A multi-objective resource allocation problem in dynamic
PERT networks
Amir Azaron
a,*, Reza Tavakkoli-Moghaddam
baDepartment of Computer Science, Cork Constraint Computation Centre, University College Cork, Cork, Ireland bDepartment of Industrial Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran
Abstract
We develop a multi-objective model for the resource allocation problem in a dynamic PERT network, where the activity durations are exponentially distributed random variables and the new projects are generated according to a Poisson pro-cess. This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a Poisson process with the generation rate of new projects. It is assumed that the mean time spent in each service station 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. To evaluate the distribution function of total duration for any particular project, we apply a longest path technique in networks of queues. Then, the problem is formulated as a multi-objective optimal control problem that involves three conflicting objective functions. The objective functions are the project direct cost (to be min-imized), the mean of the project completion time (min) and the variance of the project completion time (min). Finally, the goal attainment method is applied to solve a discrete-time approximation of the original optimal control problem. We also computationally investigate the trade-off between accuracy and the computational time of the discrete-time approximation technique.
Ó2006 Elsevier Inc. All rights reserved.
Keywords: Multiple objective programming; Queueing; Optimal control; Project management
1. Introduction
Since the late 1950s, Critical Path Method (CPM) techniques have become widely recognized as valuable tools for the planning and scheduling of large projects. In a traditional CPM analysis, the major objective is to schedule a project assuming deterministic durations. However, project activities must be scheduled under available resources, such as crew sizes, equipment and materials. The activity duration can be looked upon as a function of resource availability. Moreover, different resource combinations have their own costs.
Ulti-0096-3003/$ - see front matter Ó2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.01.027
*
Corresponding author.
E-mail address:[email protected](A. Azaron).
Applied Mathematics and Computation xxx (2006) xxx–xxx
mately, the schedule needs to take account of the trade-off between project direct cost and project completion time. For example, using more productive equipment or hiring more workers may save time, but the project direct cost will increase.
In CPM networks, activity duration is viewed either as a function of cost or as a function of resources com-mitted to it. The well-known time–cost trade-off problem (TCTP) in CPM networks takes the former view. 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.
Studies on TCTP have been done using various kinds of cost functions such as linear[10,12], discrete [7], convex[14,5], and concave[9]. When the cost functions are arbitrary (still non-increasing), the dynamic pro-gramming (DP) approach was suggested by Robinson[15]and Elmaghraby[8]. Tavares[17]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 [18] studied this problem using optimal control theory and assumed 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 time is considered as a continuous variable. Azaron et al. [1] proposed an approximation technique to deal with time–cost trade-off in classical PERT net-works.
Recently, some researchers have adopted computational optimization techniques such as genetic algorithms to solve TCTP. Chau et al.[6]and Azaron et al.[2]proposed models using genetic algorithms and the Pareto front approach to solve construction time–cost trade-off problems.
Although project scheduling and management has been investigated by many researchers, one cannot find many models regarding dynamic project scheduling in the literature. Actually, as the classical definition of project indicates, it is a one-time job which consists of several activities. Therefore, the models representing the project scheduling, including the above models, are all static. In reality, during the implementation of a project some new projects are generated, in which the activities associated with successive projects contend for resources.
Dynamic PERT does not take into account the time–cost trade-off. Therefore, combining the aforemen-tioned concepts to develop a time–cost trade-off model under uncertainty and dynamic situations would be beneficial to scheduling engineers in forecasting a more realistic project completion time and cost.
In this paper, we develop a multi-objective model for the time–cost trade-off problem in a dynamic PERT network. In fact, in real world, there are many jobs with similar structure of activities sharing the same facil-ities. We consider a service center serving various projects with the same structure. Thus, although each one acts individually as a project represented as a classical PERT network, they cannot be analyzed independently since they share the same facilities. Like every other PERT project, the completion time is stochastic since the processing time of each activity is random.
