Activity Subdivision and PERT Probability Statements

Activity Subdivision and PERT Probability StatementsAuthor(s): Thomas L. Healy, Charles E. Clark and Herbert S. MillsteinSource: Operations Research, Vol. 9, No. 3 (May - Jun., 1961), pp. 341-350Published by: INFORMSStable URL: http://www.jstor.org/stable/167566 .

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ACTIVITY SUBDIVISION AND PERT

PROBABILITY STATEMENTS*

Ti?oitnas L. I-lealy

Office of the Deputy Commander AFSC for Aerospace Systems, Lhs Angeles, California

(Received August 2,1960)

The Program Evaluation and Review Technique (PERT) has been de- veloped and implemented as a technique for planning and evaluating the progress of research and development programs. Probability theory is used in PERT to calculate a measure of the uncertainty in meeting scheduled dates. These computed probabilities can then be used by managers to aid their judgment in evaluating the validity of schedules for accomplishing program objectives. This paper investigates the effect of subdividing activities in PERT flow diagrams on computed probabilities for meeting scheduled dates. It is shown that the mechanics of activity subdivision can increase or decrease the computed probabilities, depending upon whether the scheduled date for an event is later or earlier in time than the expected date. Management personnel who are using PERT probabilities to aid their judgment in evaluating schedules should be aware of the potential biases inhereint in the computation of probabilities.

T HE Program Evaluation and Review Technique (PERT) has been de- veloped and implemented as a technique for measuring, evaluating, and

controlling the progress of research and development programs.[' Within the PERT system probability theory is used to calculate a measure of the uncertainty in meeting existing or proposed scheduled dates. These computed probabilities can the be used by technical management personnel to aid their judgment in evaluating the validity of proposed and established schedules for accomplishing program objectives.

The purpose of this paper is to investigate the effect of subdividing activities in a PERT flow diagram on the computed probabilities for meeting scheduled dates. More specifically, the problem with which this paper is concerned can be stated as follows:

Given an activity or sequence of activities that terminates in an objective event for which a scheduled date has been set, it is desired to investigate the effect of activity subdivision on the computed probability for meeting the scheduled date of the objective event.

For example, the activity sequence in Fig. 1 beginning with event A and

* The opinions expressed in this paper are those of the author and do not neces- sarily reflect the policy or views of the Air Force Systems Command (AFSC).

341



342 Thomas L. Healy

terminating in the objective event C for which a scheduled date has been set, might be further subdivided to appear as shown in Fig. 2. In going from the flow diagram of Fig. 1 to that of Fig. 2, activity subdivision has increased the amount of detail as represented by the number of activities. By imposing some rather mild restrictions on the manner in which the activity subdivision is accomplished, we wish to investigate its effect on the computed probability of meeting the scheduled date for the objective event. The analysis presented below is applicable to series and series-parallel suibdivisions, but not to strictly parallel subdivisions.

-~~~~~~j e r

Fig. 1. Initial flow diagram. Fig. 2. Subdivided flow diagram.

ANALYSIS OF ACTIVITY SUBDIVISION

WE ASSUME that a single activity with optimistic, most likely, and pessimistic elapsed time estimates a, m, and b, respectively, is divided into n serially connected subactivities with optimistic, most likely, and pessimistic elapsed time estimates ai,mi, and bi, respectively, where i= 1, 2, ,n. It is further assumed that

EiZ , ai=a, Zi-1 mi=m, M Ei=b,=b, (1)

and that the subactivity elapsed times are mutually independent. Equa- tions (1) state that the difference between the pessimistic and optmistic elapsed time estimates for the original activity is equal to the sum of the differences between the Ressimistic and optimistic elapsed time estimates of the subactivities; i.e.,

d = in di, (2)

where d=b-a and di=b,-ai for i=1,2, ,2 n. Using the PERT equations for the expected value and variance of an

activity time, we have for the original activity prior to subdivision

E(t) =16 (a+4m+b), (3a)

V (t) =[ (b -_a)] =y6 d2. (3b)

After the subdivision has been made, we have

En(t) = 1 E(ti) = 6 Zi1 (ai+4mi+bi) =3/ (a+4m+b) (4a)

Vn(t s = v. =l (bh - a) V3 6 2= i - v8nd2 (A



PERT Probability Statements 343

The subscript n is used on the left side of equations (4) to denote the expected value and variance of the original activity time after the activity has been divided into n serially connected subactivities. From equations (3a) and (4a), we see that E(t) =En(t).

