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
baDepartment of Artificial Complex Systems Engineering, Graduate School of Engineering, Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima, Hiroshima 739-8527, Japan
bDepartment 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 prob-ability 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.
Ó2005 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 manage-ment 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 there-fore, PERT was proposed to determine the distribution of the total duration,T.
0377-2217/$ - see front matter Ó2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.11.018
* Corresponding author. Tel.: +81 824 24 7701; fax: +81 824 22 7195. E-mail address:katagiri@msl.sys.hiroshima-u.ac.jp(H. Katagiri).
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 computeF(T). Fatemi Ghomi and Rabbani[11] presented a structural mechanism, which changes the structure of network to a series–parallel network, to estimateF(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 estimateF(t). Several authors have used conditional sampling to achieve var-iance reduction, see Burt and Garman[3], Garman[15], and Sigal et al.[26]. Fishman[12]achieved fur-ther 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 timetis 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 sto-chastic 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 two-station network of queues for the objective of maximum system throughput under a system time-delay con-straint 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
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 vari-able. The direct cost of each activity is also assumed to be a non-decreasing function of the amount of re-source 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 quanti-ties. We use the surrogate worth trade-off method, see Hwang and Masud[16], which is an interactive ap-proach 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 Kulk-arni and Adlakha[19]to obtain the project completion time distribution in PERT networks with Erlang dis-tributed 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 inde-pendent random variables with exponential or Erlang distributions. To do that, we extend the one of Kulk-arni 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 activitya2A 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 withna= 1,
and consequently a random variable with Erlang density function and the parameters (ka,na) can be
decom-posed tonaseries of exponentially random variables with the parameterka.
First, we transform the original PERT network into a new one, in which all activity durations have expo-nential distributions. To do that, we substitute each Erlang activity with the parameters (ka,na) withna
ser-ies of exponential activitser-ies with the parameterka. Now, LetG0= (V0,A0) be the transformed network, in
whichV0andA0represent the sets of nodes and arcs of this transformed network, respectively. The source and sink nodes are denoted bysandt, respectively. Fora2A0, leta(a) be the starting node of arca, and b(a) be the ending node of arca.
Definition 1. Let I(v) andO(v) be the sets of arcs ending and starting at nodev, respectively, which are defined as follows:
OðvÞ ¼ fa2A0:aðaÞ ¼vg ðv2V0Þ: ð2Þ
Definition 2. IfXV0, such thats2Xandt2X ¼V0X, then an (s,t) cut is defined as:
ðX;XÞ ¼ fa2A0:aðaÞ 2X;bðaÞ 2Xg: ð3Þ
An (s,t) cutðX;XÞis called an 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), becauseV0is divided into two disjoint subsetsXandX,
wheres2Xandt2X. 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 admis-sible 2-partition, ifI(b(a))XF, fora2F.
To illustrate this definition, consider Example 1 again. 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, the cut is an admissible 2-partition, because I(b(3)) = {3, 4}XF and also I(b(6)) = {5, 6}XF. However, if E= {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 inI(b(a)). If an activity is dormant at timet, then its successor activities inO(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 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.Econtains all active whileFincludes all dormant activities.
Now, letSdenote the set of all admissible 2-partition cuts of the network, andS ¼S[ fð/;/Þg. Note
thatX(t) = (/,/) implies thatY(t) =/andZ(t) =/, i.e. all activities are idle at timetand hence the
pro-ject is completed by timet. It can be proved that {X(t),tP0} is a continuous-time Markov process with state spaceS, see Kulkarni and Adlakha[19]for the details of proof.
1 3 5 2 4 b t d c s 6 Fig. 1. Example network.
