595

0

9.00 10.20 11.u0 12.60 13.80 15.00

TIME

RESULTRNT

1 9.00 10.20 11.40 12.60 13.80 15.00

TIME

Figure I&, Bimodal Resultant of Two A c t i v i k 3 e s Cornhilied in Parallel

In Chapter IV a computer model which is a direct

realization of this idea is introduced, Actual distributions

will be associated with both activities and events as labels,,

2.2 Activity Distributions

The examples chosen here to demonstrate the combina-

torial anomalies of PERT include activities whose pdf's are

other than the PERT-assumed beta distribution. While the

beta distribution is a convenient form for representing the

estimated time to perform many activities, there may be in-

stances in which the best estimate of activity duration is

better represented by some other distribution form.

An example of a bimodal distribution might arise in

the case of an activity whi-ch requires the utilization of

equipment which could be made available on either of two

dates; if the material is not ready for use of the equipment

by the earlier date, it is possible that progress on the

activity might be suspended until the later date, In this

case we would expect the activity pdf to be concentrated

around two times based on these availability dates, and the

concept of a "mean time" would probably be meaningless.

Another bimodally distributed activity might, be characterized

by one set of time estimates based on the assumption that a

closed form solution will be found to a particular problem

and another set associated with the need to resort to some

time-consuming enumerative technique, Some o t h e r activ'kv

42

might 5-rn7ve a seri-ss of m t - a n d - t r y attenpts; t h e manager

may prefer t o use a multi-modal o r a uniform distribution

t o represent the pdf in this instance.

It is not our intention t o develop a number of 'Istandard1l

distributions for PERT activities, but rather t o demonstrate

that the requirement that all activity estimates conform t o

one form of distribution 1) isunnecessary, wd 2 ) CRY! hp

unduly restrictive, possibly frustrating attempts t o apply

one's best estimate to an activity pdf. The implementation

which is described in Chapter IV accepts pdf's of any form,,

h

CHAPTER 111

L STATISTIC L DWELOPMENT

3.1 Procedures for Calculating Event pdfrs

In order to calculate the pdf#s associate& with each

of the events in the network, we will define two operations

which may be performed to produce a resultant distribution

from two arbitrary input distributions,

1) The pdfEs of two activities ( o r an activity and an event) can be composed in series to yield the pdf of the completion of the two activities performed in sequence.

2 ) The pdf*s of two activities ( o r sequences of activities) can be composed in parallel to obtain the pdf of the time when both activities have been completed if they are performed simultaneously ,

We have two basic cases which might be diagramatically

represented as follows:

Case 1 (series)

Case 2 (parallel)

a

In both cases we are given the pdf*s for the event i and the

activities a and b, As we have indicated in the previous

43

44

chapters, we can compose the pdf f with the composition of i

3 . fa and f b to get f

Further, to denote the Itcomposition processtt we will

use the symbolism fa .b in the first case and fa + b in the

second. where these may be called the 'tseri.altt and. l 'para,llel ' t

composition, respectively. This j s the same symbolism that

was used in. Chapter 11, but without formal introduction.

For the algorithms used in the implementation it was

foinnd to be more convenient to represent the distributions in

the form of the cumulative distribution function (cdf). The

cdf of a, denoted Fa, is related to the pdf by t

From the definition of fa and the restriction that fa(ta) = 0

for ta < ma, we observe that Fa(t) is the probability that the

duration of activity a is no greater than the interval t

and that Fi(t) is the probability that event i w i l l occur at

a time no later than t.

Series Composition, If' Fa and Fb are composed in

series to obtain Fa ,b , the end points of the resultant di9-

tribution are

= m + m b , ma b a M a e b = M a + M i , e

The probability that' ta 6 tb is no greater than t is the

b sum of the probabilities of all combinations of t a,nd t

which are less than or equal to t, a

' I I

,

We can differentiate the expression with respect to t csing

the method given by Hildebrand ([7] page 360) to demonstrate

that it is equivalent to the expression introduced in

Chapter 11: t '"b

ma b (t) = A F (t) = I (A [fa(taflFb(t-ta)

dt dt a *..,b

I Parallel Composition, If Fa and Fb are composed in

parallel, the end points of the resultant distribution are

I = max(ma, mb) "a + b

. M a + b = max(Ma, Mb).

The probability that both activities a and b are completed

I

46

on or before time t is the probability that a is completed on

o r before time t and that b is completed on or before time t, t h u z

pr(t a- < t and tbjt) = pr(tait)

(t) = Fa(t) * Fb(t). F

pr(t&t), or

a + b Differentiating,

fa I- b (t) = fa(t) * Fb!t) 4 fb(t) F2(’’,

which is the exprp??’rn used i n Chaptzr T T to perform tb.e

parallel composition.

3 * 2 Cofibinatorial Techniques of PERT

The series operation used in ordinary PERT analysis

is the summation of means and variances from two distri-

butions to obtain these values f o r the derived distribution;

the parallel operation consists of assigning the greatest

mean value from among the input distributions and the

variance of this distribution as the mean and variance of the

derived distribution, We can now investigate the validity

of these simplifications.

Mean and Variance of SePies Combination, Define the

mean of the distribution f, t o be Ma

a = E(ta) = tafa(ta)dta,

ma

I

47

the mean of the distribution of sums from f and fb will be a

m m a b m ma b

.. - -

l"a . - ,b e +

-_

48

So t h e mean of the distribution of the sum is equal t o t h e

sum of t h e means of the i nd iv idua l distributions, as is

assumed in PERT. , To find the variance of this sum we define

F 1

49

m b m -' a

1 I

These results demonstrate that the method used in

PERT of summing means and variances along a path is quite

valid, and if there were no parallel paths in a network, the

PERT analyst would be completely justified in discarding the

activity distributions and using only their means and variances

to calculate the expected event times, It is inthe parallel

composition that errors are introduced.

Mean of a Parallel Composition. As has been noted

previously, the time of occurrence of an event j is the latest

time of completion of the activities terminated by event j,

and the mean time of completion of both activities a and b

TO evaluate this sum, let us is Na4b make the substitution

= E {max (ta#tb)) . max(t,,tb) = &(t,+tb+(t,-tbl 1,

This may be proved by considering first the case of t >tb,

in which the right hand side reduces to t,, and the case of

tb> t

a-

in which it reduces to tbe Using the result derived a' above for the expected value of a sum, we fipd that

max (tajtb) a 4% 4 E(lta - tb])] 0

S o , to determine a+b j e must evaluate E( Ita - tb I 1.

Let us first assume that the distributions fa and

fb do not overlap, because ma 3 Mbe Pictorially,

51

Then E ( I ta - tb I ) becomes E(ta - tb), which is Ba -l"b9 SO

that - -

a+b e

So here the standard PERT ssmption that the composite mean

of the two paths in parallel is equal to the mean value of the

longer path introduces no errors in detemniningi(,

I f , however ibutions do overlapg the situation

changes. Let us assume th fb = fa is normal d with a mean 2 pa and variance o . 6 a first step in the evaluation of

E (Ita - tbi), we note that f(ta - tb) is normal with a mean

Also,

Let X =

f(X) = 0 x < 0,

0 5

81 fa can be made very nearly normal, so long as we retain i rn 2 0 and Ma finite, good approximation is obtained i? we let f (t> = n(ya, fa(t) = 0 easewhere,

'a) f o r pa - 3'azt< "a + 3oa, - -

b

The expected value of X is then

m

4Tr So we find that

bla+b = p a + “a/&- pa + e 5 6 ‘a. In this case of normal distributions, the approximation

c1 a+b max ( pb)$ which is ordinarily used in PERT

analysis, intyoduces an error slightly greater than one-

half a standard deviation, This r e s u l t can be represented

pictorially,

I

t

53

This result could have been obtained directly from the

cdf tables found in Dixon and

value of t for

Fbt SO Fa(t) =

assey [6Ie The a+,b is that

We know that Fa = Fa(t)ePb(t) = 0,5.

= O,q%, Using the cdf table for the

normal distribution, we find that Fa(t) = 0,71 for t = v a + 0 . 56 aas

hen fa and fb are arbitrary pdf’s, there is no con-

venient closed form representation of

example, Chapter I1

However, these calculations with normal distributions, to-

gether with the results of Chapter I1 show that the standard

PERT assumptions to the effect that

a+b or “a+b (see, for

ere both pdf’s are known to be uniform).

and

can be quite misleading, The only accupate method is to re-

turn to the original definition

b satisfies Fagb(ta Evaluation of the points t given this

open form is usual1 imppactical by hand but relatively easy

by computer-if one approximates the integral by finite

summation techniques, However, in order to calculate Va+b

we must MOW have the actual distributions Fa and Fb (or fa

and fb) readily available,

standard PERT implementations ich retain only p and 0

as approximate distribution parameters.