With the aid of equation (2), equation (3b) can be written as

V(t) = Y36 d = Y6 E_ ) = Y3 6 (E=1 di=+ d V+ E,_ j di dj) . (5) Since d > 0 for all i, a comparison of equations (4b) and (5) shows that

V(t) > Vni (t) . (6a)

Equality holds in equation (6a) only for the degenerate case in which either n = 1 or d = di for at most one i; that is,

fdi = d (i=k) ds =0. (i# k)

Henceforth we will consider only the nondegenerate cases and write

V (t) >Vn (t) . (6b)

In the PERT system the probability of accomplishing an event on or ahead of its scheduled date Ts is found by calculating

y-[Ts-E(T)]/VV(T), (7a)

where E(T) and V(T) are the expected value and variance, respectively, of the event time, and obtaining the corresponding probability from tables of the standard normal distribution function. Formally, we have

Prob(TTs)=,4T f exp(-? z2) dz. (7b)

Assume now that the activity which has been subdivided is on the activity path used to calculate E(T) and V(T). Without loss of generality, we can further assume that the subdivided activity is the first activity. Now let V(t) be the variance of the first activity time prior to subdivision and a be the sum of the activity time variances of the succeeding activities on that path. Then the variance of the objective event time is

V(T)-=V(t) + . (8)

From equations (7a) and (8),

y= [Ts-E(T)]/VV(t)+o2 (9a)

prior to subdividing the first activity and



344 Thomas L. Healy

after the first activity has been subdivided. From equations (6b) and (9),

1ynl > |yl, (10)

and there are three possible cases:

I. For Ts >E(T), the computed probability for accomplishing the event on or ahead of schedule will be increased.

II. For Ts <E(T), the computed probability for accomplishing the event on or ahead of schedule will be decreased.

III. For Ts =E(T), the computed probability for accomplishing the event on or ahead of schedule will not be changed.

It should be noted that if activities are combined or summed subject to the conditions of equations (1) prior to computing probabilities, then the in- equality signs are reversed in Cases I and II.

We now wish to determine more precisely the effect of activity subdivision on the computed probability for meeting a scheduled event. From equations (7) the computed probability can be affected by changes in Ts, E(T), or V(T). First, we consider changes in V(T) only and, more specifically, the change in V(t) that results from activity subdivision.

From equation (4b) the variance of the activity time after subdivision is

Vn(O =Y6 Ei-- di2.

The sum on the right side of this equation can be written as

Z- dV2=Zz (xid)2=d2 =d X (11)

where the following conditions are imposed on the xi's:

O<xi<1 forall i, E x=-1 . (12)

Differentiating, using the inethod of Lagrange multipliers, it is easily seen that for a given n-part subdivision,

Xi= 1/n (i=-1)2,..,l

will minimize equation (11) subject to the conditionls of equations (12). Hence the minimum variance for an activity that has been divided into n subactivities is

minVn (t)--6 d' 1: i= n(1/n )2=Y6 d 2/n. (13)

To find the maximum variance for an n-part subdivision we assume that the activity time ranges (i.e., the difference between the pessimistic and optimistic activity time estimates) of the original activity and of the subactivities are integral valued. This restriction does not seem too severe wheen the unit of time is a week or a day. Suppose that the time range of




the activity prior to subdivision is d. We now divide this activity into two subactivities with subactivity time ranges of (d -1) and 1. By considering the graph of the function f(x)= x2, it is clear that this particular 2-part subdivision has the largest variance of any other 2-part subdivision. Simi- larly, the 3-part subdivision with subactivity time ranges of (d-2), 1, and 1 has the largest variance of any other 3-part subdivisioni. Continuing

0.70

0. 40

0.30

0.10 ini o

00 . 1 , 1 1 IV ? t S '3 5 7 r 7 10 II 1 2 13 JY (5 /* (7 J8

Fig. 3. Upper and lower bounds on the ratio Vn(t)/V(t) as a function of n and d.

with this heuristic argument we conclude that the maximum variance for an n-part subdivision is

maxVn(t) =136 [(d-n+1).,+ (n-1)]-=?6 [d2 (2d-n) (n-1)]. (14)

Implicit in the argument leading up to equation (14) is the restriction

n<d,

which does not seem to be very severe. Equations (13) and (14) provide lower and upper bounds, respectively,

on the total variance, V(t), for an n-part subdivision; i.e.,

136 d2/n<V (t)<Y36 [d2-(2d-n)(n-I1) (15)

From equations (3b) and (15), the variance ratio, Vn(t)/V(t), can be bounded as follows:

1/n?V,(t)/V(t) <1-(2d-n)(n-1)/d2 (16)

Figure 3 gives the graph of equation (16) for various values of d and n.



346 Thomas L. Healy

As an example to illustrate the possible effect of activity subdivision on a computed probability statement, consider the following:

(a) Suppose that in a flow diagram there are 100 activities, each with an activity time variance of 6.25 (i.e., d = 15), on the critical path leding to the objective event and that T -E(T) -13; then V(T) = 625 and from equations (7),

Prob(T ' Ts) =0.30.

