Statistical background for development PERT project management technique software tool.
Majstorovic, Vlado ; Glavas, Marijana Bandic ; Rosic, Ivana Milinkovic 等
1. INTRODUCTION
Method of evaluation and review program was developed in 1958 by W.
Fazara and experts from the Special Project Office of the United States
Navy, Lockheed Aircraft and the consulting firm Booz, Allen and
Hamilton. Implementation began in the development of military project
Polaris (weapons systems), where a reduction of almost two years was
achieved. It is a graphical method based on networks, and it is useful
in cases where the exact time of certain activities in the project is
not known or it depends on some still unsolved issues. Statistical model
of analysis of the project implementation time by PERT method is of
stochastic nature, as a result of time estimation uncertainty of
duration of individual activities (Majstorovic, 2001). Seeing how
complicated PERT method application in practice is, we have decided to
create programming tool that will help us in the future providing us
faster and more efficient work.
2. STATISTICAL DISTRIBUTIONS
2.1 The concept and basic properties
Some terms: measurable space, probabilistic space, elementary
event, the space of elementary events, sigma algebra, complex event, the
function of probability, the probability of events, axioms of
probability theory and Borel [sigma]-algebra as defined in (Sarapa,
1993).
[FIGURE 1 OMITTED]
Let the ([OMEGA],F,P) is probabilistic and the (R,B) measurable
space. For a function X : [OMEGA] [right arrow] R with property that for
every set B [member of] B its inverse image [X.sup.-1] [B] is element of
F we say that the random variable is defined on the probabilistic space
([OMEGA], F,P) or that a measurable function is defined on the
measurable space ([OMEGA], F). Set X [[OMEGA]] = [R.sub.x] [subset or
equal to] R we call the space of results of random variable. Using
random variable X and probability P : F [right arrow] [0,1] we define
the probability [P.sub.x] : B [right arrow] [0,1] with the formula ([for
all] B [member of] B) [P.sub.x] (B) = P([X.sup.-1] [B]). Function
[P.sub.x] is often called the law of distribution X .
Probability theory knows two main types of random variables,
discrete and continuous. Distribution function of random variable X is
function F : R [right arrow] [0,1] defined by the formula F(x) =
P(-[infinity] < X [less than or equal to] x), [for all] x [member of]
R. We say that X is continuous random variable if its place of results
is some interval and if its distribution function F(x) in this interval
can continuously be derived.
Let the X continuous random variable with probability function f, H
: R [right arrow] R is measurable function on (R, B) and H(X) new random
variable. If [R.sub.x] is placement of the results of random variable X,
then number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is
called the expectation of random variable H (X ) related to the random
variable X.
Let the terms: [m.sub.p]--initial p-th moment of random variable X
; [m.sub.1]--expectation of random variable X, [M.sub.p] central p -th
moment of random variable X, [M.sub.2]--dispersion (variance) of random
variable X, [sigma]--standard deviation of random variable X,
[[alpha].sub.3]--coefficient of asymmetry of the distribution X,
[[alpha].sub.4]--coefficient of flattened of the distribution of random
variable X (and marks [mu], [[mu].sub.x], [bar.x], E[x];
[[sigma].sup.2], [[sigma.sup.2.sub.x], V[x], E[[(x - [mu]).sup.2]])
defined as in (Galic, 2004).
Let the gamma distribution X ~ [GAMMA] {[alpha], [beta]} and
[beta]-distribution X ~ Beta {[alpha], [beta]}, with parameters [alpha]
and [beta], [alpha], [beta] > 0, and their probability functions
f(x) = 1/[[beta].sup.[alpha]][GAMMA]([alpha]) [e.sup.-x/[beta]]
[x.sup.[alpha]-1], x [member of] [R.sup.+],
f(x) = 1/B([alpha], [beta]) [x.sup.[alpha]-1][(1 -
x).sup.[beta]-1], x [member of] <0,1> and the properties of
distributions, minimum and maximum defined as in (Vukadinovic, 1990).
3. APPLICATION OF P-DISTRIBUTION IN THE PERT METHOD OF PLANNING
PROJECTS
PERT method of planning projects is one of the oldest traditional
methods of analysis of project duration. It is
implemented in three phases: determination of duration of
activities, time the event occurres and the duration of the project. In
determination of duration of activities in this method it is assumed
that duration of individual project activities is not known in advance.
PERT network analysis technique is used to estimate project
duration when there is high degree of uncertainty of the duration of
individual activities. That is why, three values for each activity are
calculated with defined terms: optimistic duration -a- is the time for
which the activity may be the earliest finished, normal duration -m-
most likely time in which that activity will be done in most cases,
pessimistic duration -b- the longest time to carry out certain
activities, assuming that all possible obstacles appeare during its
execution.
PERT method uses the average value of time estimation distribution.