This is our major difference from

a

3

Given Fa and F the computer procedure to calculate b’ Fa+b Or Fa e b is straightforward,

which were previously presented to support our contention

that complete distributions must be retained in the data

structure for analysis of a PERT network, were generated

by this procedure, So, too, was Figure 15 which supports-

Figures 11, 12, 13, and 14

our last calculation of pa+b where fa = fb is a near ly normal

distribution. For this plot, the input distributions have

a mean pa = 12,O and standard deviation o

zero range of these distributions is from ma = ua - 3 0 , = 9.0

to Pia = pa + 30a = 15.0,

t o obtain f,+b.,

non-zero value, no appreciable value for f,+b(t) is seen f o r any

t<10,2, This plot aemonstrates that the mean of the

resultant pa+b ,pa. The routine which generated the plot, a l s o

calculatedga+B = 12.56, which is in agreement with our

analytically calculated value, and Oa+b= 0.82.

= 1,O. The non- a

fa and fb are combined in parallel

While f,+b ( 9 . 0 ) may be assumed eo have a

55 397

0 I I I I I I I I I . oa 10. 15.00

a 10.20 13. QD 15.00

.Figure 15. Parallel Combinations of Two Nearly- Normal Distributions

CHAPTER I V

1MPLEIM.E.NTAT I O N

4 , l Basic Data Structure

The computer model which is described here demon-

strates a methodology for determining the pdf associated with

any of the events in the project, and in part,i.cular that of

ppoject, completion, once the PERT network has been establjshed.

and a pdf defined for each of the activities, This mod.el

has been implemented on the UNIVAC 1108 at the University of

Maryland, using the RSVP list processing language [ 8 ] , de- veloped by Robert Liebermann.

The basic element of RSVP is call.ed an atom, RSVP

atoms may be used to represent different types of elements

within a data structure by assigning to each atom an integer

called its . The elements of a PERT network are the

activities (arcs) and events (points); activities are repre-

\

sented by type 1 atoms and events are initially represented

by type 2 atoms, As information becomes known about the pdf

associated. with an event, its atom type will be varied. In

the accompanying figures, an atom will be pictured as

where <type>is the.integer identifying the type of element

(activity, event, etc.) represented by the atom and <name>

is the symbol (a, b, c,..., f o r activities; 1, 2, 3, .*., for events) identifying the particular element. In the

56

, ! I

57

following discussion atoms will be referred to by name; e.gal

atom 30 is the atom representing event 300 Pointers are

established Pinking the appropyiate activities and events;

the following subnetwofk

8

is Pepsssnted b 8 substructure whose appearance is

In addition, e can associate with any atom any

collection of data., Since the atoms of our data structure

correspond to the points and edges of the PERT network, we

will call s w h associated data the labels of the atom. This

teminblogy agnoees w i t h that of graph theory, where one speaks

Pabeled edgese I$ should not be confused with the common

computero usage in which a label is treated as synonymous with

identifier, Note that just as one can associate more than one

label wkth the points and/or edlges of a gmph, so we may have

bels, OP collections of data, associated with the

stoms of our data stmcture,

The ppecedi. chaptePps have establ shed that, for accu-

rate PERT analysis, the COP ect labels to assign to the events

and activities of the network w e their actual probability dis-

tributions, not a condensed version consisting of the mean

and variance alone, So these distributions are the data that

we associate t o the RSVP atoms in our computer implementation.

i

Tnitially, pfif’s are known f o r only the activities and

evevlt 0. These p d f ’ s could be stored as labels; however, we

act,i.ially store cdf Is because the are more convenjent ~ C J i ise i n

the operatiens f o r combining dirtributioq fianct.iorls i n parallel

and i n series, For continuous functiow, a s ape assiir~ect nere,

a one-to-one correspondence exists between the pdf and cd.f,

s o we may construct a distribution o f the one form from the

other, as necessary,

An individual F, is stored as a 13 place vector con- I”! - m

10 a a. + n 0 ma t M a$ F,(tO), .,., F,(tlO)) , where t, = ma

We are then approximating a continuous cd.f with 3.0 points,

interpolating with quadratic functions between the points, 2l

4,2 Calcuiation and Assignment of Event Distributions

The process of determining the parallel o r series com-

position of the two distributions Fa and Fb consists first of

finding the points t which Fab(t) will be evaluated.

saw in Chapter IIT,

A s we

The choice of 10 points to represent the cdf is purely arbitrary, a tradeoff between the accuracy with which the cd.f is approximated and timing and storage requirements, Rs- cause some cdfls may have a wide range, but wlkh F(t) highly variable over a small range of t, it may be preferable in some sgplications to vary the time interval between pcints.

59

In the selnies composi iom, as we noted in the previous chapter,

I this integral w i t h the finite sum re-

sulting from Po@

a ma and substi 10 I

obtain I

i consists of information about the

itiating and terminating

d the cdf a m specified, The existence of an event is I I

implied by its ormation, and

~ ~ p ~ e s e ~ ~ ~ n g t e a upon the first reference'

the input,. POP each activity, an a c t i v i t y

ted and link of the bracketing

d the cdf is stored and 1ilnked.to the activity

ose activities have

been labeled s ana a corlpesponaing st ucture as it

ppeanos in s t o r

In Figure 16, all of the event atoms are type 2, with

1 the exception of atom 0 hieh is designated type 3. This - -____ ~. ~- ~- - ~- _ _

i I

----I

r--------

1

Figure 16* Initial PERT Network and Storage E q u i v a l e n t

type of atom will be used to represent an event, whose crlf

has been determined, Event 0 serves as a starting pint f o r

a recursive procedure for calculating the cdf8s of all events

in the network,

starting times for each activity R initiated by event i.

ny known Fi is used as the distribution of

Fi is composed in series with each F

used in determining the cdf of the event terminating a. Let

yield Fieawhich is a

ug assume that the particulars activity a is terminated by

event j,

of activity a (as distinguished from the input Fa, which is

the distribution of the duration of a),

Fiea is the distribution of the time of completion

I f no other activity

If, however, another iea terminates at j , then F = F j

activity b is initiated by an event k, pictorially, -

in parallel with e a then to determine F we must compose Fi

Fk . b j*

, Until Fk e has been calculated, we will store Fag

as a temporary label for event j , denoted FIj and called a

"partial distribution" of j. After the cdfRs of all activities

terminated by j have been used in the calculation of F'j, it

is denoted F. and referred to as the forward distribution of j . J -

T h j s process will now be illustrated for the PERT

network pictured in Figure 16. Activities a and d are

initiated at time 0, s o FIl0 is set equal to Fa and F t 3 *

is set q u a l to Fd.

these activities, the cdf's are renamed F10 and F30 and

their atoms become type 3. a "next event" list, a list which points to the events avail-

able f o r use in the series operation.

Because there is only one path to each of

Both of the events are placed ir!

Now some event, such as 10, is chosen from the "next

event" list. F10 and Fb in series determine the distribution

F20; atom 20 is changed to type 3 and event 20 is added to

the "next event" list.. F10 and F are combined in series to g yield an F'40.

been performed t o this point.

Figure 17 reflect the processing which h a s

- .

Next, F30 and Fe are combined in series, and the re-

sultant is combined in parallel with F'40 to obtain " 0 .

Series combination of F20 and Fc gives an F', ; the resul-

tant of the series combination of f4o and fh is combined

in papallel with F' t o determine F, In deriving F, this

implementation already presents more information than a simple

I

OD

PERT analysis using the assumption of a critical path.

Given the actual distribution F,,the project manager is able

t o more realistically choose a project completion date t, , hich the necessary times of the events can be derived.

For example, a conservative manager may choose'for the project '

completion date t, a time somewhat later than p, depending

upon the shape of F, . On the other hand, if the payoff -7

- -. - Figure 17. PERT Structure After the Calculation of fzO-

I

64

appeared sufficient, he might gamble on an earlier date, but

at, least he would be apprised of the risk involved.. Lacking

interactive capabilities, the implemenkntion d.escribed h e r e

automatically assigns p, as tm . The time required to complete all activi.ties subsequpnt

t o an event i i s a function of the activity cdf*s rather than 3

fixed time, and therefore t h e latest time f o r a.n event to occur

and permit the project, to remain on schedule is a, cdf, denoted

\, Gi, and called the backward distribution of i, This cdf replaces

the single value LETi in PERT notation.

combining in parallel the cdf's of all paths (i,a ) and

Gi is obtained by

subtracting t h e time values of the resulting cdf f r o v t,,,

Returning to the example, GZ0 is found by subtracting

p

iare now designated type & * indicating,that both forward and

from tag and G4.o by subtracting Fh from t,. Atoms 2 0 and 40 C

lbackward distributions have been calculated, and events 20

and 40 are placed in the "next eventtt list. A step from event / 40 yields G30 and a G Il0; GZ0 and F a r e composed in series to

yield a cdf which is composed in parallel with GIl0 to deter- / b

I

I It would be pointless t,o specify a Go' since all 10"

,'mine G

times are measured relative t o to, and s o this backward pass i s I

f!cornpLeted. Figure 18 displays the PERT network after all I

9ctivities and event have been labeled with their cAfcs, I _ - _ _ -

m 40 t

Figure 18* PERT Stmc tu re With All pdf's Shown

66

4,3 Storage Requirements

The RSVP system all.ocates five words of storage for. w.ck

atom it creates. . T h i s five word area is known 3,s the.

a,tom hea.cI and contains such information as t h e akom nlimber

(a unique identifier assigned by RSVP) I 1 ocation of the

inward and outward links (which w i l l be explnjned below),

and a pointer to the d-ata f o p t h e etom.