(b) Now assume that (i) each of the 100 activities is subdivided into 3 activities subject to the conditions of equations (1), giving a total of 300 activities on the critical path, and (ii) Ts and E(T) are not changed; then with the use of equation (16),

208.125 ?V3(T) ?475,

and from equations (7), 0.18 <Prob(T < T,) !0.27.

(c) Now assume that (i) each of the 100 activities is subdivided into 6 activities subject to the conditions of equations (1), giving a total of 600 activities on the critical path, and (ii) Ts and E(T) are not changed; then with the use of equation (16),

104.375 ?V6(T) ?291.875

and from equations (7), 0.10 <Prob(T < Ts) ?0.22.

If in the above example the scheduled date were later in time than the expected date, for example, Ts-E( T) +13, then because of the sym- metry of the normal probability density function, the corresponding probabilities would be as follows:

(a) Prob(T? Ts) =0.70,

(b') 0.73 5?Prob(T ? Ts) !0.82,

(c') 0.78 ?Prob(T< Ts) o0.90.

The effects of activity subdivision can be further illustrated by in- vestigating the change to be made in the scheduled date Ts when V(T) changes in order to maintain the same computed probability of meeting the new scheduled date. As before, activities will be assumed to be subdivided subject to the conditions of equations (1). Thus we assume E(T) to be fixed and permnit Ts to vary. The probability of accomplishing an event on or before its scheduled date is

I1 (TS-TF /

P = 1 exp(-12 2) dz, (17)

where for notational conven'ience TE = E( T) and a- V(T). The differential of P is

dP -VIr)Y' exp- 2 ( T,-TE)2/_2)llATs, ZXa- ( Ts -T)/a-I. (18)




By setting dP = 0 in equation (18),

A TS = ( TS-TE) Aor/cI. (19)

To illustrate the change to be made in TS as a result of activity subdivision, consider the previous example:

(a) When each of the 100 activities is subdivided into 3 activities, then

-10.57 ?_ 1r_ -3.21, and hence from equation (19),

1.67 <A Ts ?5.50.

In other words, the scheduled date can be slipped by at least 1.67 and at most 5.50 time units to maintain

Prob(T? Ts) =0.30.

(b) When each of the 100 activities is subdivided into 6 activities, then

- 13.82 Ao <_ -6.36, and hence from equation (19),

3.31 <ATs <?7.19.

Thus the scheduled date can be slipped by at least 3.31 and at most 7.19 time units to maintain

Prob(T_ Ts) =0.30.

From equation (17), the total differential of P is

dP-= (e/2pr )exp[-2T ( TS-TE)2/CT2)]

[ATS-ATE-(TTS-TE) Aorid]. (20)

This equation can be used for evaluating the effect on the computed probability of changes in one or more of the quantities Ts, TE, or S.

CONCLUSIONS

THE mechanics of subdividing or summing activities in a PERT flow diagram can influence the computed probabilities for accomplishing events on or ahead of their scheduled dates. If the PERT probabilities are used to aid management's judgment in allocating resources and determining scheduled dates, the process of subdividing activities to show greater planning and scheduling detail could be rather significant in that resources might be over-allocated or scheduled dates might be unrealistically slipped. Tech- nical management personnel who are using PERT probability statements to aid their judgment in evaluating the validity of schedules should be aware of the potential biases that are inherent in the present method used to com- pute probabilities. Other features of the PERT system, such as orderli- ness and consistency in planning, which result from construction of the



348 Thomas L. Healy

flow diagram and the identification of critical and slack areas, are lnot biased by activity subdivision or summing.

It is hoped that further operations research will result in improved evaluation techniques for research and development programs. FREE-

MAN"2' has suggested a generalized PERT in which cost estimates and level of technical performance estimates are obtained for each activity in addition to elapsed time estimates. KELLEY AND WALKEI0[3] have suggested the use of complete probability density functions rather than first and second moments and the Central Limit Theorem.

REFERENCES

1. D. G. MALCOLM, J. H. ROSEBOOM, C. E. CLARK, AND W. FAZAR, "Application of a Technique for Research and Development Program Evaluation," Opns. Res. 7, 646-669 (1959).

2. RAOTJL J. FREEMAN, "A Generalized PERT," Opns. Res. 8, 281 (1960). 3. JAMES E. KELLEY, JR. AND MORGAN R. WALKER, "Critical-Path Planning and

Scheduling," 1959 Proceedings of the Eastern Joint Computer Conference, pp. 160-173.

COMMENTS ON THE PRECEDING PAPER

Charles E. Clark*

System Development Corporation, Santa Monica, California

W1t rE WISH to indicate how the PERT team faced the problem discussed in At V CAPTAIN HEALY'S interesting paper.

