A multi-objective resource allocation problem in dynamic PERT networks

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1 Applied Mathematics and Computation xxx (6) xxx xxx A multi-objective resource allocation problem in dynamic PERT networks Amir Azaron a, *, Reza Tavakkoli-Moghaddam b a Department of Computer Science, Cork Constraint Computation Centre, University College Cork, Cork, Ireland b Department 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 process. 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 minimized), 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. Ó 6 Elsevier Inc. All rights reserved. Keywords: Multiple objective programming; Queueing; Optimal control; Project management 1. Introduction Since the late 195s, 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- * Corresponding author. address: [email protected] (A. Azaron) /$ - see front matter Ó 6 Elsevier Inc. All rights reserved. doi:1.116/j.amc.6.1.7

2 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) 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 committed 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 [1,1], discrete [7], convex [14,5], and concave [9]. When the cost functions are arbitrary (still non-increasing), the dynamic programming (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 networks. Recently, some researchers have adopted computational optimization techniques such as genetic algorithms to solve TCTP. Chau et al. [6] and Azaron et al. [] 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 aforementioned 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 facilities. 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 differential 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).

3 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx 3 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 Section, we compute the project completion time distribution in dynamic PERT networks with exponentially distributed activity durations, analytically. Section 3 presents the multi-objective resource allocation formulation. Section 4 presents the computational experiments, and finally we draw the conclusion of the paper in Section 5.. Project completion time distribution in dynamic PERT networks In this section, we present an analytical method to compute the distribution function of the project completion 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 of k. 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 of k. 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 the ith node, then the distribution of time spent (activity duration plus waiting time in queue) in this M/M/1 queueing system is w i ðtþ ¼ðl i kþe ðli kþt t > ; ð1þ where k and l i are 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 (l i k). Step 1.. If there are infinite servers in the service station settled in the ith node, then the time spent in this M/ M/1 queueing system would be exponentially distributed with parameter l i, because there is no queue. Step. Transform the dynamic PERT network into an equivalent classical PERT network represented as an Activity-on-Arc (AoA) graph. Step.1. Replace each node with a stochastic arc (activity) whose length is equal to the time spent in the particular service station.let us explain how to replace node k in the network of queues with a stochastic activity. Assume that b 1,b,...,b n are the incoming arcs to this node and d 1,d,...,d m are the outgoing arcs from it. Then, we substitute this node by activity (k,k ), whose length is equal to the time spent in the corresponding queueing system. Furthermore, all arcs b i for i =1,...,n end up with k while all arcs d j for j =1,...,m start from node k. The indicated process is opposite of the absorption an edge e in a graph G in graph theory (G.e), see Azaron and Modarres [3] for more details. Step.. 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., using the method of Kulkarni and Adlakha [13]. Let G =(V,A) be the transformed classical PERT network with set of nodes V ={v 1,v,...,v m } and set of activities A ={a 1,a,...,a n }. The source and sink nodes are denoted by s and y, respectively. Length of arc a A is an exponentially distributed random variable with parameter c a. For a A, let a(a) be the starting node of arc a, andb(a) be the ending node of arc a.

4 4 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx Definition 1. Let I(v)andO(v) be the sets of arcs ending and starting at node v, respectively, which are defined as follows: IðvÞ ¼ OðvÞ ¼ fa A : bðaþ ¼vg; ðv V Þ; ðþ fa A : aðaþ ¼vg; ðv V Þ. ð3þ Definition. If X V such that s X and y X ¼ V X, then an (s,y) cut is defined as ðx ; X Þ¼fa A : aðaþ X ; bðaþ X g. ð4þ An (s,y) cut ðx ; X Þ is called a 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, ) is a uniformly directed cut (UDC) because V is divided into two disjoint subsets X and X, where s X and y X. The other UDCs of this network are (, 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 -partition, if for any a F, we have I(b(a)) 6 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, this cut is an admissible -partition, because I(b(3)) = {3,4} 6 F and also I(b(6)) = {5,6} 6 F. However, if E = {6} and F = {3,4}, then the cut is not an admissible -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 -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. s y Fig. 1. The example network. Table 1 All admissible -partition cuts of the example network 1. (1,) 5. (1,4 *,6) 9. (3 *,4,6) 13. (3,4 *,6 * ) 17. (/,/). (,3) 6. (1,4,6 * ) 1. (3,4 *,6) 14. (5,6) 3. (,3 * ) 7. (1,4 *,6 * ) 11. (3,4,6 * ) 15. (5 *,6) 4. (1,4,6) 8. (3,4,6) 1. (3 *,4,6 * ) 16. (5,6 * )