Each event in the network is represented by a single

aton and in addition requires 13 words f o r its cdf, If both

forward and backward cdf’s were retained, as would be the case i n

an interactive system which allowed the project manager to

retrieve the forward and backward d5.stribirti.on r3r the slack

of any event in t h e network, 31 words of storage would be

require4 f o r each event in the network. However, in this i m -

plementation we keep only one distribution at a time. In

addition, four w0rd.s are used to stare the mean and variance

of the forward and backward distributions. Two additional

w o r d s are used per event f o r a list whi.ch links (PERT) event

number t,o (RSVP) atom number. In this implementation, then,

the storage required for Ne events i s Ne ( 5 -t 13 + 4 + 2 ) =

24 Ne words.

Each activity also requires five words for its atom

head an3 thirteen f o r its cdf. In additi.on, each activity i s

linked to two events’. Linkage is accomplished in RSVP throuch

use of outrivlgs and. inrings, which are lists of pointers (links)

to and fi-on, other atoms. Fnr path qctlvi-ty, there will be

67

an out:'link attached to the atom of the initiating event,

an in link and an out link for the activity atom, and an in

link for the terminating event, Na activities, therefore,

require Na (5 + 13 $. 4) = 22 Na words of core.

The total storage requirement is 23Ne 4 22 Na words,

If the ratio of activities to events in the network is about

3 to 2, a relatively large 500 event network would require

about 28,500 words. This number might be considered t o be

near the limit of the "fast" space available, although the

virtual memory capability of RSVP makes the total space

available practically limitless. -

List proeessors that operate in a v i r t u a l memory space

map (or page) the excess over fast core capacity to some

peripheral device such as tapes, disk, o r drum. In general,

the resultant access or page swapping can be expected to

significantly degrade execution time, However, if we consider

an actual PERT network we see that most processing is local

I

in the Sense that only a small portion of the entire network

is necessary for processing at any single time. There is no

need, f o r example, to retain in core all those activities

or events for which the forward ( o r backward) pdf has been

calculated and which are not involved in current calculations. I The auxilliary storage facility of RSVP was designed I t o take advantage o f just such situations, Atoms are not I

1 paged, but treated on an individual basis, Individual atoms - . - __ . - -.-

are not mapped i n t o fas t core u n t i l needed ( i .e . , r*eferenced)

nnil they are i n d i v i d u a l l y ma,pped out orl t h e basis of usage,

Consequently, a t any time w e may expect t h a t the data sstrgcture

i n fast memory c l o s e l y corresponds t o t h e actual collection

of events and a c t i v i t i e s we-*are then working w i t h , and t h a t

only one access t o pe r iphe ra l storage w i l l be required

event o r a c t i v i t y f o r any given pass through the network, per

Consequently, core requirements might be e f f e c t i v e l y

reduced t o about 2,000 words (for any s i z e network) without

unduly fiegrading performance.

4.4 Execution T imes

Once RSVP space has been i n i t i a l i z e d and the PERT

network b u i l t , the time r e q u i r e t o s o l v e t h e netwerk is

approximately p ropor t iona l t o t h e number of a c t i v i t i e s i n

the network,

is found by t ak ing t h e serial composition of the a c t i v i t y cdf

and the cdf of t h e i r i i t i a t i n g even+,,*

then assigned t o t h e te rmina t ing event o r , i f a cdf has been

previously ass igned t o t h e te rmina t ing event , the two cdf 's

are composed i n parallel. Therefore , f o r each event , the

number of parallel opera t ions performed is one less than the

inward degree of' the event.

network is equal t o the number of a c t i v i t i e s ,

of p a r a l l e l opera t ions performed i n c a l c u l a t i n g forward dis-

For each a c t i v i t y , t h e cdf of time of completion

This composite cdf is

The t o t a l inward degree of t he

The t o t a l number

-

tributions is Na - (Ne - l), because the inwarr'l degree of event 0 is 0, Similar analysis yields Na series and Na - Ne + 1 parallel operations in calculating the backward distributions.

Total time consumed in solving the network is then:

T = 2[NaTs + (Na - Ne + 1) Tp], where Ts is the time required for a series composition and

Tp is the time required f o r a parallel composition.

Chapter V presents examples of_'output from..this model;

The total time required for an 11 event 15 activity network,

including RSVP initialization, model input, network solution

and output was 2.9 seconds.

I

CHAPTER V

- SAMPIX OUTPUT

The model which we described i n t h e previ.ous chapter

has been exerc ised t o demonstrate the calcuZa.tion o f event

d i s t r i b u t i o n s i n a small PERT network. The results o f these

runs a r e presented i n F izures 19 through 21. For simplicity,

a l l o f t h e a c t i v i t y d i s t r i b u t i o n s are of the. nea r ly normal

type introd.uced i n t h e examples i n Chapter 111. For each

a c t i v i t y a , given p ando,, t he end po in t s of t h e cd.f are

ma = pa - ?a, and Ma = pa + 3 ~ ~ . a

I n these i l l u s t r a t i o n s , t h e input to t h e model i s shown

under the head-ing "ACTIVITY DTSTRIBUTTONS1t . The output i s

l i s t e d under lt'GVENT DISTRIBUTTONS" . The network i s displayed

w i t h t h e a c t i v i t i e s labeled w j t h t h e i r d i s t r i b u t i o n means and

variances.

In the input , l i s t . i n g , t h e braCketing events of each-activ-

i t y are shown under " E N I T I A T I N G EVENTi1 and ltTERMIl~ATING EVEfSTtl.

The d i s t r i b u t i o n parameters pa a n d o , 2 , are not i npu t ; in-

s t e a d , t he 13 place vec tor described i n Chapter TV i s used t o

s p e c i f y Fa.

under "DURATIONtt, ftMEAN1t, and "VARTANCE" . F i n a l l y , Fa as

input ( i n vec tor form) i s l i s t e d under "INPUT DISTRTRUTIONtt..

The model c a l c u l a t e s u and oa2 and l i s t s them a

Output from the model ( "EVENT DISTRIBUTIONS" ) c o n s i s t s

of t h e derived information about each event , l i s t ed under

'IEVENT" i n o rde r of i nc reas ing expected time u i . The mean

70

h -7 1

and variLancea

BUTTON1t, followed by t h e mean and var iance of Gi lander

"RACKWARD DISTRTBUTION" e The llSLACK1t 1 i s t d is the d i f f e r e n c e

between the mean of Gi and t h q t of F i , and may be t.hoiight of

as t h e mean of slack d i s t r i b u t i o n . The concept of a slack

d i s t r i b u t i o n w i 3 . I be considered i n greater de ta i l j v l t h q

fol lowing chapter ,

of Fi are l i s t e d under "FORWA8D DTSTRT- i

I n performing ordinary PE3T a n a l y s i s on the network

shown i n Fjgure 19, w e would note t h a t t h e longest path 5s

0 (0 ,20 ,~0,80 ,100) , w i t h L g ( ) = 28.0, and applyin% t h e c r i t i -

c a l path assumption, we would a s s i g n ploo = 28.0 ( i n t h i s

implementation event has beer! numbered event 100) . However,

t h i s network contains another path, o ( 0 , 10, 40, 70, I O O ) ,

on which LM. ( 0 ) is 37.0. As w e h a v e seen, t h e presence of

a second path whose d i s t r i b u t i o n overlaps t h a t of t h e " c r i t i c a l "

path ( w e might ca l l t h i s a l l n e s r - c r i t i c a l l t p a t h ) tends t o

s h i f t t h e d i s t r i b u t i o n of pred ic ted p r o j e c t completion time

toward a la te r date, and f o r t h P s p a r t i c u l a r network the s h i f t

i n p loo is about 0.5; t h e model

28.5,

c a l c u l a t e s ploo t o be about

An unexpected consequence of using ind iv idua l a c t i v i t y

p d f g s t o c a l c u l a t e t h e event p d f 8 s is revealed i n t h e column

l i s t i n g s lack . We r e c a l l t h a t one c h a r a c t e r i s t i c o f a c r i t i c a l

path i s t h a t a l l events on t h a t path con ta in zero slack.

