Abstract
Full Text
UDC 519.95
CYBERNETICS AND CONTROL THEORY
Yu. P. KRIVENKOV
ON THE THEORY OF NETWORK SYSTEMS OF PLANNING AND CONTROL
(Presented by Academician V. M. Glushkov on 8 I 1966)
The direct use of the hypotheses of the method of estimates and plan revision for processing network models of projects having stochastic estimates for the durations of individual activities leads to a considerable error in determining the random variable \(\zeta\), expressing the completion time of all project work \((^{1,2})\).
Replacing the hypothesis of the exclusive role of the critical path in forming \(\zeta\) by the hypothesis that \(\zeta\) is the maximum of random variables expressing the lengths of a certain group of the longest paths in the network, in view of the very great complexity of the analytical method \((^3)\), leads: a) to the necessity of using the method of statistical tests; b) to rejection of the hypothesis that project activities are small and close to infinitely subdivided; c) to the use of every possibility for aggregating activities.
All that has been said ultimately imposes more serious requirements on the distribution functions for the random variables \(\tau\), expressing the completion times of project activities. In this connection it should be noted that the system adopted as the basis in the method of estimates and plan revision, by its very nature, in principle does not permit construction of distribution functions for \(\tau\), since it is contradictory.
Indeed, this system consists of a four-parameter law of the \(\beta\)-distribution for each \(\tau\); three estimates characterizing this distribution: the optimistic \(t_1\), the most probable \(\hat{\tau}\), and the pessimistic \(t_2\); and two fixed formulas for the mathematical expectation \(\bar{\tau}\) and the variance \(D(\tau)\):
\[ \bar{\tau} = (t_1 + 4\hat{\tau} + t_2)/6,\qquad D(\tau) = (t_2 - t_1)^2/36. \tag{1} \]
Taking into account that the first and third estimates are specified quantiles \(\left(^{1}/_{100},\ ^{99}/_{100}\right)\), while the second is the mode of the distribution, we find that this system, generally speaking, is overdetermined, and hence contradictory.
In the present work a description is given of a certain model of an activity leading to a \(\beta\)-distribution. The estimates characterizing the distribution are analyzed, and a new definition is proposed for the principal estimates, permitting the traditional use in the system noted above and, moreover, a noncontradictory construction of the distribution function from these estimates. For the special case of research, prospecting, and design-and-development work in unexplored fields, the limiting case of the \(\beta\)-distribution in the form of a \(\Gamma\)-distribution is considered.
1. Model of an activity in network systems. Suppose that the beginning of an activity refers to the moment \(T_0\), and that the completion of the activity is a random variable \(\tau\), varying in the interval \((T_1, T_2)\) and possessing the following properties.
A. The entire time interval for performing the work \((T_0, \tau)\) consists of stages \(\rho_i\) \((i = 1,\ldots,n)\) of duration \((T_1 - T_0)/n\), pertaining to the work proper, and of \(m\) intervals \(\lambda_k\) \((k = 1,\ldots,m;\ 0 \leq m \leq n)\) of duration \((T_2 - T_1)/n\), pertaining to delays (i.e., of intervals
of the time required to identify and eliminate the disturbances, failures, difficulties that have arisen, for unplanned waiting, etc.).
B. If a difficulty has arisen on \(\rho_i\), then after \(\rho_i\) there is necessarily one \(\lambda_k\).
C. The event consisting in the fact that a difficulty has arisen at the \(i\)-th stage is determined by the \(i\)-th sample from some population.
D. Initially the unit population contains a fraction \(P\) favorable to difficulties.
E. With each stage the population increases by the amount \(\nu\), and, if difficulties arose at the preceding stage, then \(\nu\) is favorable to difficulties, and otherwise is not favorable.