As mentioned before,EandFcontain active and dormant activities of a UDC, respectively. When activ-ityafinishes (with the rate ofka), and there is at least one unfinished activity inI(b(a)), it moves fromEto 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 these sets. Thus, the elements of the infinitesimal generator matrixQ= [q{(E,F), (E0,F0)}], (E,F) andðE0;F0Þ 2S, are calculated as follows:
qfðE;FÞ;ðE0;F0Þg ¼ ka if a2E;IðbðaÞÞ 6F[ fag;E0¼E fag; F0¼F[ fag; ðaÞ ka if a2E;IðbðaÞÞ F [ fag;E0¼ ðE fagÞ [OðbðaÞÞ; F0¼FIðbðaÞÞ; ðbÞ P a2E ka if E0¼E;F0¼F; ðcÞ 0 otherwise: ðdÞ 8 > > > > > > > > > > > > < > > > > > > > > > > > > : ð4Þ
In Example 1, if we considerE= {1, 2},F= (/),E0= {2, 3} andF0= (/), thenE0= (E{1})[O(b(1)), and thus from (4b), q{(E,F), (E0,F0)} =k
1.
{X(t),tP0} is a finite-state absorbing continuous-time Markov process. Sinceq{(/,/), (/,/)} = 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 ¼jSj. State 1 is the initial state, namely (O(s),/); and stateNis the absorbing state, namely (/,/).
Let Trepresent 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,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; ð5Þ
thenF(t) =P1(t).
The system of differential equations for the vector P(t) = [P1(t),P2(t),. . .,PN(t)]Tis given by
P0ðtÞ ¼QPðtÞ;
Pð0Þ ¼ ½0;0;. . .;1T; 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 ofQ, the differential equations (6)can be easily solved.
Table 1
All admissible 2-partition cuts of the example network
1. (1, 2) 10. (3, 4, 6) 2. (2, 3) 11. (3, 4, 6) 3. (2, 3) 12. (3, 4, 6) 4. (1, 4, 6) 13. (3, 4, 6) 5. (1, 4, 6) 14. (5, 6) 6. (1, 4, 6) 15. (5, 6) 7. (1, 4, 6) 16. (5, 6) 8. (3, 4, 6) 17. (/,/) 9. (3, 4, 6)
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 comple-tion time to be increased, because these objectives have the confliccomple-tion 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 func-tions 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 objec-tive functions associated with the minimization of project completion time, which has a random nature.
The direct cost of activitya2Ais assumed to be the non-decreasing functionda(xa) of the amount of
resourcexaallocated to it. Therefore, the total direct costs of the project would be equal to
P
a2AdaðxaÞ. The mean duration of activitya2A, which is equal tona/ka(na= 1 for exponential andna> 1 for
Er-lang distribution), is assumed to be the non-increasing functionga(xa) of the amount of resourcexa
allo-cated to it. LetUa 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 activitya. It is also assumed that the mean duration of activitya 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 ofda(xa) andga(xa) as the dependent variables, for the different values ofxaas the unique
inde-pendent 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 EðTÞ ¼ Z 1 0 tP01ðtÞdt¼ Z 1 0 ð1P1ðtÞÞdt; ð7Þ
whereP01ðtÞis the density function of project completion time. The variance of project completion time is
VarðTÞ ¼
Z 1
0
t2P01ðtÞdt ½EðTÞ2: ð8Þ
The probability that the project completion time does not exceed a given thresholduis
PðT 6uÞ ¼P1ðuÞ: ð9Þ
Taking into account the above assumptions, the infinitesimal generator matrixQis a function of the con-trol vectork= [k1,k2,. . .,kn]T. Therefore, the dynamic model is
P0ðtÞ ¼QðkÞPðtÞ;
Pið0Þ ¼0; i¼1;2;. . .;N1; PNðtÞ ¼1:
Theorem 1. The appropriate multi-objective optimal control problem is Min f1ðX;kÞ ¼X a2A daðxaÞ; ð11aÞ Min f2ðX;kÞ ¼ Z 1 0 tP01ðtÞdt; ð11bÞ Min f3ðX;kÞ ¼ Z 1 0 t2P01ðtÞdt Z 1 0 tP01ðtÞdt 2 ; ð11cÞ Max f4ðX;kÞ ¼P1ðuÞ; ð11dÞ s:t: P0ðtÞ ¼QðkÞPðtÞ; ð11eÞ Pið0Þ ¼0; i¼1;2;. . .;N1; ð11fÞ PNðtÞ ¼1; ð11gÞ ka6na=ca; a2A; ð11hÞ ka6na=gaðxaÞ; a2A; ð11iÞ xa6Ua; a2A; ð11jÞ xaPLa; a2A; ð11kÞ kaP0; a2A: 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 a2A andga(xa) exceedsca, thenkagets its maximum value and becomes equal tona/ga(xa), or the mean duration of activityabecomes equal toga(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 con-straints(11h), ifga(xa) does not exceedca, then because of the same reason,kagets its maximum value and becomes equal tona/ca, or the mean duration of activityabecomes equal toca. 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 the details. For simplicity, we consider only one objective function, for examplef2ðX;kÞ ¼R01ð1P1ðtÞÞdt, in the model.
