European Journal of Operational Research 172 (2006) 838–854
www.elsevier.com/locate/ejor
Discrete Optimization
A multi-objective resource allocation problem
in PERT networks
Amir Azaron a, Hideki Katagiri a,*, Masatoshi Sakawa a,
Kosuke Kato a, Azizollah Memariani b
a
Department of Artificial Complex Systems Engineering, Graduate School of Engineering, Hiroshima University, Kagamiyama 1-4-1,
Higashi-Hiroshima, Hiroshima 739-8527, Japan
b
Department of Industrial Engineering, Bu-Ali Sina University, Ghobare Hamadani Boulevard, Hamadan, Iran
Received 14 July 2004; accepted 25 November 2004
Available online 12 February 2005
Abstract
We develop a multi-objective model for resource allocation problem in PERT networks with exponentially or Erlang
distributed activity durations, where the mean duration of each activity is a non-increasing function and the direct cost
of each activity is a non-decreasing function of the amount of resource allocated to it. The decision variables of the
model are the allocated resource quantities. The problem is formulated as a multi-objective optimal control problem
that involves four conflicting objective functions. The objective functions are the total direct costs of the project (to
be minimized), the mean of project completion time (min), the variance of project completion time (min), and the probability that the project completion time does not exceed a certain threshold (max). The surrogate worth trade-off
method is used to solve a discrete-time approximation of the original problem.
Ó 2005 Elsevier B.V. All rights reserved.
Keywords: Multiple objective programming; Project management and scheduling; Optimal control; Markov processes
1. Introduction
Project Scheduling has been a major objective of most models proposed to aid planning and management of projects. The most important method to schedule a project assuming deterministic durations is
the well-known CPM—Critical Path Method. However, most durations have a random nature and therefore, PERT was proposed to determine the distribution of the total duration, T.
*
Corresponding author. Tel.: +81 824 24 7701; fax: +81 824 22 7195.
E-mail address: [email protected] (H. Katagiri).
0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2004.11.018
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
839
This method is based on the substitution of the network by the CPAD—critical path assuming that
each activity has a fixed duration equal to its mean (critical path using average durations). The mean
and the variance of the CPAD are given by the sum of the means and of the variances of its activities,
respectively, and therefore these results considered the mean and the variance of the total duration of the
network.
Unfortunately, this is an optimistic assumption as the real mean, E(T), is greater than or equal to such
estimate. Thus, many authors have studied:
1. Analytical approximations of the cumulative distribution function of T, F(T). Charnes et al. [4]
developed a chance-constrained programming approach to PERT problems. They assumed exponential
activity durations. Martin [21] provided a systematic way of analyzing PERT networks through series–
parallel reductions. Fatemi Ghomi and Hashemin [10] generalized the Gaussian quadrature formula to
compute F(T). Fatemi Ghomi and Rabbani [11] presented a structural mechanism, which changes the
structure of network to a series–parallel network, to estimate F(T). Kulkarni and Adlakha [19] developed
a continuous-time Markov process approach to PERT problems with exponentially distributed activity
durations.
2. Upper or lower bounds of F(T). Elmaghraby [7] provided lower bounds for the true, expected project
completion time. Fulkerson [14], Clingen [5], Robillard [23] and Perry and Creig [22] have done similar
works.
3. Monte-Carlo simulation to estimate F(t). Several authors have used conditional sampling to achieve variance reduction, see Burt and Garman [3], Garman [15], and Sigal et al. [26]. Fishman [12] achieved further variance reduction by using a combination of quasirandom points and conditional sampling to
estimate the distribution and mean of project completion time.
In CPM networks, activity duration is viewed either as a function of cost or as a function of resources
committed to it. The well-known time–cost trade-off problem (TCTP) in CPM networks takes the former
view. Studies on TCTP have been done using various kinds of cost functions such as linear (Fulkerson [13]
and Kelly [18]), discrete (Demeulemeester et al. [6]), convex (Lamberson and Hocking [20] and Berman [2]),
and concave (Falk and Horowitz [9]). When the cost functions are arbitrary (still non-increasing), a
dynamic programming (DP) approach was suggested by Robinson [24]. A powerful approach for solving
this problem using dynamic programming was presented by Elmaghraby [8].
In the TCTP, the objective is to determine the duration of each activity in order to achieve the minimum
total direct and indirect costs of the project. Tavares [28] has presented a general model based on the
decomposition of the project into a sequence of stages and the optimal solution can be easily computed
for each practical problem as it is shown for a real case study.
Weglarz [30] studied this problem using optimal control theory and assuming that the processing speed
of each activity at time t is a continuous, non-decreasing function of the amount of resource allocated to the
activity at that instant of time. This means that also time is here considered as a continuous variable.
Unfortunately, it seems that this approach is not applicable to networks with a reasonable size (>10).
The stochastic assumption introduces additional difficulties and then the experimental approach has to
be adopted. There are also some other papers related to the applications of optimal control theory in stochastic networks, but we could not find any paper corresponding to the application of this issue in PERT
networks. For example, Jordan and Ku [17] investigated the optimal control of two multiserver loss queues
with two types of customers. Tseng and Hsiao [29] analyzed the optimal control of the arrival rate to a twostation network of queues for the objective of maximum system throughput under a system time-delay constraint optimality criterion. Shioyama [27] developed an optimal control problem in a queueing network
system with two types of customers and two stages. Azaron and Fatemi Ghomi [1] developed a new model
for the optimal control of service rates and the arrival rates to the service stations of a class of Jackson
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A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
networks, where the expected shortest time of the network and also the total operating costs of the service
stations per period, as two conflicting objective functions, are minimized.
In this paper, we develop a multi-objective model for the time–cost trade-off problem via optimal control
theory in PERT networks. It is assumed that the activity durations are independent random variables with
exponential or Erlang distributions. It is also assumed that the amount of resource allocated to each activity
is controllable, where the mean duration of each activity is a non-increasing function of this control variable. The direct cost of each activity is also assumed to be a non-decreasing function of the amount of resource allocated to it.
The problem is formulated as a multi-objective optimal control problem, where the objective functions
are the total direct costs of the project (to be minimized), the mean of project completion time (min), its
variance (min) and the probability that the project completion time does not exceed a given level (max).
For solving this problem, we do the discretization of time and convert the optimal control problem into
an equivalent nonlinear optimization problem.
We can apply one of the multiple objective techniques to obtain the optimal allocated resource quantities. We use the surrogate worth trade-off method, see Hwang and Masud [16], which is an interactive approach with explicit trade-off information given, to solve this multi-objective problem.