Each dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a Poisson process with the generation rate of new projects. All projects have the same activities and the same sequences.
In our proposed method, first we transform each network of queues into a proper stochastic network. Then, the distribution function of the longest path in this stochastic network, which would be equal to the project completion time distribution in the original dynamic PERT network, is determined through solving a system of linear differential equations. By applying a continuous-time Markov process technique, this system of dif-ferential equations is constructed.
Then, we develop a multi-objective model for the time–cost trade-off problem in dynamic PERT networks. It is assumed that the activity durations are independent random variables with exponential distributions. It is also assumed that the amount of resource allocated to each activity is controllable, where the time spent in each service station (activity duration plus waiting time in queue) 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 project direct cost (to be minimized), the mean of the project completion time (min) and its variance (min).
Then, we apply the goal attainment technique, which is a variation of the goal programming technique, to solve this multi-objective problem.
It is proved that solving the resulting multi-objective optimal control problem using the standard optimal control tools is impossible. Therefore, we use a discrete-time approximation technique to solve it. We also computationally investigate the trade-off between accuracy and the computational time of the discrete-time approximation technique.
In Section2, we compute the project completion time distribution in dynamic PERT networks with expo-nentially distributed activity durations, analytically. Section3presents the multi-objective resource allocation formulation. Section 4 presents the computational experiments, and finally we draw the conclusion of the paper in Section5.
2. Project completion time distribution in dynamic PERT networks
In this section, we present an analytical method to compute the distribution function of the project com-pletion time in a dynamic PERT network. A project is represented as an Activity-on-Node (AoN) graph, where an activity begins as soon as all its predecessor activities have finished. It is also assumed that the new projects, including all their activities, are generated according to a Poisson process with the rate ofk. Each activity is processed at a dedicated service station settled in a node of the network. The activities associated with successive projects contend for resources on a FCFS basis.
This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a Poisson process with the rate ofk. Moreover, the arc lengths are all equal zero. The number of servers in each service station is assumed to be either one or infinity, while the service times (activity durations) are exponentially distributed. The main steps of our proposed method are as follows:
Step 1.Compute the density function of the time spent in each service station.
Step 1.1.If there is one server in the service station settled in theith node, then the distribution of time spent (activity duration plus waiting time in queue) in this M/M/1 queueing system is
wiðtÞ ¼ ðlikÞeðlikÞtt>0; ð1Þ
wherekandliare the generation rate of new projects and the service rate of this queueing system, respectively.
Therefore, the distribution of time spent in this service station would be exponential with parameter (lik). Step 1.2.If there are infinite servers in the service station settled in theith node, then the time spent in thisM/
M/1queueing system would be exponentially distributed with parameterli, because there is no queue.
Step 2.Transform the dynamic PERT network into an equivalent classical PERT network represented as an
Activity-on-Arc (AoA) graph.
Step 2.1.Replace each node with a stochastic arc (activity) whose length is equal to the time spent in the
par-ticular service station.Let us explain how to replace nodekin the network of queues with a stochastic activity. Assume thatb1,b2,. . .,bnare the incoming arcs to this node andd1,d2,. . .,dmare the outgoing arcs from it.
Then, we substitute this node by activity (k0,k00), whose length is equal to the time spent in the corresponding
queueing system. Furthermore, all arcsbifori= 1,. . .,nend up withk0while all arcsdjforj= 1,. . .,mstart
from nodek00. The indicated process is opposite of the absorption an edgeein a graphGin graph theory (G.e), see Azaron and Modarres [3]for more details.
Step 2.2.Eliminate all arcs with zero length.
Step 3.Obtain the distribution function of the longest path in the classical PERT network with exponentially
distributed activity durations obtained in Step 2.2, using the method of Kulkarni and Adlakha [13].