Considering k as the control vector of this problem, which is also dependent on the vector x= [x1,x2,. . .,xn]T,Kas the set of allowable controls, which consists of the constraints from (11h)–(11l)
(k2K), andn-vectorl(t) as the adjoint vector function, the Hamiltonian function would be
HðlðtÞ;PðtÞ;kÞ ¼lðtÞTQðkÞPðtÞ þ1P1ðtÞ: ð12Þ
Then, we write the adjoint equations and terminal conditions, which are
l0ðtÞT ¼lðtÞTQðkÞ þ ½1;0;. . .;0;
lðTÞT ¼0; T ! 1: ð13Þ
If we could compute l(t) from(13), we could minimize the Hamiltonian function subject to k2Kin order to get the optimal controlk, and solve the problem optimally. Unfortunately, the adjoint equations
If we could also minimize the Hamiltonian function(12), subject tok2K, for an optimal control func-tion in closed form as k=f(P(t),l(t)), then we could substitute this into the state equations, P0(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 obtainkby differentiatingHrespect to k, because the minimum of Hoccurs 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), opti-mally, 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 con-vert 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 intoKequal portions with the length ofDt. IfDtis sufficiently small, it can be assumed thatP(t) varies only in times 0,Dt,. . ., (K1)Dt.
Corollary 1. Considering bu/Dtc as the integer part of u/Dt, the appropriate multi-objective nonlinear optimization problem is Min f1ðX;kÞ ¼X a2A daðxaÞ; ð14aÞ Min f2ðX;kÞ ¼X K1 k¼0 kDtðP1ðkþ1Þ P1ðkÞÞ; ð14bÞ Min f3ðX;kÞ ¼X K1 k¼0 ðkDtÞ2ðP1ðkþ1Þ P1ðkÞÞ X K1 k¼0 kDtðP1ðkþ1Þ P1ðkÞÞ " #2 ; ð14cÞ Max f4ðX;kÞ ¼P1bu=Dtc; ð14dÞ s:t: Pðkþ1Þ ¼PðkÞ þQðkÞPðkÞDt; k¼0;1;. . .;K1; ð14eÞ Pið0Þ ¼0; i¼1;2;. . .;N1; ð14fÞ PNðkÞ ¼1; k¼0;1;. . .;K; ð14gÞ ka6na=ca; a2A; ð14hÞ ka6na=gaðxaÞ; a2A; ð14iÞ xa6Ua; a2A; ð14jÞ xaPLa; a2A; ð14kÞ PiðkÞ61; i¼1;2;. . .;N1; k¼1;2;. . .;K; ð14lÞ kaP0; a2A: ð14mÞ
Proof. The integral terms in(11b) and (11c), are easily transformed into the equivalent summation terms
(14b) and (14c), by consideringP(k) instead ofP(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,. . .,N1, k= 1, 2,. . .,K is a distribution function, then the constraints (14l) should also be included in this nonlinear programming problem. h
If we consider the initial and terminal state conditions forP(k) implicitly, and substitute eachPi(0) and
PN(k) with zero and one, respectively, the multi-objective nonlinear optimization problem(14)would have
2K(N1) + 5nconstraints including the constraints(14m), andK(N1) + 2ndecision variables. For estimating the length of the time interval in this problem, first we assume that the estimated amount of resource allocated to the activityaisDa= 0.5(La+Ua). In this case, the mean duration of this activity
would be equal toga(Da) and its variance is equal toga(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. In this case, a good estimation for the length of the time interval would be equal to T^ ¼KDt¼Mþ2pffiffiffiffiV, because in most distributions, for example normal distribution, an appropriate upper bound, in which the probability that the random var-iable 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 considerK= 10 andDt¼T^=10. A feasible solution should
posses this property thatP1(K)P1e, in whicheis 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 P
1(t) by solving the system of
differ-ential 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 be-tween 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 discrete-time 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 replaceKwithK+ 10 and each new value forDtshould be se-lected, such that the length of the time interval remains unchanged, in order to investigate the trade-off be-tween optimality and computation time.