The remainder of this paper is organized in the following way. In Section 2, we extend the method of Kulkarni and Adlakha [19] to obtain the project completion time distribution in PERT networks with Erlang distributed activity durations. Section 3 presents the multi-objective resource allocation formulation. In Section
4, we describe about the application of the surrogate worth trade-off method to our formulation. Section 5
presents the computational experiments, and finally we draw the conclusion of the paper in Section 6.
2. Project completion time analysis in PERT networks
In this section, we present an analytical method to obtain the distribution function of project completion
time in PERT networks, or in fact the distribution function of the longest path from the source to the sink
node of a directed acyclic stochastic network, where the arc lengths or activity durations are mutually independent random variables with exponential or Erlang distributions. To do that, we extend the one of Kulkarni and Adlakha [19], because this method is an analytical one, simple, easy to implement on a computer
and computationally stable.
Let G = (V, A) be a PERT network with set of nodes V = {v1, v2, . . ., vm} and set of activities
A = {a1, a2, . . ., an}. Duration of activity a 2 A is either an exponentially distributed with the parameter
ka, or Erlang distributed with the parameters (ka, na).
We know that an exponential density function is a special case of Erlang density function with na = 1,
and consequently a random variable with Erlang density function and the parameters (ka, na) can be decomposed to na series of exponentially random variables with the parameter ka.
First, we transform the original PERT network into a new one, in which all activity durations have exponential distributions. To do that, we substitute each Erlang activity with the parameters (ka, na) with na series of exponential activities with the parameter ka. Now, Let G 0 = (V 0 , A 0 ) be the transformed network, in
which V 0 and A 0 represent the sets of nodes and arcs of this transformed network, respectively. The source
and sink nodes are denoted by s and t, respectively. For a 2 A 0 , let a(a) be the starting node of arc a, and
b(a) be the ending node of arc a.
Definition 1. Let I(v) and O(v) be the sets of arcs ending and starting at node v, respectively, which are
defined as follows:
IðvÞ ¼ fa 2 A0 : bðaÞ ¼ vg ðv 2 V 0 Þ;
ð1Þ
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
OðvÞ ¼ fa 2 A0 : aðaÞ ¼ vg
ðv 2 V 0 Þ:
841
ð2Þ
Definition 2. If X V 0 , such that s 2 X and t 2 X ¼ V 0 X , then an (s, t) cut is defined as:
ðX ; X Þ ¼ fa 2 A0 : aðaÞ 2 X ; bðaÞ 2 X g:
ð3Þ
An (s, t) cut ðX ; X Þ is called an uniformly directed cut (UDC), if ðX ; X Þ is empty.
Example 1. Before proceeding, we illustrate the material by an example. Consider the network shown in
Fig. 1.
Clearly, (1, 2) is a uniformly directed cut (UDC), because V 0 is divided into two disjoint subsets X and X ,
where s 2 X and t 2 X . The other UDCs of this network are (2, 3), (1, 4, 6), (3, 4, 6) and (5, 6).
Definition 3. Let D = E [ F be a uniformly directed cut (UDC) of a network. Then, it is called an admissible 2-partition, if I(b(a)) X F, for a 2 F.
To illustrate this definition, consider Example 1 again. As mentioned, (3, 4, 6) is a UDC. This cut can be
divided into two subsets E and F. For example, E = {4} and F = {3, 6}. In this case, the cut is an admissible
2-partition, because I(b(3)) = {3, 4} X F and also I(b(6)) = {5, 6} X F. However, if E = {6} and
F = {3, 4}, then the cut is not an admissible 2-partition, because I(b(3)) = {3, 4} F = {3, 4}.
Definition 4. During the project execution and at time t, each activity can be in one of the active, dormant
or idle states, which are defined as follows:
(i) Active: an activity is active at time t, if it is being executed at time t.
(ii) Dormant: an activity is dormant at time t, if it has finished but there is at least one unfinished activity
in I(b (a)). If an activity is dormant at time t, then its successor activities in O(b(a)) cannot begin.
(iii) Idle: an activity is idle at time t, if it is neither active nor dormant at time t.
The sets of active and dormant activities are denoted by Y(t) and Z(t), respectively, and
X(t) = (Y(t), Z(t)).
Consider Example 1, again. If activity 3 is dormant, it means that this activity has finished but the next
activity, i.e. 5, cannot begin because activity 4 is still active.
Table 1, presents all admissible 2-partition cuts of this network. We use a superscript star to denote a
dormant activity. All others are active. E contains all active while F includes all dormant activities.
Now, let S denote the set of all admissible 2-partition cuts of the network, and S ¼ S [ fð/; /Þg. Note
that X(t) = (/, /) implies that Y(t) = / and Z(t) = /, i.e. all activities are idle at time t and hence the project is completed by time t. It can be proved that {X(t), t P 0} is a continuous-time Markov process with
state space S, see Kulkarni and Adlakha [19] for the details of proof.
b
1
3
5
d
s
2
4
c
6
Fig. 1. Example network.
t
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A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
Table 1
All admissible 2-partition cuts of the example network
1.
2.
3.
4.
5.
6.
7.
8.
9.
(1, 2)
(2, 3)
(2, 3 )
(1, 4, 6)
(1, 4 , 6)
(1, 4, 6 )
(1, 4 , 6 )
(3, 4, 6)
(3 , 4, 6)
10.
11.
12.
13.
14.
15.
16.
17.
(3, 4 , 6)
(3, 4, 6 )
(3 , 4, 6 )
(3, 4 , 6 )
(5, 6)
(5 , 6)
(5, 6 )
(/, /)
As mentioned before, E and F contain active and dormant activities of a UDC, respectively. When activity a finishes (with the rate of ka), and there is at least one unfinished activity in I(b(a)), it moves from E to a
new dormant activities set, i.e. to F 0 . Furthermore, if by finishing this activity, its succeeding ones, O(b(a)),
become active, then this set will also be included in the new E 0 , while the elements of I(b(a)), which one of
them belongs to E and the other ones belong to F, will be deleted from these sets. Thus, the elements of the
infinitesimal generator matrix Q = [q{(E, F), (E 0 , F 0 )}], (E, F) and ðE0 ; F 0 Þ 2 S, are calculated as follows:
qfðE; F Þ; ðE0 ; F 0 Þg ¼
8
ka
>
>
>
>
>
>
>
>
>
>
>
>
< ka
>
>
>
>
P
>
>
>
ka
>
>
a2E
>
>
>
:
0
if a 2 E; IðbðaÞÞ 6 F [ fag; E0 ¼ E fag;
F 0 ¼ F [ fag;
ðaÞ
0
if a 2 E; IðbðaÞÞ F [ fag; E ¼ ðE fagÞ [ OðbðaÞÞ;
F 0 ¼ F IðbðaÞÞ;
ðbÞ
if E0 ¼ E; F 0 ¼ F ;
ðcÞ
otherwise:
ðdÞ
ð4Þ
In Example 1, if we consider E = {1, 2}, F = (/), E 0 = {2, 3} and F 0 = (/), then E 0 = (E {1}) [ O(b(1)),
and thus from (4b), q{(E, F), (E 0 , F 0 )} = k1.