The decision to subdivide or not to subdivide an activity is determined by the nature of the activity; the decision is not a matter of analysis. To illustrate this point let us consider the following hypothetical, over-simplified example. A test vehicle is to be fabricatyed by a sequence of two operations. First a metal chamber is to be made, and then the propellant is to be inserted. Suppose that these two operations are done independently. The metal is processed in one shop, and the propellant introduced in another, and the two shops operate independently of each other. In this case the analyst must subdivide the complete job into two independent activities. Suppose on the other hand, that the metal and propellant were processed in the same shop under the same foreman. The foreman would be charged with completing the test vehicle within a given time interval. If the metal processing were completed at an unexpectedly early time, the foreman might delay the propellant activity in order to use his labor on some other, more pressing task. On the other hand if the metal processing were delayed, the foreman might expedite the propellant activity to cover up his failure to keep the metal processing on schedule. In such a situation it would be better to ask the foreman how long it takes

* EDITOR'S NOTE: Charles Clark was one of the referees of the preceding paper by Healy and submitted these commiiiients for publicationi with it.



Comments by Charles E. Clark 349

to fabricate the complete test vehicle; if times were estimated for the two component activities, these times would be correlated, and the correlation might be impossible to measure. The estimated variability of the time to completion is different in the two cases, but this difference reflects a real difference.

Suppose that the two component activities are independent but the mechanics of the reporting procedure requires that a single estimate be made for the complete process. Even this situation, which occurs in practice, might introduce less bias than Captain Healy's paper shows to be possible. The independence of the two components implies that the variance of the sum of the two component times is less than the sum of the variances. The person making the estimate of variability (PERT uses the range as a measure of variability) might make an accurate appraisal of the over-all variability; the estimate of the range might be less than the sum of estimates of ranges for the component activities. Although the validity and seri- ousness of Captain Healy's remarks must not be overlooked, the PERT team felt that these biases did not vitiate the conclusions of the Polaris study.

As the PERT team faced various activities, it was not always easy to determine the appropriate amount of subdivision. This determination should be referred to the information source, not the analyst.

It was interesting to observe a pressure by management in the direction of greater detail. This was due partly to inadequate appreciation of the statistical facts discussed so effectively by Captain Healy. But there was another important consideration. The greater the detail, the better management can exercise con- tinued surveillance.

COMMENTS ON THE PRECEDING PAPER

Herbert S. Millstein*

Operations Research Incorporated, Los Angeles, California

AT THE outset, we might omment that while CAPTAIN HEALY'S remarks are III valid from a theoretical viewpoint, there are practical operating considerations that were not discussed in the article that have a profound bearing on the problem. Therefore, the following comments will enumerate the practical operating considerations to be associated with the problem in order to achieve a real- world application in the use of probability estimates for management decision making.

Initially, the probability estimates should be used by management in an interdependent manner with the variances for TE (expected time) and TL (latest time) and slack and Ts (scheduled time) for each event. To illustrate this utilization by management, we might consider the following examples:

CASE 1. Event No. 57 has a probability of 0.42 that, when analyzed with the following facts-slack is 8 weeks, variances of 23 (TE) and 10 (TL) exist with TE

* EDITOR'S NOTE: Herbert Millstein was one of the referees of the preceding paper by Healy and submitted these comments for publication with it.



350 Comments by Herbert S. Milstein

(56) and TL (64) > TS (55) with Time Now = 0-would dictate that managerent should watch the progress on this event as the value of variances and the fact that TB and TL > TS are the significant factors.

CASE 2. Event No. 57 has a probability of 0.42 when analyzed with the following facts: slack is 8 weeks, variances of 23 (TE) and 10 (TL) exist with TB (45) TL (53) < TS (55) with Time Now = 0, would dictate that management should disregard

the probability estimate as significant at this time as the other interdependent factors override it.

Using an increased density of activities leading to Event No. 57 as described in the article, the probability estimate in Case 1 would decrease approximately 27 per cent, which gives us an approximate probability of 0.31. However, this change in probability would not affect management's decision.

Using the same logic in Case 2 as described above would result in Event No. 57 having an approximate increase of 28 per cent in the probability estimate, which gives a probability of 0.54, which again would not have an effect on management's decision.

It is of paramount importance to stress that in both Case 1 and Case 2, the probability estimate acted as an index to management for decision making only when coupled with an interdependent manner with the other factors enumerated.

It should be further noted that in normal operating applications of PERT, an increase in the density of activities from 3 to 6 times the original planned network is abnormal. This would further tend to minimize the significance of the change in probability estimates.

Captain Healy's article does poilnt up a need for a greater understanding of the use of PERT as a program management tool and the necessity for proper training of analysts and management personnel involved in PERT applications.

The interest in PERT techniques as a program management tool is ever in- creasing, and articles such as this contribute greatly to its utility and understanding.



Documents

Activity Subdivision and PERT Probability Statements