4. Surrogate worth trade-off method
For solving the multi-objective nonlinear optimization problem(14), we use the surrogate worth trade-off method, which is an interactive approach with explicit trade-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:
Min f1ðX;kÞ
s:t: fjðX;kÞ6ej; j¼2;3;
f4ðX;kÞPe4;
X;k2S ðfeasible region of problem (14)Þ:
ð15Þ
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 functionwj, or the DMÕs assessment of how
much (from10 to 10) he prefers tradinggjmarginal units of the first objective for one marginal
unit of thejth objectivefj(X,k), given the other objectives remaining at their current values.
Step 4. Isolate the indifference solutions. The solutions, which havewj= 0 for allj, are said to be
indiffer-ence solutions. If there exists no indifferindiffer-ence solution, develop approximate relations for all worth functionswj¼wj_ ðfj; j¼2;3;4Þ, by multiple regressions.
5. Computational experiments
For showing the numerical stability of the theoretical developments of the paper, we solve three numer-ical examples, and investigate the trade-off between the accuracy and the computation time in each prob-lem. Case I is depicted inFig. 2.
Table 2shows 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 inFig. 3. In this network, the durations of activities 3, 4 are exponentially distributed with the parameterk3. The stochastic
process {X(t),tP0} has seven states in the order of S¼ fð1Þ;ð2;3Þ;ð2;3Þ;ð2;4Þ;ð2;4Þ;ð2;4Þ;ð/;/Þg.
Table 3shows matrixQ(k).
The ideal solutions are f1¼11 at (x1= 1,x2= 1,x3= 1), f2¼10:116 at (x1= 4,x2= 6,x3= 6),
f3¼38:487 at (x1= 4,x2= 6,x3= 6) and f4¼0:999 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 asT^¼50. Therefore, we considerK= 10 andDt= 5, at the beginning. Then,ej,j= 2, 3, 4, are varied parametrically to obtain several non-dominated
1 2 3 t b s Fig. 2. Case I. Table 2
Characteristics of the activities in Case I
a Distribution Parameters da(xa) ga(xa) ca La Ua
1 Exponential k1 3x1+ 2 245x1 5 1 4
2 Exponential k2 2x2+ 1 203x2 4 1 6
3 Erlang (k3,n3= 2) x2
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 ofKand Dtas: (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 com-pute 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)P0.95).
Now, according to any pair ofKandDt, the DM can select any of the above non-dominated solutions, if the surrogate worth functionswjfor allj, assessed by the DM, for that non-dominated solution are
consid-ered equal to zero. Otherwise, we can obtain an indifference solution, approximately, using multiple regres-sions. For example, if the surrogate worth functions for the non-dominated solution corresponding with e2= 24, e3= 55,e4= 0.95, K= 100 andDt= 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 x¼ ½x
1;x2;x3 T
¼ ½3:8;5:23;2:975T. 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 var-iance 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 inFig. 1. This network has six activities with exponentially distributed activity dura-tions. Table 5shows the characteristics of the activities. The threshold value is equal to 5.
The length of the time interval is approximated asT^ ¼6.Table 6shows the trade-off between the accu-racy 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 ofKandDt.