5 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx 5 Let S denote the set of all admissible -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 is proven that {X(t), t P } is a continuous-time Markov process with state space S, refer to [13] for details. As mentioned, E and F contain active and dormant activities of a UDC, respectively. When activity a finishes (with the rate of k a ), 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. 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, 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 the particular sets. Thus, the elements of the infinitesimal generator matrix Q =[q{(e,f),(e,f )}], (E,F) and ðe ; F ÞS, are calculated as follows: 8 c a if a E; IðbðaÞÞ 6 F [fag; E ¼ E fag; F ¼ F [fag; ðaþ c a if a E; IðbðaÞÞ F [fag; E ¼ðE fagþ [ OðbðaÞÞ; >< qfðe; F Þ; ðe ; F Þg ¼ F ¼ F IðbðaÞÞ; ðbþ P c a if E ¼ E; F ¼ F ; ðcþ ae >: otherwise. ðdþ ð5þ In Example 1, if we consider E={1, }, F =(/), E ={,3} and F =(/), then E =(E {1}) [ O(b (1)), and thus from (5b), q{(e,f), (E,F )} = c 1. {X(t), t P } is a finite-state absorbing continuous-time Markov process and since q{(/,/), (/,/)}=,itis concluded that this state is 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; ;...; N ¼jSj. State 1 is the initial state, namely X(t) =(O(s), /), and state N is the absorbing state, namely X(t) =(/,/). Let T represent the length of the longest path in the network, or the project completion time in the PERT network. Clearly, T = min{t > : X(t) = N/X() = 1}. Thus, T is the time until {X(t), t P } 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 ¼PfX ðtþ ¼N=X ðþ ¼ig; i ¼ 1; ;...; N ð6þ then, F(t) =P 1 (t). The system of linear differential equations for the vector P(t) =[P 1 (t),p (t),...,p N (t)] T is given by P ðtþ ¼QPðtÞ; ð7þ PðÞ ¼½; ;...; 1Š T ; where P (t) represents the derivation of the state vector P(t) and Q is the infinitesimal generator matrix of the stochastic process {X(t), t P }. In Section 3, 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 activities 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 completion 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.

6 6 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx 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 completion time (min). The direct cost of activity a A in the transformed classical PERT network is assumed to be a non-decreasing function d a (x a ) of the amount of resource x a allocated to it. Therefore, the project direct cost (PDC) would be equal to PDC ¼ P aa d aðx a Þ. The mean time spent in the service station a is assumed to be a non-increasing function g a (x a ) of the amount of resource x a allocated to it. As explained in Section, the mean time spent would be equal to 1 l a k, if there is one server, and equal to 1 l a, if there are infinite servers in the corresponding service station. Let U a represent the amount of resource available to be allocated to the activity a, and L a represent the minimum amount of resource required to achieve the activity a. In reality d a (x a ) and g a (x a ) can be estimated using linear regression. We can collect the sample paired data of d a (x a )andg a (x a ) as the dependent variables, for different values of x a as the independent variables, from the previous similar activities or using the judgments of the experts in this area. Then, we can estimate the parameters of the relevant linear regression model. The mean and the variance of project completion time are given by EðT Þ¼ VarðT Þ¼ Z 1 Z 1 ð1 P 1 ðtþþdt; t P 1 ðtþdt Z 1 tp 1 ðtþdt ; ð9þ where P 1ðtÞ is the density function of project completion time. The infinitesimal generator matrix, Q, is a function of the control vector l =[l a ; a A] T. Therefore, the nonlinear dynamic model is ð8þ P ðtþ ¼QðlÞPðtÞ; P i ðþ ¼ 8i ¼ 1; ;...; N 1; P N ðtþ ¼1. ð1þ Representing B as the set of nodes including M/M/1 service stations and C as the set of nodes including M/ M/1 service stations in the original dynamic PERT network (A =(B [ C)), the relations (11) should be satisfied to exist the response in the steady-state. l a > k; a B; l a > ; a C. ð11þ We do not have such constraints in the mathematical programming. Therefore, we use the constraints (1) instead of the above constraints in the final multi-objective problem l a P k þ e; a B; l a P e; a C. ð1þ Accordingly, the appropriate multi-objective optimal control problem is Min Min Min f 1 ðx; lþ ¼ X aa f ðx; lþ ¼ f 3 ðx; lþ ¼ Z 1 Z 1 d a ðx a Þ ð1 P 1 ðtþþdt t P 1 ðtþdt Z 1 tp 1 ðtþdt