LetG= (V,A) be the transformed classical PERT network with set of nodesV= {v1,v2,. . .,vm} and set of
activitiesA= {a1,a2,. . .,an}. The source and sink nodes are denoted bysandy, respectively. Length of arc
a2Ais an exponentially distributed random variable with parameter ca. For a2A, leta(a) be the starting
Definition 1. LetI(v) andO(v) be the sets of arcs ending and starting at nodev, respectively, which are defined as follows:
IðvÞ ¼fa2A:bðaÞ ¼vg; ðv2VÞ; ð2Þ
OðvÞ ¼fa2A:aðaÞ ¼vg; ðv2VÞ. ð3Þ
Definition 2. IfXVsuch thats2Xandy2X ¼V X, then an (s,y) cut is defined as
ðX;XÞ ¼ fa2A:aðaÞ 2X;bðaÞ 2Xg. ð4Þ
An (s,y) cutðX;XÞis called a uniformly directed cut (UDC), ifðX;XÞis 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) becauseVis divided into two disjoint subsetsXandX, wheres2Xandy 2X. The other UDCs of this network are (2, 3), (1, 4, 6), (3, 4, 6) and (5, 6).
Definition 3. LetD=E[Fbe a uniformly directed cut (UDC) of a network. Then, it is called an admissible
2-partition, if for anya2F, we haveI(b(a)) 6 F.
To illustrate this definition, considerExample 1again. As mentioned, (3, 4, 6) is a UDC. This cut can be divided into two subsetsEandF. For example,E= {4} andF= {3, 6}. In this case, this cut is an admissible 2-partition, because I(b(3)) = {3, 4}6 F and alsoI(b(6)) = {5, 6} 6F. However, ifE= {6} and F= {3, 4}, then the cut is not an admissible 2-partition, becauseI(b(3)) = {3, 4}F= {3, 4}.
Definition 4. During the project execution and at timet, 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 timet, if it is being executed at timet.
(ii) Dormant: an activity is dormant at timet, if it has finished but there is at least one unfinished activity in
I(b(a)). If an activity is dormant at timet, then its successor activities in O(b(a)) cannot begin. (iii) Idle: an activity is idle at timet, if it is neither active nor dormant at time t.
The sets of active and dormant activities are denoted byY(t) andZ(t), respectively, andX(t) = (Y(t),Z(t)). ConsiderExample 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.Econtains all active whileFincludes all dormant activities.
1 3 2 4 s 1 2 3 5 6 y
Fig. 1. The example network.
Table 1
All admissible 2-partition cuts of the example network
1. (1, 2) 5. (1, 4*, 6) 9. (3*, 4, 6) 13. (3, 4*, 6*) 17. (/,/)
2. (2, 3) 6. (1, 4, 6*) 10. (3, 4*, 6) 14. (5, 6)
3. (2, 3*) 7. (1, 4*, 6*) 11. (3, 4, 6*) 15. (5*, 6)
Let S denote the set of all admissible 2-partition cuts of the network, and S ¼S[ fð/;/Þg. Note that
X(t) = (/,/) implies thatY(t) =/and Z(t) =/, i.e. all activities are idle at time t and hence the project is completed by time t. It is proven that {X(t),tP0} is a continuous-time Markov process with state space
S, refer to[13]for details.
As mentioned,EandFcontain active and dormant activities of a UDC, respectively. When activitya fin-ishes (with the rate ofka), and there is at least one unfinished activity in I(b(a)), it moves fromE to a new
dormant activities set, i.e. toF0. Furthermore, if by finishing this activity, its succeeding ones,O(b(a)), become
active, then this set will also be included in the newE0, while the elements ofI(b(a)), which one of them belongs
toEand the other ones belong toF, will be deleted from the particular sets. Thus, the elements of the infin-itesimal generator matrix Q= [q{(E,F), (E0,F0)}], (E,F) andðE0;F0Þ 2S, are calculated as follows:
qfðE;FÞ;ðE0;F0Þg ¼
ca if a2E; IðbðaÞÞ 6F[ fag; E0¼E fag; F0¼F [ fag; ðaÞ
ca if a2E; IðbðaÞÞ F [ fag; E0¼ ðE fagÞ [OðbðaÞÞ; F0¼FIðbðaÞÞ; ðbÞ P a2E ca if E0¼E; F0¼F; ðcÞ 0 otherwise. ðdÞ 8 > > > > > > > < > > > > > > > : ð5Þ
InExample 1, if we considerE={1, 2},F= (/),E0={2, 3} andF0= (/), thenE0= (E{1})[O(b(1)), and
thus from(5b),q{(E,F), (E0,F0)} =c
1.