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 inFig. 4. This network has 10 activities with exponentially distributed activity dura-tions. In this case, the stochastic process {X(t),tP0} has 25 states.Table 7shows the characteristics of the activities. The threshold value is equal to 8.
1 2 3 4 t c b s
Fig. 3. Transformed network of Case I.
Table 3
MatrixQ(k) corresponding to Case I
State 1 2 3 4 5 6 7 1 k1 k1 0 0 0 0 0 2 0 (k2+k3) k2 k3 0 0 0 3 0 0 k3 0 k3 0 0 4 0 0 0 (k2+k3) k2 k3 0 5 0 0 0 0 k3 0 k3 6 0 0 0 0 0 k2 k2 7 0 0 0 0 0 0 0
Table 4
A set of non-dominated solutions in Case I
e2 e3 e4 f1 f2 f3 f4 PD.M (%) PD.V (%) K Dt C.T 25 70 0.9 19.809 25 70 0.929 7.17 70.01 10 5 3 25 70 0.9 23.876 21.262 70 0.963 2.81 48.23 20 2.5 17 25 70 0.9 25.871 19.754 70 0.976 1.48 36.25 30 1.7 1:08 25 70 0.9 27.23 18.842 70 0.978 1.09 29.68 40 1.3 1:23 25 70 0.9 28.072 18.243 70 0.978 1.42 25.45 50 1 4:37 25 70 0.9 30.065 17.174 70 0.981 1.27 16.29 100 0.5 19:52 30 100 0.9 18.553 25.557 88.055 0.9 10.58 67.98 10 5 11 30 100 0.9 20.931 22.739 100 0.91 9.67 51.92 20 2.5 19 30 100 0.9 22.157 21.959 100 0.928 7.28 42.50 30 1.7 49 30 100 0.9 22.872 21.606 100 0.931 5.67 35.82 40 1.3 1:25 30 100 0.9 23.116 21.131 100 0.928 6.69 33.44 50 1 1:50 30 100 0.9 23.949 20.527 100 0.934 5.79 25.29 100 0.5 13:11 20 85 0.9 23.511 20 26.766 0.987 10.05 81.16 10 5 4 20 85 0.9 25.663 20 57.501 0.98 1.06 48.56 20 2.5 24 20 85 0.9 25.509 20 72.107 0.974 1.69 36.27 30 1.7 44 20 85 0.9 25.456 20 79.403 0.966 1.92 30.13 40 1.3 1:19 20 85 0.9 25.16 20 84.337 0.958 3.15 27.72 50 1 1:40 20 85 0.9 26.347 19.112 85 0.962 2.88 19.67 100 0.5 11:48 35 150 0.75 15.362 26.911 150 0.797 19.97 60.68 10 5 21 35 150 0.75 17.215 24.354 150 0.812 20.27 52.94 20 2.5 26 35 150 0.75 17.907 23.741 150 0.83 19.23 50.02 30 1.7 1:07 35 150 0.75 18.338 23.481 150 0.829 18.11 48.25 40 1.3 2:24 35 150 0.75 18.368 22.682 150 0.82 20.76 48.13 50 1 3:12 35 150 0.75 18.763 22.099 150 0.825 20.97 46.57 100 0.5 10:51 23 90 0.9 21.701 23 77.519 0.913 11.27 71.17 10 5 6 23 90 0.9 21.84 22.384 90 0.929 7.13 50.41 20 2.5 15 23 90 0.9 23.182 21.491 90 0.946 4.79 39.52 30 1.7 1:19 23 90 0.9 23.983 20.992 90 0.949 3.51 32.46 40 1.3 2:02 23 90 0.9 24.333 20.51 90 0.948 4.22 29.44 50 1 2:06 23 90 0.9 25.42 19.679 90 0.954 3.62 21.02 100 0.5 6:14 22 80 0.9 23.511 22 26.766 0.987 1.06 81.15 10 5 4 22 80 0.9 22.806 21.867 80 0.947 4.81 49.15 20 2.5 17 22 80 0.9 24.341 20.808 80 0.962 2.80 37.23 30 1.7 26 22 80 0.9 25.354 20.074 80 0.965 1.97 30.22 40 1.3 1:50 22 80 0.9 25.92 19.511 80 0.965 2.49 26.84 50 