{X(t), t P 0} is a finite-state absorbing continuous-time Markov process. Since q{(/, /), (/, /)} = 0, this
state would be an absorbing one and obviously the other states are transient. Furthermore, we number the
states in S such this Q matrix be an upper triangular one. We assume that the states are numbered
1; 2; . . . ; N ¼j S j. State 1 is the initial state, namely (O(s), /); and state N is the absorbing state, namely
(/, /).
Let T represent the length of the longest path in the network, or the project completion time. Clearly,
T = min{t > 0 : X(t) = N/X(0) = 1}. Thus, T is the time until {X(t), t P 0} gets absorbed in the final state
starting from state 1.
Chapman–Kolmogorov backward equations can be applied to compute F(t) = P{T 6 t}. If we define:
P i ðtÞ ¼ P fX ðtÞ ¼ N =X ð0Þ ¼ ig;
i ¼ 1; 2; . . . ; N ;
ð5Þ
then F(t) = P1(t).
The system of differential equations for the vector P(t) = [P1(t), P2(t), . . ., PN(t)]T is given by
P 0 ðtÞ ¼ QP ðtÞ;
T
P ð0Þ ¼ ½0; 0; . . . ; 1 ;
ð6Þ
where P(t) represents the state vector of system and Q is the infinitesimal generator matrix. By taking
advantage of the upper triangular nature of Q, the differential equations (6) can be easily solved.
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
843
3. Multi-objective resource allocation problem
In this section, we develop a multi-objective model to optimally control the resources allocated to the
activities in a PERT network with exponentially or Erlang distributed activity durations, where the mean
duration of each activity is a non-increasing function and the direct cost of each activity is a non-decreasing
function of the amount of resource allocated to it. We may decrease the total direct costs of the project, by
decreasing the amount of resource allocated to the activities. However, clearly it causes the project completion time to be increased, because these objectives have the confliction with each other. Consequently, an
appropriate trade-off between the total direct costs, and the project completion time is required.
This is a multi-objective stochastic programming problem. Therefore, we transform this model into a
relevant multi-objective problem with four conflicting deterministic objective functions. The objective functions are the total direct costs of the project (to be minimized), the mean of project completion time (min),
the variance of project completion time (min), and the probability that the project completion time does not
exceed a certain threshold (max). The last three objective functions are the appropriate deterministic objective functions associated with the minimization of project completion time, which has a random nature.
The direct cost of activity a 2 A is assumed to be the non-decreasing function da(xa) of the
Pamount of
resource xa allocated to it. Therefore, the total direct costs of the project would be equal to a2A d a ðxa Þ.
The mean duration of activity a 2 A, which is equal to na/ka (na = 1 for exponential and na > 1 for Erlang distribution), is assumed to be the non-increasing function ga(xa) of the amount of resource xa allocated to it. Let Ua represent the amount of resource available to be allocated to the activity a, and La
represent the minimum amount of resource required to achieve the activity a. It is also assumed that the
mean duration of activity a cannot be smaller than a specific value ca, even if we allocate the maximum
amount of resource to it.
In application, da(xa) and ga(xa) can be estimated using linear regression. We can collect the sample
paired data of da(xa) and ga(xa) as the dependent variables, for the different values of xa as the unique independent variable, from the similar activities, which have done before, or using the judgments of the experts
in this area. Then, we can estimate the parameters of the relevant linear regression model.
The mean of project completion time is
EðT Þ ¼
Z
1
tP 01 ðtÞ dt ¼
0
Z
1
ð1 P 1 ðtÞÞ dt;
ð7Þ
0
where P 01 ðtÞ is the density function of project completion time.
The variance of project completion time is
VarðT Þ ¼
Z
1
2
t2 P 01 ðtÞ dt ½EðT Þ :
ð8Þ
0
The probability that the project completion time does not exceed a given threshold u is
P ðT 6 uÞ ¼ P 1 ðuÞ:
ð9Þ
Taking into account the above assumptions, the infinitesimal generator matrix Q is a function of the control vector k = [k1, k2, . . ., kn]T. Therefore, the dynamic model is
P 0 ðtÞ ¼ QðkÞP ðtÞ;
P i ð0Þ ¼ 0;
P N ðtÞ ¼ 1:
i ¼ 1; 2; . . . ; N 1;
ð10Þ
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A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
Theorem 1. The appropriate multi-objective optimal control problem is
X
d a ðxa Þ;
Min f 1 ðX ; kÞ ¼
Min f 2 ðX ; kÞ ¼
a2A
Z 1
tP 01 ðtÞ dt;
0
Min f 3 ðX ; kÞ ¼
Z
1
t2 P 01 ðtÞ dt 0
ð11bÞ
Z
1
2
tP 01 ðtÞ dt ;
ð11dÞ
0
s:t:
ð11cÞ
0
f 4 ðX ; kÞ ¼ P 1 ðuÞ;
Max
ð11aÞ
P ðtÞ ¼ QðkÞP ðtÞ;
ð11eÞ
P i ð0Þ ¼ 0;
ð11fÞ
i ¼ 1; 2; . . . ; N 1;
P N ðtÞ ¼ 1;
ð11gÞ
ka 6 na =ca ;
a 2 A;
ka 6 na =ga ðxa Þ;
a 2 A;
ð11hÞ
ð11iÞ
xa 6 U a ;
a 2 A;
ð11jÞ
xa P La ;
a 2 A;
ð11kÞ
a 2 A:
ka P 0;
ð11lÞ
Proof. The objective functions (11b)–(11d) are the appropriate deterministic objective functions associated
with the minimization of project completion time. The constraints (11e)–(11g) determine the dynamic of
this continuous-time system. According to the constraints (11i), if we allocate xa to the activity a 2 A
and ga(xa) exceeds ca, then ka gets its maximum value and becomes equal to na/ga(xa), or the mean duration
of activity a becomes equal to ga(xa), because the mean and variance of this activity will be decreased in this
case and consequently the last three objective functions would be satisfied. Otherwise, according to the constraints (11h), if ga(xa) does not exceed ca, then because of the same reason, ka gets its maximum value and
becomes equal to na/ca, or the mean duration of activity a becomes equal to ca. Constraints (11j) result from
the capacity limitation on the resources, and the minimum amount of resource required to achieve each
activity is insured by constraints (11k). h
We try to solve problem (11), optimally, using the Maximum Principle, see Sethi and Thompson
[25] for
R1
the details. For simplicity, we consider only one objective function, for example f2 ðX ; kÞ ¼ 0 ð1 P 1 ðtÞÞ dt,
in the model.