I n t h i s case, however, w i t h t h e expected du ra t ion of t h e

e n t i r e p ro jec t exceeding t h a t of any ind iv idua l pa th , a11

events a r e shown t o have some s lack . T h i s means t h a t a n y

6

INITIATING TERMINATING DURATION INPUT DISTRIBUTION EVENT

0 0 0

10 10 20 30 30 40 50 60 60 70 80 90

EVENT

0 10 30 20 60 90 50 40 80 70 100

EVENT

10 20 30 50 40 50 60 90 70 80 80 90 100 100 190

MEAN VARIANCE

3 a O O O 6 0 0 0 0 5,000 20000 10.00Q

5 0 0 0 0 3 a O O O 5.000 9 0 0 0 0 90000 3.001 20001 Sa001 8.004 4.001

,441 a441 0110 a 028 a991 0110 0 028 0110 a 441 a991 a 108 027

0 108 0970 0108

“e 1,000 4oOOO 40000 1 0 5 0 0 70000 4aOOO 20500 4,000 70000 6.000 2,000 1,500 4 o O O O 5,000 30000

Ma

8.000 5.000

6a000 2.300

13*r)oo 60000 3.500 6-000 110000 12aOOO 40000 20500 6.000 110000 5.000

EVENT OISTRIBUTIONS

FORWARD DISTRIBUTION BACKWARD DISTRIBUTION SLACK MEAN VARIANCE MEAN VARIANCE

0000 3 * O O G 5.000 6.000 8 0 0 0 0 10,253 11.000 130000 200000 22aOOO 28 0 489

,000 ,441 0110 0441 0 140 0 135 0 561

1.441 1.579 la960 1,849

0000 4,483 14 , 482 60484 17a482 24 488 110484 14 488 20 485 23,488 28 0 489

0 000 1 a 558 I 305 2 346 10101

0 108 10982

e 5 5 1 0 970 0 108 0 000

,000 1,483 90482 484

9 0 482 14.235 ,484

1.488 ,485 10488 0000

? I

73

sequence of activities in the project can consume more than

its predicted time without causing project slippage. No

individual sequence is absolutely Itcritical".

We noted previously that f o r networks containing no

ltnear-criticaltt paths, the critical path assumption holds

and project completion is indeed determined by only one path,

On the other hand, as the lengths of other paths approach

that of the longest path the duration of the project surpasses

that of any path. We can observe the sensitivity of project

duration to changes in the length of a 'tnear-criticalll

path by varying the cdf of the activity linking event 40 to

event 100.

o f the activity is increased from 9.0 to 9.9. The length of

the longest path is unchanged from the previous example, so

the PERT predicted project length is still 28,O; however, the

expected duration calculated by the model is now 28.9. An

increase of 0.9 in a near-critical path has resultsd in arl

increase of about 0.4 in the overall project duration.

In Figure 20, Ma has been increased so that p a

M of this same activity is again lengthened, so %hat a in Figure 21 its mean is 10,0, Now LD of' (0, 10, 40, 70, 100) is 28.0, so there are t w o "critical" paths. As before,

the increase in the overall project length (from 28.90 t o

28.96) calculated by the model is about one-half as great as

the increase in the mean length of a near-critical path,. If

A C T I V I T Y DTSTRXPUTIONS

'~NITIATING TERMINATING DURATION EVENT EVENT MEAN VARIANCE

u 0 0

10 10 20 30 30 40 5 0 6 0 6 0 70 80 90

10 20 30 50 40 50 60 90 90 80 00 90 100 100 100

3,000 6 m O O O 5 e O O O 2.000

1 0 ~ 0 0 0 5.000 30000 5 .000 90900 9 , 0 0 0 3 . 0 0 0 20000 5,000 8,000 4.000

,441 ,441 ,110

028 ,991 ,110 0 038 ,110 , 926 , 991 ,110 , 028 ,110 ,991 ,110

ma 1 ,000 4 ,000 4 . 0 0 0 1 ,500 7 ,000 4 , 0 0 0 2 , 5 0 0 4 , 0 0 0 7 ,000 6 , 0 0 0 28000 1 ,500 4 .000 5,000 3,000

INPUT RTSTRTRIJT T O N

Ma 5.000 0,00n 6,000 2,5017

13 ,000 6,000 3 , 5 0 0 6,oon

12,ROfl 12,001) 4, OOf l 2 , 5 0 0 6,000

1s , 000 5,6100

EVENT DISTRIBUTIONS

EVENT FOHWARO OISTRIRUTION BACKWARD DISTRIRUTION SLACK MEAN WAR IANCE MEAN VARIANCE

0 ,000 10' 3.000 30 5,000 20 6,000 6 0 8 , 0 0 0 90 10,252 50 11 ,000 4 0 1 3 , 0 0 0 8 0 20 ,000 70 22,900

100 28 ,900

, 000 , 4 4 1 e l 1 0 ,441 , 1 4 0 a335 e 5 6 1

1 e441

2 . 4 4 7 1 ,925

1 , 5 7 9

, 0 0 0 3 , 998

14 ,900 6 , 9 0 0

19 ,900 24 ,300 l l e 9 0 0 14 e 000 2 0 , 9 0 0 25.900 2 8 , 9 0 0

,000 2 a 064' 1,333 2,3?0 1 .130

,110 2,006 1 , 0 4 2

,991 ,110 ,000

, 000 a 998

9,900 ,900

9 , 9 0 0 14,64F1

,900 1 , 000

, 9 0 0 1 , 000 , 000

Figure 20. Network After Lengthening a Near-Critic81 PaLh

E

A C T I V I T Y D T S T P P R ~ l T TONS

EVEiJT

U 11 0

1 I) 10 20 3 0 30 40 50 61) 6 0 70 RU 90

-

EVENT

0 10 3 0 20 6 0 90 50 40 A0 70 100

EVENT

10 20 30 50 40 50 60 YO 70 80 8 0 9 0 100 100 100

D U R A T I O N MEAN VARIANCE

30000 6.000 5.000 2.000

l O o O 0 0 5.000 30000 !5*000

10.000 9.000 3e001 2.001 5,001 R e O O Y 4.001

f O H WARD D I S T H THUT I WJ MEAN VARIANCE

,000 3 ,000 5 , 0 0 0 6 , 0 0 0 8 , 0 0 0 10,253 11,000 13 000 20,000 23.000 28 , 958

,000 e441 ,110 ,441 ,140 , 135 .5hl

1.441 1,579 2,511 1,941

,441 ,441 ,110 ,028 ,991 , 110 e 028 ,110 ,991 ,991 ,108

027 , 108 .970 0 108

m a 1,000 4,009 4.000 1.500 7,000 4,000 2,500 4,000 7.000 6,000 2,000 1 ,500 4,nuo 5 ,000 3,000

RACKWAR@ DISTRIRCJTION WEAN VARIANCE

,000 3,955 14,352 6,954 17.951 24,957 11,954 13,957 20.954 23,959 28,958

,000 2.129 1.305 2 e 346 1,101

108 1.982 le104 ,970 ,108 ,000

SI, ACK

. 0 0 0 . 955 9,952

e 954 9,951 14.704

954 , 957 ,954 D 957 ,oon

Figure 21e Network With Two Cri t ica l P a t h s

76

'is assumed to be approximately normal, we find. tha,t th.e F I O O probability of actually completing the project in the PERT-

predicted time (28.0) is only about 0.25.

These examples demonstrate that i t is not necessa,ry

to make the simplifying assumption of B critical path i y , per-

forming the PERT calculations and that by using finite

scamnine techniques it, i s possible to acciirately calculate

event prlf's based on the activity pdf's that are .in.put, More-

over , thees examples have cast further doubts 017 t h n ""p1j7;p-

t i o r thst e.11~ s ; r ~ l ~ F a t h need n a x s s ? - i ? y m o w I * y f t i c q l

than any of the otheys. Thc procedures demonstrated he^^ f o r

calculating event times might readily be incorporated Sn a

system f o r project managemeqt; one approach to des ign l r lg t h i s

system w i l l be explored in t h e foll.owing chapter

f

CHAPTER VI

SUMMARY

In this thesis we have exm+ned the typical PEW'

implementation nnd the errors which can result from some of

the standard PERT calculations. We have demonstrated a model

for project management in which these calculations a?e per-

formed correctly. In this chapter ne will. discuss the in-

corporation o f a model such as the one introduced here in an

information system for project; management.

The basic input to any PERT system consi-sts of activity

information - for each activity, the bracketir_g events aqd

distribution of durations would be input. The distribLition

could be enumerated, as in the case of our model, or, if the

user felt that some standard form of pdf best represented

activity duration, the d.istributions he desired. could be

generated by the system from appropriate input parameters (such

as the mean and variance). In order to associate actual

dates with events rather than merely times, it would be

necessary to input the project starting date and to provide

a calendar and the working schedule (number of working days

i.n the week, holidays, etc. ). This information could be

input with the particular project, or it might be maintained

internal to the system.

77

As a. first step in the project, initistion, giver1 t h e

activity information, the system would "calcul~to the forward

distribution of all events and would present f, in yaphic

form on a CRT console, on-line plotter, or whatevsr 4evic .P

was available. As we discussed previously, the prgsentstion

of the entire f m , rather than merelyy,,provides the project

manager more information on which to base his specification

of t, . After he has chosen t, on the basis of whatever

criteria he has used (pm,95$ confidence level, etc.), the

system would perform the backward pass calculating the back-

ward and slack distributions of all events.

With this project initiation run, the initial schedu1.e

would be produced and the network retained in storage. Ty-

pical output would include a list of all events along with

information about their forward and backward distributions.

The event information presented by our model, consisting of

the means and variances of the forward and backward distri-

butions along with the slack, might be representative of a

minimum set of output; in addition the times associated with

95% confidence levels could be printed, the probability of

occurrence of an event by some input target date, or even

plots of the entire pdf's could be output. Another useful

outpiit would be an illustration of the PERT network, showing

jmportant time (e,g., Figure 9 ) . This illustration would

serve as a visual display of the project flow, and a l s o it

could be compared with a manually prepared original to ascertain

that the network had been input properly,

i

7?