If by \(A_i^k\) we denote the event consisting in the fact that at the \((i+1)\)-st stage a difficulty arose, provided that in the preceding \(i\) stages \(k\) difficulties arose, then
\[ P(A_i^k)=(P+k\nu)/(1+i\nu). \qquad (1\leq k\leq i\leq n). \]
Defining the probability density of the random variable
\[ p_{m,n}=C_n^m \frac{ p(p+\nu)\ldots[p+(m-1)\nu](1-p)(1-p+\nu)\ldots[1-p+(n-m-1)\nu] }{ 1(1+\nu)(1+2\nu)\ldots[1+(n-1)\nu] } \]
and taking into account that \(\tau=T_1+(T_2-T_1)x\), where \(x=\lim_{n\to\infty} m/n\), with the notation \(p/\nu=\alpha\), \((1-p)/\nu=\beta\), we obtain for \(\tau\) the distribution function
\[ F_{\tau}(t)=\frac{1}{K(\alpha,\beta)} \int_{T_1}^{T_2}(\tau-T_1)^{\alpha-1}(T_2-\tau)^{\beta-1}\,d\tau,\quad K(\alpha,\beta)=(T_2-T_1)^{\alpha+\beta-1}B(\alpha,\beta), \tag{2} \]
the mathematical expectation, mode, and variance:
\[ \overline{\tau}=\frac{\beta T_1+\alpha T_2}{\alpha+\beta},\qquad \hat{\tau}=\frac{(\beta-1)T_1+(\alpha-1)T_2}{\alpha+\beta-2},\qquad D(\tau)=\frac{\alpha\beta}{(\alpha+\beta)^2}\frac{(T_2-T_1)^2}{\alpha+\beta-1}. \tag{3} \]
2. Analysis of estimates characterizing the random variable \(\tau\). The mode of the distribution of \(\tau\), which has the meaning of the most probable estimate, is sufficiently clearly perceived in every task. It is very convenient to characterize \(\tau\) by the quantities \(T_1\) and \(T_2\), which, in light of the foregoing, are purely technological parameters that in some cases can be discerned. For example, \(T_1=T_0-\) waiting, \(T_2=\infty\)—research and exploratory work. Using the above-mentioned estimates \(t_1\) and \(t_2\) to describe \(\tau\), one must keep the following in mind:
A. The quantities \(t_1,t_2\) must not be identified with \(T_1,T_2\). Indeed, comparing the expressions for the variances (1) and (2), in the case of zero skewness we shall have \(\alpha=\beta\geq4\), whence, estimating the quantiles of the distribution, we obtain an error of \(26\%\), for
\[ t_1-T_1=T_2-t_2\gtrless \left(\frac{\alpha B(\alpha,\alpha)}{100}\right)^{1/\alpha}(T_2-T_1) =0.13(T_2-T_1)\quad \text{for }\alpha=\beta=4; \]
\[ t_1-T_1=T_2-t_2\gtrless0.16(T_2-T_1)\quad \text{for }\alpha=\beta=6. \tag{4} \]
B. The dependence of the quantiles \(t_1\) and \(t_2\) on the quantities \(q_1=1/100\) and \(q_2=99/100\) that define them is very strong; in other words, small changes in \(q_1\) and \(q_2\) lead to very large changes in \(t_1\) and \(t_2\).
C. The use of quantiles determined by very rare probabilities creates difficulties in forming estimates. Indeed, any expert more easily imagines quantiles \(1/3, 2/3\) than quantiles \(1/100, 99/100\), which in some cases are not even discernible.
3. Construction of a consistent system of estimates.
Consider the quantity \(\tau_c=\bar{\tau}+\sigma(\bar{\tau}-\hat{\tau})\), where \(\sigma\) is equal to two or to a number close to it. The quantity \(\tau_c\) in a certain sense characterizes the middle of the region where the values of the probability density are practically different from zero.
Examples. 1) \(T_1=0,\ T_2=1,\ \alpha=5,\ \beta=2,\ \bar{\tau}=0.714,\ \hat{\tau}=0.8,\ \sigma=1.5,\ \tau_c=0.58;\) 2) \(\alpha=9,\ \beta=2,\ \sigma=2,\ \tau_c=0.687.\)
Let us give a general definition: \(t_1\) and \(t_2\) are, respectively, the lower and upper (optimistic and pessimistic) estimates of the perceptible limits of the distribution of \(\tau\).
More concretely, we formulate this definition in the form
\[ \text{I. } (t_1+t_2)/2=\tau_c. \qquad \text{II. } t_2-t_1=\mu\sqrt{D(\tau)}. \tag{5} \]
Here \(\mu\) is an arbitrary constant subject to certain restrictions.
From the conditions \(t_1,t_2\in(T_1,T_2)\), \(t_1<\min\{\bar{\tau},\hat{\tau}\}\), \(t_2>\max(\bar{\tau},\hat{\tau}\}\), we obtain the conditions on \(\mu\):
\[ \frac{|\beta-\alpha|}{\alpha+\beta-2} \sqrt{\frac{\alpha+\beta+1}{\alpha\beta}} <\mu< \frac{2}{\alpha+\beta-2} \sqrt{\frac{\alpha+\beta-1}{\alpha\beta}} \min \left\{ \begin{array}{l} \alpha^2+\alpha\beta-\alpha(2+\sigma)+\beta\sigma\\ \beta^2+\alpha\beta-\beta(2+\sigma)+\alpha\sigma \end{array} \right\}. \]
In the case \(\alpha=\beta\), this condition takes the form \(0<\mu<\sqrt{2\alpha+1}\), and for \(\alpha\ge 2\) gives \(\mu<\sqrt{20}\cong 4.47\). In the case \(\beta=\infty\) we obtain \(1<\mu<2(2+\sigma)/\sqrt{\alpha}\); if \(\alpha\ge 2,\ \sigma=2\), then \(1<\mu<5.7\). Taking into account estimate (4), corresponding to the cases \(\alpha=\beta=4\) and \(\alpha=\beta=6\), and the condition \(t_2-t_1=\mu\sqrt{D}\), we shall have \(\mu\approx 4.44\) and \(\mu\approx 4.9\).