Considering k as the control vector of this problem, which is also dependent on the vector
x = [x1, x2, . . ., xn]T, K as the set of allowable controls, which consists of the constraints from (11h)–(11l)
(k 2 K), and n-vector l(t) as the adjoint vector function, the Hamiltonian function would be
T
H ðlðtÞ; P ðtÞ; kÞ ¼ lðtÞ QðkÞP ðtÞ þ 1 P 1 ðtÞ:
ð12Þ
Then, we write the adjoint equations and terminal conditions, which are
T
T
l0 ðtÞ ¼ lðtÞ QðkÞ þ ½1; 0; . . . ; 0;
T
lðT Þ ¼ 0;
T ! 1:
ð13Þ
If we could compute l(t) from (13), we could minimize the Hamiltonian function subject to k 2 K in
order to get the optimal control k , and solve the problem optimally. Unfortunately, the adjoint equations
(13) are dependent on the unknown control vector, k, and therefore they cannot be solved directly.
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
845
If we could also minimize the Hamiltonian function (12), subject to k 2 K, for an optimal control function in closed form as k = f(P (t), l (t)), then we could substitute this into the state equations,
P 0 (t) = Q(k)P(t), P(0) = [0, 0, . . ., 1]T, and adjoint equations (13) to get a set of differential equations, which
is a two-point boundary value problem. Unfortunately, we cannot obtain k by differentiating H respect to
k, because the minimum of H occurs on the boundary of K, and consequently k cannot be obtained in
closed form.
According to the two mentioned points, it is impossible to solve the optimal control problem (11), optimally, even in the case of single objective. Relatively few optimal control problems can be solved optimally.
Therefore, we try to solve this problem, numerically. To do that, we do the discretization of time and convert the multi-objective optimal control problem into an equivalent multi-objective nonlinear programming
one. In other words, we transform the differential equations to the equivalent difference equations as well as
transform the integral terms into equivalent summation terms. To follow this approach, the time interval is
divided into K equal portions with the length of Dt. If Dt is sufficiently small, it can be assumed that P(t)
varies only in times 0, Dt, . . ., (K 1)Dt.
Corollary 1. Considering bu/Dtc as the integer part of u/Dt, the appropriate multi-objective nonlinear
optimization problem is
X
Min f 1 ðX ; kÞ ¼
d a ðxa Þ;
ð14aÞ
a2A
Min
f 2 ðX ; kÞ ¼
K 1
X
kDtðP 1 ðk þ 1Þ P 1 ðkÞÞ;
ð14bÞ
k¼0
Min
"
#2
K 1
K 1
X
X
2
ðkDtÞ ðP 1 ðk þ 1Þ P 1 ðkÞÞ kDtðP 1 ðk þ 1Þ P 1 ðkÞÞ ;
f 3 ðX ; kÞ ¼
k¼0
Max
f 4 ðX ; kÞ ¼ P 1 bu=Dtc;
s:t:
P ðk þ 1Þ ¼ P ðkÞ þ QðkÞP ðkÞDt;
P i ð0Þ ¼ 0;
ð14cÞ
k¼0
ð14dÞ
k ¼ 0; 1; . . . ; K 1;
i ¼ 1; 2; . . . ; N 1;
ð14eÞ
ð14fÞ
P N ðkÞ ¼ 1;
k ¼ 0; 1; . . . ; K;
ð14gÞ
ka 6 na =ca ;
a 2 A;
ð14hÞ
ka 6 na =ga ðxa Þ;
a 2 A;
ð14iÞ
xa 6 U a ;
a 2 A;
ð14jÞ
xa P L a ;
a 2 A;
ð14kÞ
P i ðkÞ 6 1;
ka P 0;
i ¼ 1; 2; . . . ; N 1; k ¼ 1; 2; . . . ; K;
a 2 A:
ð14lÞ
ð14mÞ
Proof. The integral terms in (11b) and (11c), are easily transformed into the equivalent summation terms
(14b) and (14c), by considering P(k) instead of P(kDt). The objective function (11d) is transformed to the
new one (14d), because of the same reason. The differential Eqs. (11e)–(11g) are also easily transformed
into the equivalent difference Eqs. (14e)–(14g), by the same approach. Since each Pi(k), for
i = 1, 2, . . ., N 1, k = 1, 2, . . ., K is a distribution function, then the constraints (14l) should also be
included in this nonlinear programming problem. h
846
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
If we consider the initial and terminal state conditions for P(k) implicitly, and substitute each Pi(0) and
PN(k) with zero and one, respectively, the multi-objective nonlinear optimization problem (14) would have
2K(N 1) + 5n constraints including the constraints (14m), and K(N 1) + 2n decision variables.
For estimating the length of the time interval in this problem, first we assume that the estimated amount
of resource allocated to the activity a is Da = 0.5(La + Ua). In this case, the mean duration of this activity
would be equal to ga(Da) and its variance is equal to ga(Da)2/n. Then, it is assumed that each activity has a
fixed duration equal to its mean. The mean and the variance of the critical path are given by the sum of the
means (M) and of the variances (V) of its activities, respectively. pInffiffiffiffi this case, a good estimation for the
length of the time interval would be equal to T^ ¼ KDt ¼ M þ 2 V , because in most distributions, for
example normal distribution, an appropriate upper bound, in which the probability that the random variable exceeds this bound is less than 0.05 would be equal to its mean plus two times of its standard
deviation.
A computer program was written in order to evaluate our algorithm on some problems with different
sizes and investigate the trade-off between the accuracy (correctness) and the computation time in each
problem. At the beginning of the algorithm, we consider K = 10 and Dt ¼ T^ =10. A feasible solution should
posses this property that P1(K) P 1 e, in which e is assumed to be equal 0.05 in this paper. If a solution
does not have the mentioned property, the value of Dt is increased in order to satisfy this necessary
condition.