The form i n which t h e network is s to red wi th in t h e

system would be dependent upon a t r adeof f between time a n d

s to rage requiremen.ts. For moat a p p l i c a t i o n s it, would prob-

a b l y be more economical t o s t o r e only the iqpt in.formstion

and r e -ca l cu la t e t h e event d i s t r i b u t i o n s w h e m v e r a user d.e-

s i red t o update o r query t h e fi1.e. However, i n a 1a.rg-e

s o p h i s t i c a t e d multiprogramming sys tem it m i g h t be preferable

t o s t o r e more of the information about event d i s t r i b u t i o n s

and to provide a query module t o r e t r i e v e t h i s information.

The more powerful c a l c u l a t i n g rou t ines would be ca,l.led upon

only as needed.

With t h e p a s s i n s of time t h e system would monitor

actual progress by compari.ng it wi th the pre4ictions. When-

ever any event occurred, o r fa i led t o occiir on schedule, all

event 8 - i s t r ibu t ions , as w e 2 1 as t h e new proba.bi3.ity of achieving

t, would be r e c a l c u l a t e d , and. whenever desired (peri .od.icslly

o r on an exception b a s i s ) an updated schetlu3.e would be pro-

diiceflo As c r i t i c a l times approached, warnings coi1l.d he pro-

diiced 9l.on.g w i t h requests f o r ripd-ated information., and when-

ever any event became overdue, a r e p o r t of t h e f a c t a long w i t h

its consequences worild be genera te4 (s ince event ti.mes are

PW repwsevtpd by d 5 s t r i b u t i o n s rather t h a n f ixed t imes, i t

would be mcessary f o r t he u s e r t o spec i fy the meaning of

I t overdue" ) . The system could i.nclu4e a t lplanningll mode of operatien.

which w o u l d accept t r i a l inpu t s , Cerform the necessary calcu-

l a t i o n s t o bui ld a. tanporary. network s t r u c t u r e and m s p w d

F

t o que r i e s without a . l t e r i n g t h e permaaent project rec .0~4 a

Usins th i - s mode the manager could update t h e coylseqilenceF:

of t h e occurrence of some event. on a parti.ci.ilm? d a t e . O r h e

may wish t o cons ider s h i f t i n g some resource f m m one rr*cti..vi-e.y

t o another , by e s t ima t ing t h e e f f e c t s of' t h i s ship't. on t h e

a c t i v i t y d i s t r i b u t i o n , he may observe the effects on t h e

o v e r a l l p r o j e c t ( i n c r e a s e or decrease of slack a t va.ri.om

poi~?!t.s, rhanges ir F r o b s b i l i t y of meet. inz t, ' ; Giv9:n

t h i s p lannine t o o l , he can examl-ne t h e consequenw o f a large

v a r i e t y of proposed ac t ions .

With.our model the concept of slack i s no longer t h e

simple d i f f e rence between two f i x e d p o i n t s i n time as i t is

normally treated i n PERT ana lys i s . There i s f o r any event i.

a d i s t r i b u t i o n f i represent ing t h e time a t w h i c h t h e event,

is expected t o occur and another d i s t r i b u t i o n g reprqsent ing

the t i m e a t which it must occur i n o rde r t h a t t h e p r o j e c t be

cornnleted. by t, e There i s then f o r any ti.me t a probability

t h a t event i occurs t units o f time ahead of i t s reqLii.red

time. We w i l l . c a l l t h e func t ion a s s o c i a t e d w i t h t h i s prob-

ability the slack d i s t r i b u t i o n , denoted s The p r o b a b i l i t y

t h a t the slack of event i is exac t ly t is given by

i

i'

'de note t h a t t h i s expression i s equiva len t t o perform t h e

ser ia l composition, and t h e r e s u l t s de r ived . previ oiisly, w e

see t h a t u ( s i ) =

given t h e model as slack, and i f a s i n g l e parameter is t o be

used t o r ep resen t the slack, t h i s i s probably t h e m o s t mean-

i n g f u l , On the o ther hand, i f t h e manager f e l t t h a t addj-

t i o n a l information woirld be use fu l , t h e p r o j e c t management

system would be capable of present ing t h e e n t i r e slack d i s -

tplibution func t ion , i n t h e same manner that, 1.p; ppesents $he

forward and backward d i s t r i b u t ion.

(pi) - v ( f i ) . Th i s value, p ( s i ) , is chat

T h i s d e s c r i p t i o n of a. Fro jec t rnanegement, information

sys tem i n d i c a t e s one poss ib l e means of u t i l i z i n g the mociel

described i n Chapter TV, The system described is not inher-

ent1.y dependefit upon our model, and there are REBT systems

i n ex i s t ence which include many of t h e features whi.ch we have

descr ibed. However, these systems are based upon c r i t i c a l

path analysis, and w e have seen tha t there are networks f o r

which t h e c r i t i c a l path c a l c u l a t i o n s introduce errors, The

model which w e have introduced can be u t i l i z e d t o ca . lculate

event d i s t r i b u t i o n s which accu ra t e ly re f lec t t h e times pre-

d i c t e d t o perform the a c t i v i t i e s , no matter what t h e form of

t h e network,

i

REFERENCES

1.

2 .

3.

4.

5.

6.

7 .

8.

Department of D,?fense Di rec t ive 3200,9, I n s t i t u t i o n of Operat ional Systems Development,

MacCrimmon, R , 2. and R. A . Ryavec: "An Analyt ic S tudy of t h e PERT Assumptionsf1, Operations Research, 12 : 1.6-17 ( 1 9 6 4 ) .

Ringer, L. J.: Simulat ion of PERT Completion Times & S t r a t i f i e d Sam l i n Methods, Defense Documentation Center, AD 63?-830,*

Rusacker,.R. G.and T . L. Saaty: F i r i i t e Graphs -- and Net- works, M c G r a w - H i l l , N e w York, 1 9 r -

Knuth, D, E. : The A r t , of Computer Programming, V o l . 1, "Fundamental Algorithms", Addison-Wesley, Reacting, Mass,, 1968,

Dixon, W.,T. and F.J. Massey: In t roduc t ion _I t o Statistical Analysis , M c G r a w - H i l l , N e w York, 1947.

Lieberman, R.N.: Re la t iona l S t r u c t u r e Vertex Processor , Univers i ty of Maryland Technical Repor-'7, March, 1969.

Hi.ldebrand, F.B. : Advanced Calculus f o r Appl.ications, P ren t i ce Hall, Engl-ewood. C l i f f s , N . J T l 9 6 2 .

I I

APPENDIX A

PROGRAM LISTING

83

H G fY a 0 a a

a zi H

13 cy s FY cv 0

L

2 z 0 r z 0 V

Y z A a m .- (v \D 4 3 0 C - 0

I- C 0

0 9 9 0 0 0

4

W 0 0 V

0 W v) 3 W 0

U 0 c VI

- a

...

- ..

a

.. v) Y V 0 J

2 0 z z 0 u

m

d 0 0 5 0 Q

v) J c

r) 0 0 0

W 5 a. z Y V 0 A

- m, m LJ u z W K w LL iJ c J

z II: W I- X W

a *in\3+04(ur)t J-I .Or-O.4Nmfnn9 0000*4 .44 .4d4-4NnJNNNNN C C d O C O C c C C c C C c C C e C e 0000000000000000000

0 0 0 0 0 0 0 N 0 0 0 0 000000000000

a r-rr-a

4-l4~~00cu0000 o o o o o o c o o o o o 000000000000 000000000000

r - m g d d r - m m o a c y o 9 t - N N * O n - l o N m N

d N ~ ~ 9 d 0 0 0 0 0 0 000000000000 c o o o o c o o o o o c o o o o o c o o o o o o

a w w . l n w w

d d d * d o o o m o o o 00000000.4000 000000000000 000000000000

c o o a o o a c o ~ o o m 0 3 3 9 3.3 '3 3 J c3 3 3 3 c o c o c o o c c o c o o

u

o c u j ~ m o o o o o t o o O O O O O O O n O O n O O 0 0 0 c, c, 0 0 0 0 0 (u 0 0 0000000D00000

o N FI 3 n o o P- o .o o o o C? e c c C? C: CI. 4 t ne cj c3 O O O O O O O O O O O O O 0000003000000

K U H U U l l :

4 4 . 4 . 4 ~ 0 0 n J 0 n J 0 0 ~ 0000000000000 c c c c c c c o c c c c o O O O O O O 0 O G O O O O

v) H

cn - - a m W

r 3 2

f 0 I- < X t- n I m e z w 7 W

A J 4

U. 0

I- u)

-I

c

H

. w z &- I> 0 CL

H

W > H t- 3 0 W X W

z n d z

U V

0 0 o..o 4-4 C Q C Q

? I

W K

m

I-

> I- V

0 z v) #- z W > W

LI 0

v) 2 0

I- 3

!Y I-

0 W E I-

z > W

I n . Arn

2 0 M I I-I-

w z X U

a

2 U

U

a

a

m

m U

2

U

t-

W

U

a

a I- < 0

4 c 2

9, a . 1

0 0 0 IC in

z

- W n U

5 n 0 I- I 4

OI

L

;*, 0 0 m I- 11) J Z . > W

u

L

(u- In

w N

-I

I-

U

a U

U

5

in z 0 P-

az W T. I- W I-

w V

in P- z W > W

k- m U J

0 z I- m U n II

a

W

a a 'L

a

n a W K

. Y C 0 3 I- W z I- C w a W I k-

0

3 0:

s',

a m 4 0 m

V U U v v v v v v v v v v vvb) v v v u v

0 (u

0 !-

0 - " 2;