{X(t),tP0} is a finite-state absorbing continuous-time Markov process and sinceq{(/,/), (/,/)} = 0, it is concluded that this state is an absorbing one and obviously the other states are transient. Furthermore, we number the states inSsuch thisQmatrix be an upper triangular one. We assume that the states are numbered 1;2;. . .;N¼ jSj. State 1 is the initial state, namelyX(t) = (O(s),/), and stateNis the absorbing state, namely
X(t) = (/,/).
LetTrepresent the length of the longest path in the network, or the project completion time in the PERT network. Clearly,T= min{t> 0 :X(t) =N/X(0) = 1}. Thus,Tis the time until {X(t),tP0} gets absorbed in the final state starting from state 1.
Chapman–Kolmogorov backward equations can be applied to computeF(t) =P{T6t}. If we define
PiðtÞ ¼PfXðtÞ ¼N=Xð0Þ ¼ig; i¼1;2;. . .;N ð6Þ
then,F(t) =P1(t).
The system of linear differential equations for the vectorP(t) = [P1(t),P2(t),. . .,PN(t)]T is given by P0ðtÞ ¼QPðtÞ;
Pð0Þ ¼ ½0;0;. . .;1T; ð7Þ
whereP0(t) represents the derivation of the state vectorP(t) andQis the infinitesimal generator matrix of the
stochastic process {X(t),tP0}. In Section3, the project completion time distribution is obtained, numerically.
3. Multi-objective resource allocation problem
In this section, we develop a multi-objective model to optimally control the resources allocated to the activ-ities in a dynamic PERT network, representing as a network of queues, where the mean time spent in each service station 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 project direct cost, by decreasing the amount of resource allocated to the activities. However, clearly it causes the mean completion time for any particular project to be increased, because these objectives are in conflict with each other. Consequently, an appropriate trade-off between the total direct costs and the mean project completion time is required. The variance of com-pletion time for any particular project should also be considered in the model, because when we only focus on the mean time, the resource quantities may be non-optimal if the project completion time substantially varies because of randomness.
Therefore, we have a multi-objective stochastic control problem. The objective functions are the project direct cost (to be minimized), the mean of project completion time (min) and the variance of project comple-tion time (min).
The direct cost of activitya2Ain the transformed classical PERT network is assumed to be a non-decreas-ing functionda(xa) of the amount of resourcexaallocated to it. Therefore, the project direct cost (PDC) would
be equal to PDC¼Pa2AdaðxaÞ.
The mean time spent in the service stationais assumed to be a non-increasing functionga(xa) of the amount
of resourcexaallocated to it. As explained in Section2, the mean time spent would be equal tol1
ak, if there is one server, and equal to 1
la, if there are infinite servers in the corresponding service station. LetUarepresent the amount of resource available to be allocated to the activity a, and Larepresent the minimum amount of
resource required to achieve the activitya.
In realityda(xa) andga(xa) can be estimated using linear regression. We can collect the sample paired data
ofda(xa) andga(xa) as the dependent variables, for different values ofxaas the independent variables, from the
previous similar activities or using the judgments of the experts in this area. Then, we can estimate the param-eters of the relevant linear regression model.