After solving the problem (14) and obtaining k , we compute P1(t) by solving the system of differential equations with constant coefficients (6), analytically. Then, we compute the real mean and the
variance of the project completion time from (7) and (8), respectively. The percentage difference between the approximated mean taken from (14b) and the real mean taken from (7), PD.M, and also
the percentage difference between the approximated variance taken from (14c) and the real variance
taken from (8), PD.V, or the absolute differences between the approximated values and the real values
divided by the real ones can be considered as two important criteria for the accuracy of the discretetime approximated solution. As K is increased and Dt is decreased, PD.M and PD.V approach zero.
Therefore, the approximated discrete project completion time distribution approaches to the analytical
distribution, because of matching the first two moments, and consequently, the optimal solution of the
discrete-time problem (14) approaches to the optimal solution of the original optimal control problem
(11).
In the next steps of the algorithm, we replace K with K + 10 and each new value for Dt should be selected, such that the length of the time interval remains unchanged, in order to investigate the trade-off between optimality and computation time.
4. Surrogate worth trade-off method
For solving the multi-objective nonlinear optimization problem (14), we use the surrogate worth tradeoff method, which is an interactive approach with explicit trade-off information given, see Hwang and
Masud [16] for the details. The method consists of two phases: Phase 1. Identification and generation of
non-dominated solutions which form the trade-off functions in the objective surface. Phase 2. Search for
a preferred solution in the non-dominated solutions. The preferred solution is located by interaction with
the decision maker ‘‘DM’’.
The major steps in the SWT method to solve the multi-objective programming problem (14) are
Step 1. Determine the ideal solution for each of the objectives of problem (14), or the optimal value for
each objective function subject to the set of constraints. Then, set up the multi-objective problem
in the following form:
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
Min f 1 ðX ; kÞ
s:t:
f j ðX ; kÞ 6 ej ;
847
j ¼ 2; 3;
ð15Þ
f 4 ðX ; kÞ P e4 ;
X ; k 2 S ðfeasible region of problem (14)Þ:
Step 2. Identify and generate a set of non-dominated solutions by varying eÕs parametrically in problem
(15).
Step 3. Interact with the DM to assess the surrogate worth function wj, or the DMÕs assessment of how
much (from 10 to 10) he prefers trading gj marginal units of the first objective for one marginal
unit of the jth objective fj (X, k), given the other objectives remaining at their current values.
Step 4. Isolate the indifference solutions. The solutions, which have wj = 0 for all j, are said to be indifference solutions. If there exists no indifference solution, develop approximate relations for all worth
_
functions wj ¼ wj ðfj ; j ¼ 2; 3; 4Þ, by multiple regressions.
5. Computational experiments
For showing the numerical stability of the theoretical developments of the paper, we solve three numerical examples, and investigate the trade-off between the accuracy and the computation time in each problem. Case I is depicted in Fig. 2.
Table 2 shows the characteristics of the activities. The given threshold value, u, is equal to 40. We want
to obtain the optimal allocated resource quantities. The transformed network is shown in Fig. 3. In this
network, the durations of activities 3, 4 are exponentially distributed with the parameter k3. The stochastic
process {X(t), t P 0} has seven states in the order of S ¼ fð1Þ; ð2; 3Þ; ð2 ; 3Þ; ð2; 4Þ; ð2 ; 4Þ; ð2; 4 Þ; ð/; /Þg.
Table 3 shows matrix Q(k).
The ideal solutions are f1 ¼ 11 at (x1 = 1, x2 = 1, x3 = 1), f2 ¼ 10:116 at (x1 = 4, x2 = 6, x3 = 6),
f3 ¼ 38:487 at (x1 = 4, x2 = 6, x3 = 6) and f4 ¼ 0:999 at (x1 = 4, x2 = 6, x3 = 6).
The single objective optimization problem for generating a set of non-dominated solutions is formulated
according to (15). The length of the time interval is approximated as T^ ¼ 50. Therefore, we consider K = 10
and Dt = 5, at the beginning. Then, ej, j = 2, 3, 4, are varied parametrically to obtain several non-dominated
1
s
2
b
t
3
Fig. 2. Case I.
Table 2
Characteristics of the activities in Case I
a
Distribution
Parameters
da(xa)
ga(xa)
ca
La
Ua
1
2
3
Exponential
Exponential
Erlang
k1
k2
(k3, n3 = 2)
3x1 + 2
2x2 + 1
x23 þ 2
24 5x1
20 3x2
15 2x3
5
4
3
1
1
1
4
6
6
848
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
1
2
s
b
t
3
4
c
Fig. 3. Transformed network of Case I.
Table 3
Matrix Q(k) corresponding to Case I
State
1
2
3
4
5
6
7
1
2
3
4
5
6
7
k1
0
0
0
0
0
0
k1
(k2 + k3)
0
0
0
0
0
0
k2
k3
0
0
0
0
0
k3
0
(k2 + k3)
0
0
0
0
0
k3
k2
k3
0
0
0
0
0
k3
0
k2
0
0
0
0
0
k3
k2
0
solutions. For investigating the trade-off between accuracy and computation time on a PC Pentium IV
2.1 GHz Processor, C.T. (hh:mm:ss), we also solve the problem for some different pairs of K and Dt as:
(K = 20, Dt = 2.5), (K = 30, Dt = 1.7), (K = 40, Dt = 1.3), (K = 50, Dt = 1), (K = 100, Dt = 0.5) and compute the values of PD.M and PD.V in each case. Table 4 shows the results for 10 different sets of ej,
j = 2, 3, 4. In this table, all solutions are non-dominated and satisfy the necessary condition (P1(K) P 0.95).
Now, according to any pair of K and Dt, the DM can select any of the above non-dominated solutions, if
the surrogate worth functions wj for all j, assessed by the DM, for that non-dominated solution are considered equal to zero. Otherwise, we can obtain an indifference solution, approximately, using multiple regressions. For example, if the surrogate worth functions for the non-dominated solution corresponding with
e2 = 24, e3 = 55, e4 = 0.95, K = 100 and Dt = 0.5 are all equal to zero, or this solution is considered as
an indifference solution by the DM, then the optimal allocated resource quantities are
T
T
x ¼ ½x1 ; x2 ; x3 ¼ ½3:8; 5:23; 2:975 . In this case, the approximated mean project completion time is almost
the same as the real one and the percentage difference between the approximated variance and the real variance is about 14%. Therefore, the accuracy of this solution is acceptable, but access to this level of accuracy
needs a long computation time (about 17 minutes).