In 4

I- z W > w -I U

I K W c L 0

z 0 I- 3 ffi n CL C

0 tl I-

U 3:

z,

U

2

U

0 (u

z 0

I- 3 E4

2 a c

P

t- z W > W

-I U- z P C w I-

n

40, 00, 00,

00 II II

. *

- I

uu -1

c'", o n T I CI-

-IO 0. 0 4

Uc(

--

2 Y U 0 3 I- W 2

W I I- . n W t- 6 J 3 - urn

c 84 + U

v) I - 9 4 4 - 4 9 9

- - 3 I- X w z I- VI J z > - u r n

- .. 0 -e LY

U

w, n

X 0 z

.t-U X - w v l 2- I1 0 -I II- X - Oil:

U> -C

0-1 I-J

u v

,.-.I-

z a

,n II

z a

U

F ) cv 0 I-

E O 00 I- a- -3.4

o w

- - . g5 u1dZ

a I - w C X 3

z J - C J Z

w w v I- (u

a k o

#-

z W E z-

m I- z W > w W I I-

I- U 0 v)

a

rn . 0 II

3- o u 4 - II 84 u-

I- corn (uJ z 0) O W

v v w v v u v

W 2 2 U

t L

0 V

Q (u

* IC .+ 4

+ r C Ul- u r n

(u rr)

n r)

. VI I- z W > W

-1 J

W t- n K 3

a

a m rr)m

U W O

- 4 4

9, 9

W I- n CY 3

-

2 0 n I- 3

&: I- m 0

m I4

w

a 3

0 I&

I- z w > kl X 3

-

9,

I- C Y I O O J

-0I-l-

I----+&

0 (u a 0 a =? 4 9 \Q

.

e \ D 9 r ) C \ l r r ) n too ruo s d O 0 O S O o o o o m o 0 0 0 o w o 0 0 0 0 0 0

e c w w w

d O O O c u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

+ + * r ) O O i -r l lccui-0

O f 9 0

ooo(voo 0 0 0 0 0 0

0“s o m l r l o

w w

c c r l o ( Y r u r u 000000 0 0 0 0 0 0 0 0 0 0 0 0

0 W X W I- Z - 3 Y ov Zt wv) Z W w x W C m

LL l n 0

0 IC V - w X I X C

I-0 Z C W >a

va

a x a

w e Xf;

a= n

vo W

00 ZI-

I- v l c U K -lid

Y Z 0 5 3

c LnE 0

uu

m

a x

at a v) U W

I- Z U > w x tt- lnU A =

a C e 0 ?

0,

a a I- . - 0 0 0 f-

VI 0

I-

0 b

Y

U

E - a

U . n

4 0 0 In

e - a l n J W

>

- z n W E r C

(u-m 0-0 OlnI- In- 2 Y O U . ZI-- ln - W + I W w 3

c I *

r 4 u

> z a r)

+ - u n a v)zz a - - l o w w 2 -,l x

Z I n U 0 2 ) w w I w - t- I- Z E D B-0 n ouo a ec v o w 3 I

m cu s n cu 0

s m I n r ( r ( 0 m o P ) r ( e - o r ( P ) O O 9 0 0 0 0 0 ~ 0 000030 0 0 0 0 0 0

M U U

P(-4oocuo O O O O t O Qci0039 O C 3 0 Q 0 0

- z 0 P 1E 0 V

1

w I z

2. t- U z e V 4

w I I-

X 0 LL

L z 0

c a V 0 d

c(

Y 2 a J m - 0 -4 d 3 0 a

0

I-

D

Y

a a

w z d M r3 0 0 0

CI L

W

>- c n

*

- f- m M 0 0 0

b

Y V 0 -1 m,

Y V 0 J m -

0 L - cu In

VI W

W

W L L Id

z” a

a

I- z w r z 0

In v) G

U

O J

53 U

a

V .. In 0

P I

3\ E 9 0 I

v)oI u o -ln [L O K LLO @LA

r l g v v v v v v v v v v v v O V U ..

n W ln 1s)

S b d O 0 0 ~ d ~ O C 3 0 0 9 oooc?coin 0 0 0 0 0 0 0 0000000

W z U ,l a z a

W C x w

w 0

K 0 I- m

a Id 0

tx 0 f- m

r w - a 4r)o000cu 0 0 0 0 0 0 0 c o o o o c ) o 0 0 0 0 0 0 0

M * U I S i - Q c ( ( U O f 000004dddr ) c c c c c c o c ~ o 0000000000

- \ \

VI z 0 n I- =, m, a

n

I- Ln Y

> I- - n I- U a T - f (v

X m .

e. X t- -4

z 0 I-

*

U

a Q c"

(3 z U I- .= z- I a W Z t-0

U

H

L

\

W u z U U * >

z U W

U

c L

I- Z W > u

I- z w w W

k-4-

O E - - 3 - II lnl- 6 - 4 I-C r O W 0 YELL

Q m

aool

a=za

8-i

a Q > G 5 -I J

V a

U b na on I - -3 m II a AI- J-Ia ea V' '3

0 II I-

z H

I 4

Lo tu (\J t 0 4 4 P

0 0 0 l- t- a 0 0 - a (3 0

r r 0 I- 'a 3 W z a w b

W K V

I W z VI

.)

w

5 W 9 W

0 z I- a I-

c1

U

W

5

4 + il 0 I-

I I

0 k-

- a

l-p

. r 0 I-

3 Irl z

a

a W b- U W

V

3 W Z

a

2 I- z W > W

(3

b- z, E a

I K W k-

t f

b-

I 0 t- .= (3

t- z, z, a

I U w c z 0

I- z 3 0 V z 0 t- 0 0

m

a In N

V U V V U

* v) V

I- v) 0 2 . 0

0

H

a U

0 z

.e z t- J 0. 5 0 0

LL 0

0 z w

0, a Y

. . I .

c u t w d d c v d O O Q O 00000 0 0 0 0 0 6aOQO6a

&' d 4 0 0 a I- 2

0

> a c z w

U

a

a 3 LA a i- VI

w, * m c 3 0 a 3 VI m

".

P3 cv * M cu 0

'v, z 0 I L 0 V

Y 2 4 J m - M t 0 0 0 0

0

I-

i3

(\I In 4 0 O 0

- a a -

- 4 - W 0 0 V

3 W VI 3

W (3 4 M 0 P cn

00

e. v) Y u 0 J m

4 0 0 0 0 e,

v) J I-

w b a w w

elOO0Ca ooocaa 00000 CJOOOCS

0aowMga rnOP=(\d&l d o g 0 0 o o m o 8 O C O O Q ooooa

VI 4 P 4 b . M

000000 a 000000

2 wwwL110: h

lnoP)( \ Imta PLnMNcuO O P 0 3 0 0 C)P(OMOO o o o w 0 o OOCaQQO

> t-

> I- V 4 2

w

w

a 0, - w LL

v)

x w b-

n n

.L

o w v Q V Q

+ z W > W

I-

9 U LL

0 N

0 t

a 0 I- n

II

In (u

0

4

U

n m a Ncu

w I n 3 z 0 r - C +I- z 00 c3V

0 0 b u Y

Z K 2 I-0 W Z K W

e m V

t- v) 0 2 0

Q

U

a U

0 z

.. 2 0

+ J

w

a

a U

a v u v ouu 0 u

O * N S 3 0 MMCUNO r-oEboo d o m o o O O N O O 00000

U P B

nJ0tuON 00000 00000 ooooa

- + d - o L n \ Q f m a 4 4 N 00000 00000 00000 00000

a t 4 n w w

00008 00000 00000 00000

e P 2- W t 4

5 P z w 9. W

z 0

tL

w b m W z 0

0 r ==E 3 Y u <

m II kl I- m W z n I- 3 0 K 03 3 m

a

a

m

rr) cu d M cu 0

2! OOoNb dOPNCV -40900 307V)oo 0 0 0 0 0 o c o o o

b~ w w

- 4 m N o o 00000 30000 0 0 O O Q

z 0 E z-z 0 V

W B 4 z P

2 0

Y a: Q J m

k b- e u 0 d W > #-

==E

U

._ F3 d 0 3 0 0 - 0

a l-

0

Y

a

0- > t? z, Y

0 Y 03

b m W z c 3 0 K c3 3 m

a

H

R % t- m > b- V

I- X w z t 3 0

W > 0 r

e 4

a

LI: (u €4 0 0 0

II w U

- W . - 3 - c \ l P -+a0009 3 m o o o m oaoo00 000000 c

W E a z

e. I- z 0 H

n

d 0 0 3 0 0

0 s - m

W u 2 W a W LL W U

0 x a I- VI w z b 3 0 a: 2 VI

m

U

m

cu m .. m z u 0 J m

Q w m 3

ln d L-

d Q. 2 Y W #- X W

W 0 U U 0 0- m

2 0 r P 0 V

W 0 n

0 c 0

3 0 0 0 d P d cccccc O O O Q O O t-

m

A

. c z W 3 W

0

0 W V w U Q

0 c W > 0 51

z,

a ffl Y 0 z, I-

Ln t- U U t-

0 N

W 3

e z 0 V

z,

K 0 U W z3 W Z Z

U P JI-3 A Z t - 0 V U K W a o w z

0 0 I C Y )

. m u c ul 0 z a

0

0 z

&-I

a U

I .