The mean and the variance of project completion time are given by
EðTÞ ¼ Z 1 0 ð1P1ðtÞÞdt; ð8Þ VarðTÞ ¼ Z 1 0 t2P01ðtÞdt Z 1 0 tP01ðtÞdt 2 ; ð9Þ whereP0
1ðtÞis the density function of project completion time.
The infinitesimal generator matrix,Q, is a function of the control vectorl= [la;a2A]T. Therefore, the
nonlinear dynamic model is
P0ðtÞ ¼QðlÞPðtÞ;
Pið0Þ ¼0 8i¼1;2;. . .;N1;
PNðtÞ ¼1.
ð10Þ
RepresentingBas the set of nodes includingM/M/1 service stations andCas the set of nodes includingM/
M/1service stations in the original dynamic PERT network (A= (B[C)), the relations(11)should be sat-isfied to exist the response in the steady-state.
la>k; a2B; la>0; a2C.
ð11Þ
We do not have such constraints in the mathematical programming. Therefore, we use the constraints(12)
instead of the above constraints in the final multi-objective problem
laPkþe; a2B;
laPe; a2C. ð12Þ
Accordingly, the appropriate multi-objective optimal control problem is Min f1ðx;lÞ ¼X a2A daðxaÞ Min f2ðx;lÞ ¼ Z 1 0 ð1P1ðtÞÞdt Min f3ðx;lÞ ¼ Z 1 0 t2P01ðtÞdt Z 1 0 tP01ðtÞdt 2
s.t. P0ðtÞ ¼QðlÞPðtÞ; Pið0Þ ¼0; 8i¼1;2;. . .;N1; PNðtÞ ¼1; gaðxaÞ ¼ 1 lak ; a2B; gaðxaÞ ¼ 1 la ; a2C; laPkþe; a2B; laPe; a2C; xa6Ua; a2A; xaPLa; a2A. ð13Þ
A possible approach to solving(13)to optimality is to use the Maximum Principle (see[16]for details). For simplicity, consider solving the problem with only one of the objective functions,f2ðx;lÞ ¼R01ð1P1ðtÞÞdt.
Clearlyxa¼ga1ðla1kÞfora2Bandxa¼g
1
a ðl1aÞfora2C. Therefore, we can considerlas the unique con-trol vector of the problem, and ignore the role of x= [xa;a2A]Tas the other independent decision vector.
Consider Kas the set of allowable controls consisting of all constraints except the constraints representing the dynamic model (l2K), and N-vector k(t) as the adjoint vector function. Then, Hamiltonian function would be
HðkðtÞ;PðtÞ;lÞ ¼kðtÞTQðlÞPðtÞ þ1P1ðtÞ. ð14Þ
Now, we write the adjoint equations and terminal conditions, which are
k0ðtÞT¼kðtÞTQðlÞ þ ½1;0;. . .;0;
kðTÞT ¼0; T ! 1. ð15Þ
If we could computek(t) from(15), then we would be able to minimize the Hamiltonian function subject to
l2Kin order to get the optimal controll*, and solve the problem optimally. Unfortunately, the adjoint Eq. (15)are dependent on the unknown control vector (l) and therefore they cannot be solved directly.
If we could also minimize the Hamiltonian function(14), subject tol2K, for an optimal control function in closed form as l*=f(P*(t),k*(t)), then we would be able to substitute this into the state equations,
P0(t) =Q(l)ÆP(t),P(0) = [0, 0,. . ., 1]T
, and adjoint Eq. (15)to get a set of differential equations, which is a two-point boundary value problem. Unfortunately, we cannot obtain l* by differentiating H with respect
to l, because the minimum of H occurs on the boundary ofK, and consequentlyl*cannot be obtained in
a closed form.