1 2:17 22 80 0.9 27.414 18.497 80 0.969 2.23 18.39 100 0.5 31:45 27 75 0.9 19.45 25.124 75 0.921 8.19 69.45 10 5 6 27 75 0.9 23.323 21.594 75 0.955 3.71 48.56 20 2.5 15 27 75 0.9 25.033 20.34 75 0.97 2.05 36.52 30 1.7 16 27 75 0.9 26.227 19.479 75 0.972 1.48 29.76 40 1.3 1:43 27 75 0.9 26.917 18.896 75 0.972 1.91 26.09 50 1 2:37 27 75 0.9 28.645 17.848 75 0.976 1.70 17.28 100 0.5 26:44 29 110 0.85 17.217 26.239 110 0.864 14.08 65.47 10 5 3 29 110 0.85 20.057 22.917 110 0.89 12.55 53.90 20 2.5 51 29 110 0.85 21.20 22.283 110 0.909 10.11 45.70 30 1.7 59 29 110 0.85 21.865 22.037 110 0.912 8.18 39.51 40 1.3 1:10 29 110 0.85 22.037 21.531 110 0.908 9.56 37.84 50 1 2:23 29 110 0.85 22.754 21.048 110 0.914 8.62 30.68 100 0.5 6:01 21 60 0.95 24.145 21 20.804 0.994 2.77 84.06 10 5 6
The length of the time interval is approximated asT^ ¼11.Table 8shows the trade-off between the accu-racy 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 ofKandDt.
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. 5–7show the computation time, PD.M and PD.V, respectively, according to the different pairs ofK and Dt for 10 non-dominated solutions of Case I and also the indifference solutions of Cases II and III, respectively. Table 4 (continued) e2 e3 e4 f1 f2 f3 f4 PD.M (%) PD.V (%) K Dt C.T 21 60 0.95 25.212 20.336 60 0.978 1.31 48.33 20 2.5 25 21 60 0.95 27.971 18.439 60 0.986 0.76 36.53 30 1.7 43 21 60 0.95 29.737 17.486 60 0.987 0.53 29.49 40 1.3 1:07 21 60 0.95 30.961 16.854 60 0.994 0.71 24.75 50 1 1:47 21 60 0.95 33.568 15.757 60 0.989 0.61 14.70 100 0.5 8:08 24 55 0.95 20.985 24 55 0.95 4.05 72.61 10 5 3 24 55 0.95 26.101 19.7 55 0.984 0.89 49.01 20 2.5 28 24 55 0.95 29.291 17.733 55 0.99 0.49 36.98 30 1.7 39 24 55 0.95 31.303 16.77 55 0.991 0.36 29.83 40 1.3 52 24 55 0.95 32.757 16.12 55 0.996 0.38 24.55 50 1 2:08 24 55 0.95 35.719 15.007 55 0.992 0.45 14.37 100 0.5 17:10 Table 5
Characteristics of the activities in Case II
a da(xa) ga(xa) ca La Ua 1 2x1 0.70.1x1 0 1 5 2 3x2+ 1 1.50.2x2 0 1 6 3 x3+ 2 10.1x3 0 1 9 4 x4 1.50.3x4 0 1 4 5 3x5+ 4 0.90.1x5 0 1 5 6 x6+ 3 1.10.1x6 0 1 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 0.9 10 0.6 * 2.5 2.5 0.9 38.411 2.5 0.623 0.988 11.31 59.88 20 0.3 2:52 2.5 2.5 0.9 36.402 2.5 1.203 0.944 3.73 47.65 30 0.2 13:53 2.5 2.5 0.9 35.599 2.5 1.371 0.937 0.76 44.96 40 0.15 23:56 2.5 2.5 0.9 35.235 2.5 1.421 0.931 0.08 42.95 50 0.12 27:48 2.5 2.5 0.9 34.986 2.5 1.454 0.93 0.43 41.67 60 0.1 49:21 2.5 2.5 0.9 34.513 2.5 1.471 0.929 1.38 41.11 70 0.086 1:20:50 * No feasible solution exists forK= 10 andDt= 0.6.