7 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx 7 s.t. P ðtþ ¼QðlÞPðtÞ; P i ðþ ¼; 8i ¼ 1; ;...; N 1; P N ðtþ ¼1; g a ðx a Þ¼ 1 l a k ; a B; g a ðx a Þ¼ 1 l a ; a C; ð13þ l a P k þ e; a B; l a P e; a C; x a 6 U a ; a A; x a P L a ; a A. 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, f ðx; lþ ¼ R 1 ð1 P 1ðtÞÞdt. Clearly x a ¼ g 1 a ð 1 Þ for a B and x l a k a ¼ g 1 a ð 1 l a Þ for a C. Therefore, we can consider l as the unique control vector of the problem, and ignore the role of x =[x a ; a A] T as the other independent decision vector. Consider K as the set of allowable controls consisting of all constraints except the constraints representing the dynamic model (l K), and N-vector k(t) as the adjoint vector function. Then, Hamiltonian function would be HðkðtÞ; PðtÞ; lþ ¼kðtÞ T QðlÞPðtÞþ1 P 1 ðtþ. Now, we write the adjoint equations and terminal conditions, which are k ðtþ T ¼ kðtþ T QðlÞþ½ 1; ;...; Š; ð15þ kðt Þ T ¼ ; T!1. If we could compute k(t) from (15), then we would be able to minimize the Hamiltonian function subject to l K in order to get the optimal control l *, 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 to l K, 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, P (t) =Q(l) Æ P(t), P() = [,,...,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 of K, and consequently l * 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 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. IfDt is sufficiently small, it can be assumed that P(t) varies only in times,dt,...,(k 1)Dt. Since each P i (k), for i =1,,...,N 1, k =1,,...,K, is a distribution function, then the constraints (16) should also be considered in the final discrete-time problem (refer to [4] for more details about the proposed technique) P i ðkþ 6 1 i ¼ 1; ;...; N 1; k ¼ 1; ;...; K. ð16þ Theoretically, when K approaches to infinity and Dt approaches 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, P 1 (K) should approach one. ð14þ

8 8 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx 3.1. Goal attainment method This method requires setting up a goal and weight, b j and c j (c j P ) for j = 1,,3, for the three indicated objective functions. The c j relates the relative under-attainment of the b j. For under-attainment of the goals, a smaller c j is associated with the more important objectives. c j, j = 1,,3, are generally normalized so that P 3 j¼1 c j ¼ 1. The appropriate goal attainment formulation to obtain x * is Min s.t. z X d a ðx a Þ c 1 z 6 b 1 ; aa Z 1 Z 1 ð1 P 1 ðtþþdt c z 6 b ; t P 1 ðtþdt Z 1 tp 1 ðtþdt c 3 z 6 b 3 ; Pðk þ 1Þ ¼PðkÞþQðlÞPðkÞDt; k ¼ ; 1;...; K 1; P i ðþ ¼; i ¼ 1; ;...; N 1; P N ðkþ ¼1; k ¼ ; 1;...; K; P i ðkþ 6 1; i ¼ 1; ;...; N 1; k ¼ 1; ;...; K; g a ðx a Þ¼ 1 l a k ; a B; g a ðx a Þ¼ 1 l a ; a C; l a P k þ e; a B; l a P e; a C; x a 6 U a ; a A; x a P L a ; a A; z P. 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 to b and c. Depending upon the values for b,it is possible that c does not appreciably influence the optimal solution. Instead, the optimal solution can be determined by the nearest Pareto-optimal solution from b. This might require that c be varied parametrically to generate a set of Pareto-optimal solutions. Solving the goal attainment formulation (17) leads to the approximated objective function value (z Approx. ). For computing the exact value of z (z Exact ), 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 compute P 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, z Exact is given by z Exact ¼ Max PDC b 1 ; EðT Þ b ; VarðT Þ b 3 c 1 c c 3 4. Numerical example. ð18þ 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