According to these points, it is impossible to solve the optimal control problem(13), optimally, even in the restricted case of a single objective problem. Relatively few optimal control problems can be solved optimally. Therefore, we do the discretization of time and convert the optimal control problem (13)into an equivalent nonlinear programming one. In other words, we transform the differential equations to the equivalent differ-ence equations as well as transform the integral terms into equivalent summation terms. To follow this approach, the time interval is divided into Kequal portions with the length ofDt. IfDtis sufficiently small, it can be assumed that P(t) varies only in times 0,Dt,. . ., (K1)Dt. Since each Pi(k), for i= 1, 2,. . .,N1,
k= 1, 2,. . .,K, is a distribution function, then the constraints (16)should also be considered in the final dis-crete-time problem (refer to[4]for more details about the proposed technique)
PiðkÞ61 i¼1;2;. . .;N1; k¼1;2;. . .;K. ð16Þ
Theoretically, whenKapproaches to infinity andDtapproaches to zero, the optimal results of the original problem will be obtained, but in this case the computational time also approaches to infinity, which is not practical in reality. Practically, we should select a finite value for K. Moreover, in an accurate solution,
3.1. Goal attainment method
This method requires setting up a goal and weight,bjandcj(cjP0) forj= 1, 2, 3, for the three indicated
objective functions. Thecjrelates the relative under-attainment of thebj. For under-attainment of the goals, a
smaller cj is associated with the more important objectives. cj, j= 1, 2, 3, are generally normalized so that P3
j¼1cj¼1. The appropriate goal attainment formulation to obtain x*is
Min z s.t. X a2A daðxaÞ c1z6b1; Z 1 0 ð1P1ðtÞÞdtc2z6b2; Z 1 0 t2P01ðtÞdt Z 1 0 tP01ðtÞdt 2 c3z6b3; Pðkþ1Þ ¼PðkÞ þQðlÞPðkÞDt; k¼0;1;. . .;K1; Pið0Þ ¼0; i¼1;2;. . .;N1; PNðkÞ ¼1; k¼0;1;. . .;K; PiðkÞ61; i¼1;2;. . .;N1; k¼1;2;. . .;K; gaðxaÞ ¼ 1 lak ; a2B; gaðxaÞ ¼ 1 la; a2C; laPkþe; a2B; laPe; a2C; xa6Ua; a2A; xaPLa; a2A; zP0. ð17Þ
Lemma 1. If x* is Pareto-optimal, then there exists a c, b pair such that x* is an optimal solution to the
optimization problem(17).
The optimal solution using this formulation is fairly sensitive tobandc. Depending upon the values forb, it is possible thatcdoes not appreciably influence the optimal solution. Instead, the optimal solution can be determined by the nearest Pareto-optimal solution fromb. This might require thatcbe varied parametrically to generate a set of Pareto-optimal solutions.
Solving the goal attainment formulation(17)leads to the approximated objective function value (zApprox.).
For computing the exact value ofz(zExact), in order to obtain the accuracy of the discrete-time approximation
technique, we should do the following approach. After solving the optimization problem(17)and obtaining
l*, we computeP
1(t) from Eq.(7). Then, the exact mean and the variance of the project completion time are
computed from(8) and (9), respectively. Finally,zExactis given by zExact¼Max PDCb1 c1 ;EðTÞ b2 c2 ;VarðTÞ b3 c3 . ð18Þ 4. Numerical example
In this section, we solve a numerical example to investigate the performance of the proposed method for the resource allocation in the dynamic PERT network, which is represented as the network of queues depicted in
Fig. 2. The activity durations (service times) are exponentially distributed random variables. Moreover, the new projects, including all their activities, are generated according to a Poisson process with the rate of
k= 10 per year. The objective is to obtain the optimal allocated resource quantities using the goal attainment technique. The other assumptions are as follows:
1. There is no service station in node 0. It means that there is no predecessor activity for the activities 1 and 2 of each project.
2. There is one server in the service stations settled in the nodes 1, 2, 3, 6 and 7. 3. There are infinite servers in the service stations settled in the nodes 4 and 5.