1 4 8 10 2 3 5 7 9 6 b d c e t f s
Fig. 4. Case III.
Table 7
Characteristics of the activities in Case III
a da(xa) ga(xa) ca La Ua 1 2x1 0.70.1x1 0 1.5 3 2 3x2+ 1 1.50.2x2 0 1.5 3 3 x3+ 2 10.1x3 0 1.5 3 4 x4 1.50.3x4 0 1.5 3 5 3x5+ 4 1.30.2x5 0 1.5 3 6 x6+ 3 1.10.1x6 0 1.5 3 7 2x7+ 5 1.50.2x7 0 1.5 3 8 4x8+ 1 10.2x8 0 1.5 3 9 5x9+ 2 0.90.1x9 0 1.5 3 10 2x10+ 3 20.4x10 0 1.5 3 Table 8
Trade-off results in Case III
e2 e3 e4 f1 f2 f3 f4 PD.M (%) PD.V (%) K Dt C.T 5.5 5 0.85 10 1.1 * 5.5 5 0.85 58.783 5.123 5 0.869 19.61 4.96 20 0.55 6:25 5.5 5 0.85 67.166 5.5 2.021 0.929 1.13 49.65 30 0.367 19:49 5.5 5 0.85 67.222 5.5 2.405 0.931 1.06 39.99 40 0.275 45:20 5.5 5 0.85 66.846 5.5 2.717 0.922 1.45 32.86 50 0.22 51:35 * No feasible solution exists forK= 10 andDt= 1.1.
0 1000 2000 3000 4000 5000 6000 10 20 30 40 50 60 70 100 C.T. (Sec.) Ι1 Ι2 Ι3 Ι4 Ι5 Ι6 Ι7 Ι8 Ι9 Ι10 ΙΙ ΙΙΙ K
According toFig. 5, computation time grows withK. Computation time is also strongly dependent on the network size, and sensitive to the values ofej,j= 2, 3, 4, or the structure of non-dominated solutions.
According toFig. 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 ofK= 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 toFig. 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 solu-tion, in almost all case, with one excepsolu-tion, 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.
0 5 10 10 20 30 40 50 60 70 100 PD.M % Ι6 Ι7 Ι8 Ι9 Ι10 ΙΙ ΙΙΙ K Ι1 Ι2 Ι3 Ι4 Ι5 25 15 20 Fig. 6. PD.M versusK. 0 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 100 PD.V% Ι1 Ι2 Ι3 Ι4 Ι5 Ι6 Ι7 Ι8 Ι9 Ι10 ΙΙ ΙΙΙ K Fig. 7. PD.V versusK.
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 func-tion, 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 non-linear programming formulation (14) can grow exponentially with the network size. As the worst case example, consider a complete transformed network with mnodes and m(m1)/2 activities. The size of the state space for this network is given byN(m) =UmUm1, where
Um¼X
m
k¼0
2kðmkÞ ð16Þ
(refer to Kulkarni and Adlakha[19]).
Consequently, the number of constraints of the nonlinear model is equal to 2K(N(m)1) + 2.5m(m1), and the number of decision variables would be equal toK(N(m)1) +m(m1). Therefore the number of constraints and variables of the nonlinear programming formulation(14)grows exponen-tially with m. In practice, the number of arcs of the PERT networks is generally much less than m(m1)/2. Even large sparse networks generally produce a reasonable size state space.
According to computational experiments, whenKgoes to infinity andDtgoes 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 ofKandDt, 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 net-work size and also between PD.V and the netnet-work 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 thea-fractile of the distribution function of project completion time with the value ofapre-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 indepen-dent exponentials. For unimodal distributions, the sum of two indepenindepen-dent exponentials is a reasonable approximation. For multimodal distributions, one must use mixtures.
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 meth-ods. 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 solu-tion in order to determine a new solusolu-tion.
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