9 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx 9 λ Fig.. The dynamic PERT network. Fig.. 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 = 1 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. It means that there is no predecessor activity for the activities 1 and of each project.. There is one server in the service stations settled in the nodes 1,, 3, 6 and There are infinite servers in the service stations settled in the nodes 4 and 5. The transformed classical PERT network is depicted in Fig. 3. The stochastic process {X(t), t P } related to the longest path analysis of this classical PERT network has 14 states in the order of S ¼fð1; Þ; ð1; 3Þ; ð1; 5Þ; ð1; 5 Þ; ð; 4Þ; ð; 4 Þ; ð3; 4Þ; ð3; 4 Þ; ð4; 5Þ; ð4 ; 5Þ; ð4; 5 Þ; ð6þ; ð7þ; ð/; /Þg. Table 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 nonlinear 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. s Fig. 3. The transformed classical PERT network. y Table Matrix Q(l) State l 1 l 1 1 l 3 1 l l 5 l l l 4 l 1 6 l 1 7 l 4 l l l 4 l 5 1 l 5 11 l 4 1 l l

10 1 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx Table 3 Characteristics of the activities a d a (x a ) g a (x a ) L a U a 1 3x 1 þ.7.1x x x x x x x x x x x x x Then, considering the goal vector b: (b 1 = 5, b =,b 3 =.75), three factorial experiments according to the following three sets of c: c1 : ðc 1 ¼ :99; c ¼ :455; c 3 ¼ :455Þ; c : ðc 1 ¼ :7693; c ¼ :769; c 3 ¼ :1538Þ; c3 : ðc 1 ¼ :899; c ¼ :178; c 3 ¼ :893Þ are designed to obtain a set of Pareto-optimal solutions in each case. For example, using the first set of c leads to the following consideration: one year deviation from the mean project completion time is as important as its variance and times as important as one thousand dollars deviation from the project direct cost, respectively. To investigate the trade-off between the accuracy (in terms of K) and computational time, we consider the following levels of K (K =, K = 5, K = 5) in our computational experiments. Moreover, P 1 (K) should be greater than.99. If a solution does not have this property, the value of Dt is increased in order to access to this level of accuracy. Thus, the following combinations of K and Dt are considered: (K =, Dt =.35), (K = 5, Dt =.14), (K = 5, Dt =.14). The value of e is also considered equal to.1 in all experiments. Finally, we use LINGO 6 on a PC Pentium IV.1 GHz to solve the problem and to compute the approximated objective function values (z Approx. ) and the related computational times for the three sets of c. The exact objective function values (z Exact ) are also computed from Eq. (18). For example, the optimal allocated resource quantities, considering the first set of c(c1), K = 5 and Dt =.14, are shown in Table 4. Table 5 shows the corresponding values of PDC, E(T), Var(T), as the three indicated criteria, P 1 (K = 5), z and the related computational time in seconds (CT). Fig. 4 shows the approximated and the exact objective functions for the three indicated sets of c, considering K =, K = 5 and K = 5. Fig. 5 shows the related computational times. According to Fig. 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 z Approx. and z Exact are decreased, when K is increased. As it is seen in Fig. 4, the approximated and the exact z are 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 compute z, in these experiments. According to Fig. 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 4 Optimal allocated resource quantities, considering c1 and K = 5 x 1 x x 3 x 4 x 5 x 6 x Table 5 Optimal criteria, considering c1 and K = 5 PDC E(T) [Approx.] E(T) [Exact] Var(T) [Approx.] Var(T) [Exact] z Approx. z Exact P 1 (K) CT

11 A. Azaron, R. Tavakkoli-Moghaddam / Applied Mathematics and Computation xxx (6) xxx xxx z 5 z(approx.) z(exact) c1(k=) c1(k=5) c1(k=5) c(k=) c(k=5) c(k=5) c3(k=) c3(k=5) c3(k=5) Fig. 4. Objective function values. 5 Computational Time (Sec.) Computational Time c1(k=) c1(k=5) c1(k=5) c(k=) c(k=5) c(k=5) c3(k=) c3(k=5) c3(k=5) Fig. 5. Computational times. 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 transformed 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 K approaches to infinity and Dt 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 problem, but the computation time goes to infinity. Therefore, we should select the proper values for K and Dt, in the realistic sized problems, so that we can solve the problem in an acceptable level of accuracy with reasonable 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 with n nodes and nðn 1Þ arcs, the size of the state space is given by N(n) =U n U n 1, where U n ¼ Xn kðn kþ ð19þ k¼ (refer to [13]).

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