Case II is depicted in Fig. 1. This network has six activities with exponentially distributed activity durations. Table 5 shows the characteristics of the activities. The threshold value is equal to 5.
The length of the time interval is approximated as T^ ¼ 6. Table 6 shows the trade-off between the accuracy and the computation time of one non-dominated solution, which is assumed to be considered as an
indifference solution by the DM, for the different pairs of K and Dt.
In the case of K = 70 and Dt = 0.086, the optimal resource quantities would be
x = [1, 1, 7.366, 4, 1, 5.147]T. In this case, PD.M and PD.V are almost equal to 1.3% and 41%, respectively.
The accuracy of this solution is relatively good, but even access to this level of accuracy takes a long time
(more than one hour).
Case III is depicted in Fig. 4. This network has 10 activities with exponentially distributed activity durations. In this case, the stochastic process {X(t), t P 0} has 25 states. Table 7 shows the characteristics of the
activities. The threshold value is equal to 8.
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
849
Table 4
A set of non-dominated solutions in Case I
e2
e3
e4
f1
f2
25
25
25
25
25
25
70
70
70
70
70
70
0.9
0.9
0.9
0.9
0.9
0.9
19.809
23.876
25.871
27.23
28.072
30.065
25
21.262
19.754
18.842
18.243
17.174
30
30
30
30
30
30
100
100
100
100
100
100
0.9
0.9
0.9
0.9
0.9
0.9
18.553
20.931
22.157
22.872
23.116
23.949
25.557
22.739
21.959
21.606
21.131
20.527
20
20
20
20
20
20
85
85
85
85
85
85
0.9
0.9
0.9
0.9
0.9
0.9
23.511
25.663
25.509
25.456
25.16
26.347
20
20
20
20
20
19.112
35
35
35
35
35
35
150
150
150
150
150
150
0.75
0.75
0.75
0.75
0.75
0.75
15.362
17.215
17.907
18.338
18.368
18.763
26.911
24.354
23.741
23.481
22.682
22.099
23
23
23
23
23
23
90
90
90
90
90
90
0.9
0.9
0.9
0.9
0.9
0.9
21.701
21.84
23.182
23.983
24.333
25.42
23
22.384
21.491
20.992
20.51
19.679
22
22
22
22
22
22
80
80
80
80
80
80
0.9
0.9
0.9
0.9
0.9
0.9
23.511
22.806
24.341
25.354
25.92
27.414
27
27
27
27
27
27
75
75
75
75
75
75
0.9
0.9
0.9
0.9
0.9
0.9
29
29
29
29
29
29
110
110
110
110
110
110
21
60
f3
70
70
70
70
70
70
f4
PD.M (%)
PD.V (%)
K
Dt
C.T
0.929
0.963
0.976
0.978
0.978
0.981
7.17
2.81
1.48
1.09
1.42
1.27
70.01
48.23
36.25
29.68
25.45
16.29
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
3
17
1:08
1:23
4:37
19:52
88.055
100
100
100
100
100
0.9
0.91
0.928
0.931
0.928
0.934
10.58
9.67
7.28
5.67
6.69
5.79
67.98
51.92
42.50
35.82
33.44
25.29
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
11
19
49
1:25
1:50
13:11
26.766
57.501
72.107
79.403
84.337
85
0.987
0.98
0.974
0.966
0.958
0.962
10.05
1.06
1.69
1.92
3.15
2.88
81.16
48.56
36.27
30.13
27.72
19.67
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
4
24
44
1:19
1:40
11:48
0.797
0.812
0.83
0.829
0.82
0.825
19.97
20.27
19.23
18.11
20.76
20.97
60.68
52.94
50.02
48.25
48.13
46.57
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
21
26
1:07
2:24
3:12
10:51
77.519
90
90
90
90
90
0.913
0.929
0.946
0.949
0.948
0.954
11.27
7.13
4.79
3.51
4.22
3.62
71.17
50.41
39.52
32.46
29.44
21.02
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
6
15
1:19
2:02
2:06
6:14
22
21.867
20.808
20.074
19.511
18.497
26.766
80
80
80
80
80
0.987
0.947
0.962
0.965
0.965
0.969
1.06
4.81
2.80
1.97
2.49
2.23
81.15
49.15
37.23
30.22
26.84
18.39
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
4
17
26
1:50
2:17
31:45
19.45
23.323
25.033
26.227
26.917
28.645
25.124
21.594
20.34
19.479
18.896
17.848
75
75
75
75
75
75
0.921
0.955
0.97
0.972
0.972
0.976
8.19
3.71
2.05
1.48
1.91
1.70
69.45
48.56
36.52
29.76
26.09
17.28
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
6
15
16
1:43
2:37
26:44
0.85
0.85
0.85
0.85
0.85
0.85
17.217
20.057
21.20
21.865
22.037
22.754
26.239
22.917
22.283
22.037
21.531
21.048
110
110
110
110
110
110
0.864
0.89
0.909
0.912
0.908
0.914
14.08
12.55
10.11
8.18
9.56
8.62
65.47
53.90
45.70
39.51
37.84
30.68
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
3
51
59
1:10
2:23
6:01
0.95
24.145
21
0.994
2.77
84.06
150
150
150
150
150
150
20.804
10
5
6
(continued on next page)
850
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
Table 4 (continued)
e2
e3
e4
f1
f2
f3
f4
PD.M (%)
PD.V (%)
K
Dt
C.T
21
21
21
21
21
60
60
60
60
60
0.95
0.95
0.95
0.95
0.95
25.212
27.971
29.737
30.961
33.568
20.336
18.439
17.486
16.854
15.757
60
60
60
60
60
0.978
0.986
0.987
0.994
0.989
1.31
0.76
0.53
0.71
0.61
48.33
36.53
29.49
24.75
14.70
20
30
40
50
100
2.5
1.7
1.3
1
0.5
25
43
1:07
1:47
8:08
24
24
24
24
24
24
55
55
55
55
55
55
0.95
0.95
0.95
0.95
0.95
0.95
20.985
26.101
29.291
31.303
32.757
35.719
24
19.7
17.733
16.77
16.12
15.007
55
55
55
55
55
55
0.95
0.984
0.99
0.991
0.996
0.992
4.05
0.89
0.49
0.36
0.38
0.45
72.61
49.01
36.98
29.83
24.55
14.37
10
20
30
40
50
100
5
2.5
1.7
1.3
1
0.5
3
28
39
52
2:08
17:10
Table 5
Characteristics of the activities in Case II
a
da(xa)
ga(xa)
ca
La
Ua
1
2
3
4
5
6
2x1
3x2 + 1
x3 + 2
x4
3x5 + 4
x6 + 3
0.7 0.1x1
1.5 0.2x2
1 0.1x3
1.5 0.3x4
0.9 0.1x5
1.1 0.1x6
0
0
0
0
0
0
1
1
1
1
1
1
5
6
9
4
5
6
Table 6
Trade-off results in Case II
e2
e3
e4
f1
f2
f3
f4
PD.M (%)
PD.V (%)
K
Dt
C.T
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
0.9
0.9
0.9
0.9
0.9
0.9
0.9
38.411
36.402
35.599
35.235
34.986
34.513
2.5
2.5
2.5
2.5
2.5
2.5
0.623
1.203
1.371
1.421
1.454
1.471
0.988
0.944
0.937
0.931
0.93
0.929
11.31
3.73
0.76
0.08
0.43
1.38
59.88
47.65
44.96
42.95
41.67
41.11
10
20
30
40
50
60
70
0.6
0.3
0.2
0.15
0.12
0.1
0.086
*
2:52
13:53
23:56
27:48
49:21
1:20:50
* No feasible solution exists for K = 10 and Dt = 0.6.