The transformed classical PERT network is depicted inFig. 3. The stochastic process {X(t), tP0} related to the longest path analysis of this classical PERT network has 14 states in the order of S ¼ fð1;2Þ;ð1;3Þ;
ð1;5Þ;ð1;5Þ;ð2;4Þ;ð2;4Þ;ð3;4Þ;ð3;4Þ;ð4;5Þ;ð4;5Þ;ð4;5Þ;ð6Þ;ð7Þ;ð/;/Þg. Table 2 shows Q(l) (diagonal elements are equal to minus sum of the other elements at the same row).
Table 3 shows the characteristics of the activities in the transformed classical PERT network. The cost unit is in thousand dollars and the time is in years. The structures of functions (different linear and nonlin-ear forms) are selected so as to represent a wide variety of problems encountered in the resource allocation problem in PERT networks. In real cases, these functions can be estimated using linear or nonlinear regression.
λ 0 1 4
3 5
2
6 7
Fig. 2. The dynamic PERT network.
Table 2 Matrix Q(l) State 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 l210 0 0 l110 0 0 0 0 0 0 0 0 0 2 0 l310 0 0 0 l110 0 0 0 0 0 0 0 3 0 0 l5 0 0 0 0 l110 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 l110 0 0 0 5 0 0 0 0 l4 l210 0 0 0 0 0 0 0 6 0 0 0 0 0 0 l210 0 0 0 0 0 0 7 0 0 0 0 0 0 l4 l310 0 0 0 0 0 8 0 0 0 0 0 0 0 0 l310 0 0 0 0 9 0 0 0 0 0 0 0 0 l4 l5 0 0 0 10 0 0 0 0 0 0 0 0 0 0 l5 0 0 11 0 0 0 0 0 0 0 0 0 0 l4 0 0 12 0 0 0 0 0 0 0 0 0 0 0 l610 0 13 0 0 0 0 0 0 0 0 0 0 0 0 l710 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 6 7 2 5 1 4 2 3 3 5 s y
Then, considering the goal vectorb: (b1= 50, b2= 2,b3= 0.75), three factorial experiments according to
the following three sets ofc:
c1:ðc1¼0:909;c2¼0:0455;c3¼0:0455Þ; c2:ðc1¼0:7693;c2¼0:0769;c3¼0:1538Þ; c3:ðc1¼0:8929;c2¼0:0178;c3¼0:0893Þ
are designed to obtain a set of Pareto-optimal solutions in each case. For example, using the first set ofcleads to the following consideration: one year deviation from the mean project completion time is as important as its variance and 20 times as important as one thousand dollars deviation from the project direct cost, respectively. To investigate the trade-off between the accuracy (in terms ofK) and computational time, we consider the following levels ofK(K= 20,K= 50,K= 500) in our computational experiments. Moreover,P1(K) should be
greater than 0.99. If a solution does not have this property, the value ofDtis increased in order to access to this level of accuracy. Thus, the following combinations of K and Dt are considered: (K= 20, Dt= 0.35), (K= 50, Dt= 0.14), (K= 500,Dt= 0.014). The value ofeis also considered equal to 0.01 in all experiments. Finally, we use LINGO 6 on a PC Pentium IV 2.1 GHz to solve the problem and to compute the approx-imated objective function values (zApprox.) and the related computational times for the three sets of c. The
exact objective function values (zExact) are also computed from Eq.(18).
For example, the optimal allocated resource quantities, considering the first set of c(c1), K= 500 and
Dt= 0.014, are shown inTable 4.Table 5shows the corresponding values of PDC,E(T), Var(T), as the three indicated criteria,P1(K= 500),zand the related computational time in seconds (CT).
Fig. 4shows the approximated and the exact objective functions for the three indicated sets ofc, consid-eringK= 20, K= 50 andK= 500.Fig. 5shows the related computational times.