0 0 0 0 0 0

40 0 0 0 0 0 0

d lu rl 0 0 0

I- Z 0 P

> C I- Z w

U

-1 J

a a a

a w, L U I- 2 0 IX

3 v)

m

'e.

0 0 0 0 0 0

2 z 0 E E 0 V

Y Z U J m - m s 0 3 0 0

0

0

!-

0

* -f 4 0 d 0

- a a -

rl

W

0 V

0 w v) 2 W (3 U 0 I- v)

- n

..

W I 4 Z

Y V 0 J m

v) W V z W C w LL w U -I U z a: W t- X w

.. -

#. W H z a 0

Z

I- V 0 J

W > I- .u A w a

w a >. I-

s V 0 J

z a

W

e.

- m, P z

trl 0

n*!n u 003 a: C C O 0 000 P

v)

J m m 4u) a 4 0 0 0 0 00 00

n

40 00 0 0 0 0

( 3 3 - 1 m a d X >

mt- 0 S4nl 000 000 300 0 0 0

CII:

r(o0 000 DCCI 0 0 0

Y Y

0 I - 0 v ) lun 40 Y

lu 04 ICON 0 0 0 000 CIOO 0 0 0

a- -loo 000 O G O 0 0 0

f", on 0z -I3 r l V 3

F ) O 9 406 000 coo 000 000

4o"lO 0 0 0 C O C 0 0 0

I

J a: b

2 52

H

-3 K c

a

a I- # n 3 9 -I 4 0:

a U

- n w z I- 3 0 a 3 ln

U

m

v)

0 I-

0 (3

a 0

I-

e

?

0 (3

0 r ( *

Y 00: I -c

0 0 W L L 2 0

5v)

a

I ki- a c I- m w n l OI-

52

w

W U n x

Z W

a c

? I

cr

e m V

C m 0 z (3

W

a a n

0 2

.. z 0

e- U. J

Q 5 0 V

lA 0

0 z w

H

U

m 4 I - mcucu 000 0 0 0 0 0 0 0 0 0

4 0 0 0 0 0 000 0 0 0

9 + 4 0 0 0

c z 0 0 > U i- P w

U

VI W

U W m

H

4! L w I- =I 0

9 ul

a m

0 0 0 0 0 0

I

'v, z 0 5 I 0 V

Y z J a m - 4 9 0 3 0 c

0

I- 4 0

Y

a

.a 4 cu cu 9 w 0

4

W 0 0 V

- - .. n W m 3 W a a K 0 I- m

n

W H z Y u 0 -I

a

L

m, m W V z w K w LL W U J * z Y W I- x W

0..

W 5 d z z 0 I- d V 0 J W

I-

J W LY

W >- I-

Y V 0 J

. U

2I a

a

n

s

m, I- 2 W z z 0

m m

W (3 4 K 0 t- VI

u

a

I -mm 4N-4 -406 0 0 0 0 0 0 0 0 0

400 0 0 0 0 0 0 0 0 0

2 a 0 mE: M53Y 401: b O Q O O N 4 0 0 0 0 0 0 0 0 0 0 0

Kp: .-(no 000 0 0 0 0 0 0

a 0 C U J W -493IY

0 0 0 0 0000 Q O O O 0 0 0 0

LX- - 4 d m m o o o w 0000 0000

P-oCO\Q W * O 4 0400 e?OC20 Q O O O 0000

xLI.r

4-400 0 0 0 0 occ?c 0 0 0 0

d K I- ul U

LZ 3 U I- ln &-4

? U K I- VI n 0

m c

w, a W ln

w z I- n - 2 0 oc 9 VI m

0 f

m ro 0 0 I- Is

0 O (3 (3

* - . a O M 0 3

w o i n t-en 8.2-

* . 1 I

. x K c

0 '",

E

5 t-

0 a. 4

0 LL >. I- W A H

a

m a

a a2 0

a W > U I- -a A 3 I 2 w t- A 3 V J Q V

a

00 V V V

HI-C m z z w o o nvu

. ln 0,

Y 0

P, Y K

0 z

M

N I- cu 0 0 0

I- z 0 a > K l- z W

U

0 - -

0 0 0 0 0 0

‘v, z 0 r ZE 0 W

Y z J a m ea N 9 0 3 0 0

n 0 - a a k

0 .a d d OI) 0 0 0

n s-4

W 0 0 W

- .. n w v) 3

W 0

0: 0 c v)

a

- W r a z Y W 0 -1

a

m, ul W V z w K w LL w K

J =z z W I- % W

a

Id > I- a -I W C

U

Y W 0 -d m,

o o f - m * 9 = ? r n r n N ( u . Z 4 0 4 0000000 0000000 c o o o o o o 0000000

KE:OZCWKBZ

0000000 0000000 0000000 0000000

cUO9NI-2\gO lcNlv lvO4- i l v o o o o o o 0000000 0000000 0000000

c e a u w ~ o c 4000000 0000000 0000000 0000000

0 + A d

J r n d 3 d O L a d Z L L a

lnrnln.-rorc(u L O r n ( u ( u S d 0 0000000 0000000 0 3 0 3 ,a0 0 G O O 0 0 0 0

n 3

n w w w ~ + n s

K K K Ccrx ~ 0 0 0 0 0 0 0000000 ~ 0 0 0 ~ 0 0 8-Y 0 0 0 0 Q 0

U N b O b r n M R ~ ~ l v O - I O 0000000 0000000 3000000 0030000~

I Y ~ X C U ~ K

4 0 D D . 3 0 3 0000000 0000000 oou0ooo

I n 4 * 0 4 m r c ) C Y . - ( m l v o o . - + o 4 00~C30000 OCIc2c2OOQCI 00000000 00000000

KUUKWtY:(bL 0

m a a d o o o o o o o 00 Y 00000000 00 0 C O C O C O O e 00 c 00000000 in

0 rb

0 c 0 (3

n cl w

K b v) m 0 w

0 -4

v v o o v w w v w v u

0

w =s a 3 V

d 9 3

LL 0 0

3: 1- 0 z

U z c v) 0

z CI

n

i- In

gL

a CI

a W

0 E -4

n a LI I b-

P 0

v) t-

0 P, n

W I c- 2 0

II X c

0 0 ZrLk o u v

I D W In a m

0 z 3 0 a C ‘ m .

ul u c ln 0 z U

0

w

a U

0 F)

0 e

0 t-

0 U O 4a X .)I- d(u . m

I 4

0 t3

irl 0 0

W

I- d. -I 3 I: 3 V

z

P c(

LL

C In d a

0 z - K I- m 0 II

I 0 0: Q

CI

H

m

W U a W

3

I- z 0 V

5 W 53

c ;z 0 0

5 m w > K 3 0 ,

4

n

+a

LI

II

.. z c J

I: 0 V

0, a

a w

0 Z w

O B n J n

h. 0

n z W

.

to >- to 0 0

(u 0 0 0 0 0

U

0

Lb w z o a v

0 0 0 0

I)

0 w 8- t 5 0

ul

b-

H

k4

w

a kl u z w X

t- z 0 z 3 0 W a:

- a

I-3

I- z w .z W B- a b

0 0 0 0 0 0

- .

4 >- d 0 0 0

- 0 w o z a a z

0 P O 2 0 0 0

b-

V 0 d

U

a

t-w a r m

Y z e. 4 m

c 3

LL w Q

s

u, l-4

?- a LL a 5 m 0 u B

W 2

I- =, 0 c m 13 m

w

N P) 0 c3 0

3 c 0 r ( - 0 0 0 - o a

a n C -

u > 3 w * b- a 0 A 0 w o a 0

0 s o

W

>- k - 3

n a t z 0 a CI

W E a z -

0 N -4 0 0 0

c) - 0 Y O u 0

0 D - in

W V z W

W LL w

a

a

W 0 0 V

b z W H z (3

ul v) a

ma

n L4-i P)

(3 W a

H P)

cz W z

a Lo 4 0 0 0 0

.. 4 z 3: w f- X u

a w 0 e K 0 f- v)

m 0 0 0

0 0 0 0

f

N O * O 00 00 00 00

w

-40 0 0 0 0 00

000 000 000 0 0 0

K a

roo0 000 000 0 0 0

0 0 0 0 0 0

(u

z 0 a E 0 V

- *4(u D O 0 0 0 0 0 0 0 - 0 0 0

Ir; 000' r z d o 0 - 000 z a 0 0 0 0-30

a = a

a 4

Lo c. . z 0 t 3

U

m n a

W

10 e o\ 0

0 LL 0 * e 0

!!! 0:

0 r(

0 I- O (3

b z 0 m I- 3

K I-

0 c Z e I- J 3 m W U LL 0 0 Z W

K w 3 0 -1

0 z LL

ffi

z

u

z 0 W b f W 0 t (3 a

3

t m 3

a

w

a H a I- m 0

5: U 0 LL W a W z I- 2 0 a 3 m

bl -

U

m

s (30 m z 33 -403

(u 0 1 0 - t o m 000 000 O D 0

4 P) 0 3 0 0 d

(u .rl 0 0 0 - _ -

0 0 0

Mp-) c z 0 u

a

W

> I- a

.-in3

0 0 0 000

oao cv d 4 0 0 0

4 .e

W o 0 V

0 w m 2

e.

a Y V 0 -1 2

* c z W

a Y V 0 -I m,

0 s u

v9 Lo

Lo E O K I OCO L O w \ K Q

0 I m m

w o - m x

O p : L O Q L L

.. In W V z W K fflti) w un w S K K 3 w v z J

LL n a

a -

c z W %: z 0 U

CU m P- 0 rl

v v v V v v w u v v u u m m

W 0

a: 0

m

a

a

000 000

W 0

U 0 I- m

a L x ns Lrl 00 I- c c x 00 w

4-80 000 o c c 000

e

u# Cl-

W 3

i- z 0 u

z,

0 N

u

bb 0

0 0 0 0 0 0

d

?! x 0 I I 0 e,

Y a U d m ... 0 N 0 3 0 0

0

arl !- e

- n .9

9 s- 0 0 0 0

sl

w 0 0 V

0 w v) =I

w 0

0 c ul

c- - *e

a a

M 3 a 5 b 0 0 0

- 0 w o r < = o - 0 2 0 0 0

c < 0 0 -4

n

w

t- e o - 4 0 W O K O

0 - 0

W

2. e o b

- 0 u o V 0 J

t l M

n w

E

w (3

F) aw.4 0 n : o 0 o c 0 e o

ul

* I- z w > w z < k 0

z 0 m P 3

K ti

P W 5

P w I

B-

w N -4 cy

c-

E

z! w

U

$4

n

f, 0 4

v o u

0 x

. . . .. A .;

m a ( u r l 0 0 0 0 00 00

a 0 0 0 0 0 0 0 0

Scm cia 00 0 0 0 0 0 0

H X

Z

W E a

0 0 00 0 0 00

W I t-

e I- v)

0 w

0 0 0 0 0 0

m 0 0:

S Q

m-4 dol 0 0 0 0 03 00

(u - Z 0 5 2E 0 V

c. W I U z arx

00 0 0 00 CY 0

L z 0

I- < u 0 J

+I

Y Z =z J m e

I- (32 aI-4 -40 40

( v 3 m o 0 0 00 00 00

U 4 m 0 0 00

oa

W > w

t-

0 z, n V'

1c 0 0 0 0

(u Q * X * 0 a + a m E 3 ul I I 0 ffl z I3 ul

m

n

I-

J W a

U 0 z - a

W I

a 4 > 4 I

cu * Q E 3

0 z .. W

>. i-

a I- z 0 H

n

W r z a

W I t

0 W V

J

ffl

a n

w

6.

Y u 0 A m -

>. I- 2 W

a W 3 z n I- z 0 0

rl 0 0 0 eo z

0 0 z W

I 4 X II X

I-B b- e J

ZE 0 u

).4

0 e -

Ir) In

4

W 0 0 V

- In w V z W oc Z H w UT) L L o x w I-= a z w J

z Y Os- w 00 I- C C x 00 W

n z a

0 0 (um

v v u u

a > H

a a

Y U

0 0 s - cuNd 00 0 oc e, 00 0 000

e.

m 0

r r B > o 29

0 I

m a \ w c

Y O K lL3 e&.

a m a \

e. n W m 3

Lr 0

w 0

a 0 i- ln

a n P W

I- 3 0 a m 3 ln

w (3 < a 0 c ln

MU

4 0 0 00 0 C C C .o 0 0

9 9 N 0 0 0

I- z 0

* c z W

84

n

a

z w (3 P z w 2

e 3 0 K

3 v)

n

w

rn

0 0 0 0 0 0

c.

2 z 0 r a: 8 V

4L z d a m - \Q m 0 3 0 0

0

4t c et

- v

n a n a 0 P) 0

0 a

CI

e9

w 0 0 V

0 W in 3 W (3 =z X 0 I- v)

v

e.

..

W 9: a z Y 0 0 J

0

m, in W V z W K W LL w K

a 2 rz W I- x W

mm64c@@ N C(r l8e .3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

K a K M K

0 0 0 0 0 O O O Q B O O O O B 0 8 0 d ) c a

# n

ncc w w oz o a

P 9 j o a cc! 0 0

W x z

z c e V 0 J

W

c J W K

w Q * b-

x V 0 J

a

c1

2

2 a

D

en

m - 8- z W I z a in in U W (3

K 0 I- m

s-4

a

a k

4 W LLo0 B vov3,Q, (Um-4S-PI (u C(d00 o o o a o 0 0 0 0 0 c o o o o 0 0 0 0 0

0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

d d O d 0 C v 0 J n J P ) d 0 0 0 0 0 ooooa 0 0 0 0 0 0 0 0 0 0

a a ~ K

0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 6 0 0 0 0

PI€4p.*ecrm 3 W r i r ( Q C 8 N O 0 0 0 8 o o o o o a 0ooooci ~ 0 0 0 0 0

P T K K W E

- 4 0 3 3 3 3 3 3

000000 0 0 0 0 0 0

o o o o a c ~

P)*f\b1mP9)9 - 4 ( u € 4 ~ 0 0 4 0 0 0 0 0 c ~ 0 0 0 0 ' 0 0 0 0 0 0 0 0 0 0 0 0

C x a z w K

r looooo 0 0 0 0 0 0 c !ocooo 0 0 0 0 0 0

e U

c z 0 P

I4

3 3 K I c cI1 z in Y,

84

a n

a m A

E 0 z * J 0

M

W t- z W z W (3

c 4

n

a

W r c IA 0

I V

W

z 0

>- I-

J

e 10 0

a

n

I4

a

a n W > t J 3 E 3 V

W I e L3

I A * in

- 3 J G

+ U 29: - 0 O Z

A I O VfL

U W w K K 01 LLI-

U

a

z,

n+

a

v v u u v v u v v v

. . . . ...

0

rl

c4) 4

w a IZ e < Z W 0 Qv, 3+ P

c3w W O m e

u W W u 3 - 4 - I - W

W d

a > o w

a m z

urn

m a m a ac a z

cr

H w I W

I- LL

W O C 3 J a a U W Y > a b z O H w o C-a

OIL -0 w P W u 4 2 a ul-

Y v )

e 0

0 s

T

a w

U U V O

0 d

0 z w

0 z

n r 0 W

e

. . > . ...

Q m 0 b - 3 B J zsy W Z

U 9 0 8 0 0 0 8 0 0 0 O B

w

O P 0 0 0 0 6 0

sy t-

W H ern na

d 0 N O 30 s r ) O c u 8 00

w

4 t=- 0

< c 4 0

a a

8-a 0

cue 0 0 0 0 00

0 0 0 f-

* u

cu m B 3 z 0 - t- b: 0

W

A -J

V

a a

a

n * 0

c . i 1

0 0

m M s 4

m z 3 I

l n l n w n E O 3 H I - w

d Od O O P O O Q OOLn - 0 0 0 u 0 0 0

H < M I L z

ON(v - 0 0 0 z 0 0 0

P sz

0, 0 0 0

V 3 E I- ln

c Y

b: e c

W

d a a

K 0 r: K W

0 ZE SE 0 V

m z 3 z n >

ln 0:

J -JN

V - In -0 O H I-+ A -

t- -0

a +

nu

*U

a -c o a

a z a 0 1

o LI. m a -El

c 4 0

0 I-

X

Y z J a m

L 0 b- a

rr) D r 3 z

ra 4 rl 0 a 0 0

0

a I- < Q

- v

LL 0 f

0 a w e z 0 ci

CI b d d d

0000 0000 0 0 0-0

ln z - 0 Z E

a - -a

m a

- a z m -a

rn La

HY)

P)o

3: ZI-

on

LZ - II

t-f-t--c z z z z 0000 b4wub-I

a n n n

W E ef Z

m a r

i, Z 3 LI.

-a

M sr)

w I I- (.

Y V 0 d m,

ij 0 z co

z 0

4 0 0 0

a W > 0 V W 0:

0 z 0 * ..a.

P) M .e

-4

w 0 0 V

- ..

u LL CA

W 0 2 w n W L L . u n

Ln 0 1 w 0

0 w ln 3

s z 0 8-a h

9 4c 2 X w

u (3 4

0 b ln

a

Ld w (3 m 3 - m a r - 0 a mN)m

ooooor ( [2: 0 0 8 o c c c c c 0 coc 0 0 0 0 0 0 b. 000

v)

iJ z 3 LL

X w a 0 IA. ~ .:

. .. ? .

..*

'".

. v) V n I- ul 0 z (3

0 a W

0 z