The length of the time interval is approximated as T^ ¼ 11. Table 8 shows the trade-off between the accuracy and the computation time of one non-dominated solution, which is assumed to be considered as an
indifference solution by the DM, for the different pairs of K and Dt.
In the case of K = 50 and Dt = 0.22, the optimal resource quantities would be
x = [1.5, 1.5, 3, 3, 1.782, 1.5, 3, 1.5, 1.5, 3]T. In this case, PD.M and PD.V are almost equal to 1.4% and
32%, respectively. The accuracy of this solution is relatively good, but access to this level of accuracy needs
a long time (about 50 minutes).
Figs. 5–7 show the computation time, PD.M and PD.V, respectively, according to the different pairs of K
and Dt for 10 non-dominated solutions of Case I and also the indifference solutions of Cases II and III,
respectively.
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
1
4
s
8
b
2
c
10
d
3 5
6
f
7
851
t
9
e
Fig. 4. Case III.
Table 7
Characteristics of the activities in Case III
a
da(xa)
ga(xa)
ca
La
Ua
1
2
3
4
5
6
7
8
9
10
2x1
3x2 + 1
x3 + 2
x4
3x5 + 4
x6 + 3
2x7 + 5
4x8 + 1
5x9 + 2
2x10 + 3
0.7 0.1x1
1.5 0.2x2
1 0.1x3
1.5 0.3x4
1.3 0.2x5
1.1 0.1x6
1.5 0.2x7
1 0.2x8
0.9 0.1x9
2 0.4x10
0
0
0
0
0
0
0
0
0
0
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
3
3
3
3
3
3
3
3
3
3
Table 8
Trade-off results in Case III
e2
e3
e4
5.5
5.5
5.5
5.5
5.5
5
5
5
5
5
0.85
0.85
0.85
0.85
0.85
f1
f2
58.783
67.166
67.222
66.846
f3
5.123
5.5
5.5
5.5
f4
5
2.021
2.405
2.717
PD.M (%)
0.869
0.929
0.931
0.922
19.61
1.13
1.06
1.45
PD.V (%)
K
Dt
C.T
4.96
49.65
39.99
32.86
10
20
30
40
50
1.1
0.55
0.367
0.275
0.22
*
6:25
19:49
45:20
51:35
* No feasible solution exists for K = 10 and Dt = 1.1.
6000
4000
3000
2000
100
K
70
60
50
40
30
0
20
1000
10
C.T. (Sec.)
5000
Fig. 5. Computation time (second) versus K.
Ι1
Ι2
Ι3
Ι4
Ι5
Ι6
Ι7
Ι8
Ι9
Ι10
ΙΙ
ΙΙΙ
852
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
25
PD.M %
20
15
10
5
0
10 20 30 40 50 60 70
K
10
0
Ι1
Ι2
Ι3
Ι4
Ι5
Ι6
Ι7
Ι8
Ι9
Ι10
ΙΙ
ΙΙΙ
0
10
K
70
60
50
40
30
20
90
80
70
60
50
40
30
20
10
0
10
PD.V%
Fig. 6. PD.M versus K.
Ι1
Ι2
Ι3
Ι4
Ι5
Ι6
Ι7
Ι8
Ι9
Ι10
ΙΙ
ΙΙΙ
Fig. 7. PD.V versus K.
According to Fig. 5, computation time grows with K. Computation time is also strongly dependent on
the network size, and sensitive to the values of ej, j = 2, 3, 4, or the structure of non-dominated solutions.
According to Fig. 6, when we increase K, the percentage difference between the approximated and the
real mean of project completion time, which is one of the important criteria for the accuracy of the solution,
in most cases, will be decreased. The exception is the case of K = 50. It seems that for decreasing PD.M in
this case, we should change the value of Dt. There is also no significant relation between PD.M and the
network size.
According to Fig. 7, when we increase K, the percentage difference between the approximated and the
real variance of project completion time, which is another important criteria for the accuracy of the solution, in almost all case, with one exception, will be decreased. There is also no significant relation between
PD.V and the network size.
6. Conclusion
In this paper, we developed a new multi-objective model for resource allocation in PERT networks with
Erlang distributions of activity durations. Our method can be applied for the management and control of
the projects with stochastic natures.
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
853
The multi-objective nonlinear optimization problem was cast into a discrete-time approximation, and
the latter model was solved using the surrogate worth trade-off method.
The major advantages of the SWT method are
1. The method requires the DM to consider only two objectives at a time while assessing the worth function, even if there are many objectives in the problem.
2. Both linear and nonlinear models can be exercised.
3. The method is applicable to both static and dynamic problems.
4. Preference information from multiple decision makers with varied opinions can be processed.
Taking into account the above advantages, the SWT method is an appropriate method to solve the
multi-objective resource allocation problem.
The limitation of this model is that the number of variables and constraints of the multi-objective nonlinear programming formulation (14) can grow exponentially with the network size. As the worst case
example, consider a complete transformed network with m nodes and m(m 1)/2 activities. The size of
the state space for this network is given by N(m) = Um Um1, where
Um ¼
m
X
2kðmkÞ
ð16Þ
k¼0
(refer to Kulkarni and Adlakha [19]).