According toFig. 4, the approximated and the exact objective function values are decreased or the accuracy of the discrete-time approximation method is increased, when we increase K. Moreover, the differences between zApprox. and zExact are decreased, when K is increased. As it is seen in Fig. 4, the approximated
and the exactzare almost the same, in most cases. The reason is that PDC, which does not change in the exact solution, is the most effective criterion among the three indicated criteria to computez, in these experiments. According toFig. 5, the computational time grows with K. Moreover, the computational time is clearly dependent on the network size, because the state space grows with the network size.
Table 3
Characteristics of the activities
a da(xa) ga(xa) La Ua 1 3x2 1þ2 0.70.1x1 1 5 2 2x2+ 1 1.50.2x2 1 6 3 x3+ 2 10.1x3 1 9 4 x4 1.50.3x4 1 4 5 3x5+ 4 0.90.1x5 1 5 6 x6+ 3 1.10.1x6 1 6 7 4x7+ 1 10.2x7 1 3 Table 4
Optimal allocated resource quantities, consideringc1 andK= 500
x1 x2 x3 x4 x5 x6 x7
1 6 8.948 4 1 6 2.422
Table 5
Optimal criteria, consideringc1 andK= 500
PDC E(T) [Approx.] E(T) [Exact] Var(T) [Approx.] Var(T) [Exact] zApprox. zExact P1(K) CT
5. Conclusion
In this paper, we developed a new multi-objective model for the time–cost trade-off problem in a dynamic PERT network with exponentially distributed activity durations. The new projects are generated according to a renewal process. The projects share the same facilities and have to wait for processing in a station if the same activity of previous project is not finished.
In the proposed methodology, the dynamic PERT network, representing as a network of queues, was trans-formed into an equivalent classical PERT network, in that the project completion time distribution could be computed analytically. Then, for obtaining the optimal resources allocated to the activities, we developed a goal attainment model with three conflicting objectives, minimization of the project direct cost, minimization of the mean of project completion time and minimization of the variance of project completion time. Then, in order to solve the resulting optimal control problem, it was transformed into a nonlinear programming.
According to the numerical example, when Kapproaches to infinity andDt goes to zero, the differences between the approximated and the exact objective function values approach zero. In this case, the optimal solution of the discrete-time problem approaches to the optimal solution of the original continuous-time prob-lem, but the computation time goes to infinity. Therefore, we should select the proper values forKandDt, in the realistic sized problems, so that we can solve the problem in an acceptable level of accuracy with reason-able computational time.
The limitation of this model is that the state space can grow exponentially with the network size. As the worst case example, for a complete transformed classical PERT network withnnodes andnðn21Þarcs, the size of the state space is given byN(n) =UnUn1, where
Un¼ Xn k¼0 2kðnkÞ ð19Þ (refer to[13]). 0 5 10 15 20 25 30 35 40 45 c1(K=20) c1(K=50) c1(K=500) c2(K=20) c2(K=50) c2(K=500) c3(K=20) c3(K=50) c3(K=500) z z(Approx.) z(Exact)
Fig. 4. Objective function values.
0 5000 10000 15000 20000 25000 c1(K=20) c1(K=50) c1(K=500) c2(K=20) c2(K=50) c2(K=500) c3(K=20) c3(K=50) c3(K=500)
Computational Time (Sec.)
Computational Time
In practice, the number of activities in PERT networks is generally much less thannðn21Þ, and it should also be noted that for large networks any alternate method of producing reasonably accurate answers will be pro-hibitively expensive.
The proposed model can be extended to the general dynamic PERT networks, where general activity dura-tions are allowed. In general networks, it is possible to approximate non-exponential distribudura-tions by mixture of sums of independent exponentials. For unimodal distributions, the sum of two independent exponentials is a reasonable approximation. For multi-modal distributions, one must use mixtures.
Another multi-objective technique like goal programming, SWT or STEM can also be applied to solve the multi-objective problem(13), refer to Hwang and Masud[11] for the details of the mentioned methods.
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