[FIGURE 2 OMITTED]
Estimation is based on calculation of the expected duration of each
activity for three estimations of duration of each activity by
mathematical and statistical, so called P-distribution, and by
determination of the variance of time which determines coarse measure of
the duration of activities, according to formulas in the table:
As for [beta]-distribution is valid E[X] = [alpha]/[alpha] +
[beta], V[X] = [alpha][beta]/[([alpha] + [beta]).sup.2]([alpha] + [beta]
+ 1), if instead of value X, whose [beta]- distribution is between 0 and
1, we put the time T with [beta]-distribution between a and b, then it
should be T = (b-a)X + a. From the previous formula we get: E(T) =
a[alpha] + b[beta]/[alpha] + [beta], V(T) = [(b-a).sup.2]
[alpha][beta]/[([alpha] + [beta]).sup.2] ([alpha] + [beta] + 1).
Especially, for values of parameters [alpha] and [beta]: [alpha] =
[beta] = 4, we get formulas which are commonly used in the PERT method
E(T) = a + 4m + b/6, V(T) = [(b - a/6).sup.2]. Since it is an
estimation, duration of each activity can be anywhere between a and b.
Labels to specification of the time when the event will happen, the PERT
method it is about the expected time: [([T.sub.E]).sub.i]--the earliest
expected start, [([T.sub.L]).sub.j]--the latest expected start,
[([T.sub.E]).sub.j]--the earliest expected finish,
[([T.sub.L]).sub.j]--the latest expected finish. Suppose that execution
of event has a normal distribution law. Then we get the probability of
execution of event (Z) as stochastic value with standardized normal
schedule [Z.sub.i] = [([T.sub.L]).sub.i] - ([T.sub.E]).sub.i]/[square
root of [summation]] [[sigma].sup.2.sub.i,j], where the
[[sigma.sup.2.sub.i,j] is the estimation of variance of activity of the
longest path which starts from the initial event, and ends in some event
(i). Expected time of project duration is equal to expected time of
execution of the last event: [([T.sub.E]).sub.n] = [([[T.sub.L]).sub.n].
Since this time is statistical value, the project can finish in shorter
or in longer time than expected time, so we talk about probability of
execution of the project. If we mark the planned period for project
completion as [([T.sub.P]).sub.n], and expected time of completion of
project as ([[T.sub.E]).sub.n], than we calculate probability factor
([Z.sub.n]) as follows: [Z.sub.n] = [([T.sub.p]).sub.n] -
[([T.sub.E]).sub.n]/[square root of [summation]]
[[sigma].sup.2.sub.i,j], where the [[sigma.sup.2.sub.ij], denotes the
variance of activities from critical path. Based on this determined
factor from the table for normal distribution we determine the
probability of project implementation [P.sub.(Z)] (Andrijic, 1982).
4. CONCLUSION
Success of the work is measured by achieving the following goals:
delivering of the product, service or project results within requested
quality, being on time and within budget.
Statistics is an area that can provide answers to the problems when
the "classical mathematics" can not accurately
"calculate" individual stages of the project. It is necessary
to know the basics of statistics that are mentioned in this paper, very
well, from which it can be seen that "classical statistical
calculation" is too long, especially in big and complex projects.
For this reason, it is necessary to have practical programme tool,
which is based on mentioned statistical foundations for PERT method, so
this method could be closer to the end users, which may not be familiar
with basics of statistic, but still they will be able to use the
mentioned program successfully, applying PERT method in practice. Our
further work will move in this direction.
5. REFERENCES
Andrijic, S. (1982). Matematicke metode programiranja u
organizacijama udruzenog rada (Mathematical methods of programming in
organizations of associated labor), Svjetlost, Sarajevo
Galic, R. (2004). Vjerojatnost (Probability), ETF, ISBN 9536032-27-9, Osijek
Heldman, W. & Cram, L. (2004). IT Project +, Sybex, ISBN
07821-4318-0, San Francisco-London
Ivkovic, Z.A. (1976). Teorija verovatnoca sa matematickom
statistikom (Probability theory with mathematical statistics),
Gradjevinska knjiga, Beograd
Majstorovic, V. (2001). Production and Project Management, DAAAM,
ISBN 3-901 509-23-2, Mostar-Wien
Sarapa, N. (1993). Vjerojatnost i statistika I.dio (Probability and
Statistics 1stpart), SK, Zagreb
Vukadinovic, S.V. (1990). Zbirka resenih zadataka iz teorije
verovatnoce (Collection of solved tasks in the theory of probability),
Privredni pregled, ISBN 86-315-0062-3, Beograd
Tab. 1. Formulas for PERT method
PERT formula Standard deviation Variance
([t.sub.i,j]): ([sigma]): ([[sigma].sup.2]):
[a.sub.i,j] + [b.sub.i,j] - [([b.sub.i,j] -
4[m.sub.i,j] + [a.sub.i,j]/6 [a.sub.i,j]).sup.2]/6
[b.sub.i,j]/6