Consequently, the number of constraints of the nonlinear model is equal to 2K(N(m) 1) +
2.5m(m 1), and the number of decision variables would be equal to K(N(m) 1) + m(m 1). Therefore
the number of constraints and variables of the nonlinear programming formulation (14) grows exponentially with m. In practice, the number of arcs of the PERT networks is generally much less than
m(m 1)/2. Even large sparse networks generally produce a reasonable size state space.
According to computational experiments, when K goes to infinity and Dt goes to zero, the percentage
differences between the approximated mean and variance, respectively, and the real ones approach zero.
Therefore, the approximated discrete project completion time distribution approaches to the continuous
distribution, because the first two moments of the approximated distribution and the analytical one are
matched with each other. In this case, the optimal solution of the discrete-time problem approaches to
the optimal solution of the original continuous-time problem, but the computation time goes to infinity.
Therefore, we should select the values of K and Dt, in the realistic sized problems, such that we can solve
each problem in a minimum level of accuracy with reasonable computation time.
According to the computational experiments, there is no significant relation between PD.M and the network size and also between PD.V and the network size, but the computation time grows with the size of the
PERT network.
Our model can be extended in the following directions:
1. The model can include the other objective functions like the minimization of total indirect costs of the
project, or the minimization of the a-fractile of the distribution function of project completion time with
the value of a pre-assigned by the DM.
2. The model can be extended to the general PERT networks, where general activity durations are allowed.
In general networks, we can approximate non-exponential distributions by mixture of sums of independent exponentials. For unimodal distributions, the sum of two independent exponentials is a reasonable
approximation. For multimodal distributions, one must use mixtures.
3. The allocated resource quantities can be considered as a function of time.
854
A. Azaron et al. / European Journal of Operational Research 172 (2006) 838–854
4. Another interactive method like the method of satisfactory goals, STEM or SEMOPS can be applied to
solve the multi-objective problem (14), see Hwang and Masud [16] for the details of the mentioned methods. In these methods, the progressive definition takes place through a DM-Analyst dialogue at each
iteration, and at each such dialogue, the DM is asked about some trade-off based upon the current solution in order to determine a new solution.
References
[1] A. Azaron, S. Fatemi Ghomi, Optimal control of service rates and arrivals in Jackson networks, European Journal of Operational
Research 147 (2003) 17–31.
[2] E. Berman, Resource allocation in a PERT network under continuous activity time–cost function, Management Science 10 (1964)
734–745.
[3] J. Burt, M. Garman, Conditional Monte Carlo: A simulation technique for stochastic network analysis, Management Science 18
(1971) 207–217.
[4] A. Charnes, W. Cooper, G. Thompson, Critical path analysis via chance constrained and stochastic programming, Operations
Research 12 (1964) 460–470.
[5] C. Clingen, A modification of FulkersonÕs algorithm for expected duration of a PERT project when activities have continuous d.f.,
Operations Research 12 (1964) 629–632.
[6] E. Demeulemeester, W. Herroelen, S. Elmaghraby, Optimal procedures for the discrete time–cost trade-off problem in project
networks, Research Report, Department of Applied Economics, Katholieke Universiteit Leuven, Leuven, Belgium, 1993.
[7] S. Elmaghraby, On the expected duration of PERT type networks, Management Science 13 (1967) 299–306.
[8] S. Elmaghraby, Resource allocation via dynamic programming in activity networks, European Journal of Operational Research
64 (1993) 199–245.
[9] J. Falk, J. Horowitz, Critical path problem with concave cost curves, Management Science 19 (1972) 446–455.
[10] S. Fatemi Ghomi, S. Hashemin, A new analytical algorithm and generation of Gaussian quadrature formula for stochastic
network, European Journal of Operational Research 114 (1999) 610–625.
[11] S. Fatemi Ghomi, M. Rabbani, A new structural mechanism for reducibility of stochastic PERT networks, European Journal of
Operational Research 145 (2003) 394–402.
[12] G. Fishman, Estimating critical path and arc probabilities in stochastic activity networks, Naval Research Logistic Quarterly 32
(1985) 249–261.
[13] D. Fulkerson, A network flow computation for project cost curves, Management Science 7 (1961) 167–178.
[14] D. Fulkerson, Expected critical path lengths in PERT networks, Operations Research 10 (1962) 808–817.
[15] M. Garman, More on conditioned sampling in the simulation of stochastic networks, Management Science 19 (1972) 90–95.
[16] C. Hwang, A. Masud, Multiple Objective Decision Making, Methods and Applications, Springer-Verlag, Berlin, 1979.
[17] S. Jordan, C. Ku, Access control to two multiserver loss queue in series, IEEE Transactions on Automatic Control 42 (1997)
1017–1023.
[18] J. Kelly, Critical path planning and scheduling: Mathematical basis, Operations Research 9 (1961) 296–320.
[19] V. Kulkarni, V. Adlakha, Markov and Markov-regenerative PERT networks, Operations Research 34 (1986) 769–781.
[20] L. Lamberson, R. Hocking, Optimum time compression in project scheduling, Management Science 16 (1970) 597–606.
[21] J. Martin, Distribution of the time through a directed acyclic network, Operations Research 13 (1965) 46–66.
[22] C. Perry, I. Creig, Estimating the mean and variance of subjective distributions in PERT and decision analysis, Management
Science 21 (1975) 1477–1480.
[23] P. Robillard, Expected completion time in PERT networks, Operations Research 24 (1976) 177–182.
[24] D. Robinson, A dynamic programming solution to cost–time trade-off for CPM, Management Science 22 (1965) 158–166.
[25] S. Sethi, G. Thompson, Optimal Control Theory, Martinus Nijhoff Publishing, Boston, 1981.
[26] C. Sigal, A. Pritsker, J. Solberg, The use of cutsets in Monte-Carlo analysis of stochastic networks, Mathematical Simulation 21
(1979) 376–384.
[27] T. Shioyama, Optimal control of a queueing network system with two types of customers, European Journal of Operational
Research 52 (1991) 367–372.
[28] L. Tavares, Optimal resource profiles for program scheduling, European Journal of Operational Research 29 (1987) 83–90.
[29] K. Tseng, M. Hsiao, Optimal control of arrivals to token ring networks with exhaustive service discipline, Operations Research 43
(1995) 89–101.