Computing the value of \(\mu\) for the normal distribution \((\alpha=\beta=\infty)\), we shall have \(t_2-t_1=2\cdot 2.33\sqrt{D(\tau)}\), i.e. \(\mu=4.66\).
The foregoing shows that, in the first approximation, one may put \(\mu=\sqrt{20}\cong 4.47\). We note that although the formal definition of the estimates \(t_1\) and \(t_2\) depends on the particular values of \(\sigma\) and \(\mu\), it should be borne in mind that these values can always be corrected with the aid of the distribution function actually constructed from the estimates \(t_1\) and \(t_2\).
4. Construction of the distribution function.
The quantities \(t_1\) and \(t_2\), in the sense of the new definition, can retain their operational meaning as the minimum and maximum duration of the work under the most favorable and, respectively, unfavorable combination of circumstances, and make it possible to determine \(\bar{\tau}\) and \(D(\tau)\) by the formulas:
\[ \bar{\tau}=(t_1+2\sigma\tau+t_2)/2(1+\sigma), \qquad D(\tau)=(t_2-t_1)^2/\mu^2. \tag{6} \]
If, in addition to the quantities \(t_1,\bar{\tau},t_2\), one more technological parameter is given, for example \(T_1\), then, analyzing expression (3), one can obtain the values for \(\alpha,\beta,T_2\) in the form
\[ \alpha=(M-1-2M\Delta)/[2-(3+M)\Delta], \qquad \beta=\alpha(1+\alpha)/(M-\alpha), \]
\[ T_2=[(\alpha+\beta)\bar{\tau}-\beta T_1]/\alpha, \]
where
\[ M=(\bar{\tau}-T_1)^2/D(\tau), \qquad \Delta=(\bar{\tau}-\hat{\tau})/(\bar{\tau}_1-T_1). \]
5. Limiting case \(T_2=\infty\).
Letting \(T_2\) and \(\beta\) tend to infinity under the condition \(\beta-1=BT_2\), where \(B=\mathrm{const}\), we obtain
\[ F_\tau(t)=\frac{B^\alpha}{\Gamma(\alpha)} \int_{T_1}^{t}(\tau-T_1)^{\alpha-1}e^{-B(\tau-T_1)}\,d\tau, \]
whence
\[ \bar{\tau}=T_1+\alpha/B,\qquad \hat{\tau}=T_1+(\alpha-1)/B,\qquad D(\tau)=\alpha/B^2. \]
Using relation (5) to determine the estimates \(t_1, t_2\), we obtain the validity of formulas (6). If the values \(t_1, \hat{\tau}, t_2\) are given, then the distribution parameters are obtained in the form
\[ \alpha=D(\tau)/(\bar{\tau}-\hat{\tau})^2,\qquad B=1/(\bar{\tau}-\hat{\tau}),\qquad T_1=\bar{\tau}-D(\tau)/(\bar{\tau}-\hat{\tau}). \]
The parameters are obtained just as simply if the sets \(T_1, \hat{\tau}, t_2\) or \(T_1, t_1, \hat{\tau}\) are given. In the case where \(T_1, \hat{\tau}\) and the quantile \(T_q\), where \(q=2/3\), are given, then to determine \(B\) one may use one of the iterative processes
\[ B_{k+1}=\frac{2}{3}\frac{B_k}{\varphi(B_k)} \quad \text{or} \quad B_{k+1}=B_k-\frac{1}{A(B_k)}\bigl(\varphi(B_k)-{}^2/{}_3\bigr), \]
in which
\[ \varphi(B)=\frac{B^\alpha}{\Gamma(\alpha)} \int_{T_1}^{T_q}(t-T_1)^{\alpha-1}e^{-B(t-T_1)}\,dt, \qquad \alpha=(\hat{\tau}-T_1)B+1, \]
\(A(B)\) is some positive function such that, in a neighborhood of the sought solution \(B_0\), the following holds:
\[ 0<\frac{d}{dB}\left[\frac{\varphi(B)-{}^2/{}_3}{A(B)}\right]<2. \]
Moscow Institute of Physics and Technology
Received
7 I 1966
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