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