resource constrained scheduling simulation model for alternative stochastic network projects

Mathematics and Computers in Simulation 63 (2003) 105–117

Resource constrained scheduling simulation model foralternative stochastic network projects

Dimitri Golenko-Ginzburga,b,∗, Aharon Gonika, Zohar Lasloca Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

b Academic College of Judea and Samaria, Ariel 44837, Israelc Department of Industrial Engineering and Management, Negev Academic College of Engineering, Beer-Sheva 84100, Israel

Received 2 March 2003; accepted 27 March 2003

Abstract

The paper presents a heuristic for resource constrained network project scheduling. A network project comprisingboth alternative deterministic decision nodes and alternative branching nodes with probabilistic outcomes is con-sidered. Several renewable activity related resources, such as machines and manpower, are imbedded in the model.Each type of resource is in limited supply with a resource limit that is fixed at the same level throughout the projectduration. Each activity in the project requires resources of various types with fixed capacities. The activity durationis a random variable with given density function.

The problem is to minimize the expected project duration by determining for each activity, which will be realizedwithin the project’s realization, its starting time (decision variable), i.e. the time of feeding-in resources. The resourcedelivery schedule is not calculated in advance and is based on decision-making in the course of monitoring the project.The suggested heuristic algorithm is performed in real time via simulation. Decision-making is carried out:

• at alternative deterministic decision nodes, to single out all the alternative sub-networks (joint variants) in orderto choose the one with the minimal average duration;

• at other essential moments when at least one activity is ready to be operated but the available amount of resourcesis limited. A competition among those activities is carried out to determine the subset of activities which have tobe operated first and can be supplied by available resources. Such a competition is realized by a combination ofa knapsack resource reallocation model and a subsidiary simulation algorithm.

A numerical example to illustrate the heuristic algorithm is presented.© 2003 IMACS. Published by Elsevier Science B.V. All rights reserved.

Keywords: Alternative decision nodes; Probabilistic branching; Joint variant; Renewable resources; Resource constrainedGERT project scheduling algorithm; Stochastic project simulation

∗ Corresponding author. Tel.:+972-8-6238322; fax:+972-8-6472958.E-mail address: [email protected] (D. Golenko-Ginzburg).

0378-4754/03/$ – see front matter © 2003 IMACS. Published by Elsevier Science B.V. All rights reserved.doi:10.1016/S0378-4754(03)00050-8

106 D. Golenko-Ginzburg et al. / Mathematics and Computers in Simulation 63 (2003) 105–117

Nomenclature

aij lower bound of valuetij (pre-given)bij upper bound of valuetij (pre-given)F ∗

ij the actual moment activity (i, j)∗ is finished,F ∗ij = S∗

ij + tijGt the remaining network project at momentt ≥ 0; G0 = G(N,A)

G(N, A) stochastic network project of CAAN type1

i(α) decision node with deterministic alternative outcomesi( ¯α) alternative node with stochastic outcomes(i, j) activity leaving nodei and entering nodej, (i, j) ⊂ A ⊂ G(N,A)

(i, j)∗ activity (i, j) which will be actually realized in the course of the project’sdevelopment (conditioned on the model’s decision)

Jrt therth joint variant of projectGt (a sub-network of PERT or GERT type),1 ≤ r ≤ mt

mt number of joint variants in projectGt

n number of different resourcesp(i, j)∗ conditional probability of activity (i, j)∗ to be on the critical path in the course

of the project’s realizationrijk capacity of thekth type resources allocated to activity (i, j),

1 ≤ k ≤ n (pre-given and fixed)Rk total available resources of typek at the project management disposal

(pre-given and fixed throughout the planing horizon)Rk(t) free available resources at momentt ≥ 0Rmax

k (t|S∗ij, J

optrt ) maximal value of thekth resource profile at momentt on condition that activities

(i, j)∗ start at momentsS∗ij and at momentt an optimal joint variantJopt

rt is chosenS∗

ij the moment resources are fed-in and activity (i, j)∗ starts (a random value)tij random duration of activity (i, j), with density functionft(i, j)T(G|S∗

ij, Joptrt ) random project’s duration, on condition that according to the resource

constrained scheduling model an optimal joint variantJoptrt will be

chosen and all activities (i, j)∗ start at momentsS∗ij

Greek letterµij average value oftij (pre-given)

1 Note that sinceG(N, A) is analternative network, the set ofactually realized activities (i, j)∗ is a subset of all activities{(i, j)} enteringG(N, A). Thus, (i, j)∗ ⊂ {(i, j)}.

1. Introduction

A number of recent papers (e.g.[1,4,6,9–12,14], etc.) present various algorithms on resource con-strained project scheduling. Besides[4,6] no published algorithm considers network projects under

D. Golenko-Ginzburg et al. / Mathematics and Computers in Simulation 63 (2003) 105–117 107

random disturbances which cover a very broad spectrum of R&D projects, e.g. PERT, GERT and VERTtype network projects with random activity durations[8], CAAN type network models with alternativeoutcomes in key nodes[2,3] and more complicated GAAN type network models[5]. However, all re-source constrained project scheduling models developed in[4,6] deal only with PERT type projects, i.e.with non-alternative network projects.

However, for a certain project its topology may implement various alternative outcomes (deterministicand stochastic), when there are several possible alternative ways for reaching the project’s target. Suchnetwork projects usually occur when an entirely new device is designed with no similar prototypes in thepast (e.g. in chemical industries, aerospace and in other defense related industries). They are faced witha great deal of uncertainty in their progress as well as with alternative outcomes in key events. Since theimportance of such projects is significant, practically all industrial developed countries have to considerand to perform the so-called goal programs or goal projects as the basic trend of technological progress.The need for high quality resource constrained scheduling models for such complicated projects becomesmore and more important. Thus, undertaking research in this area is useful both from the theoretical andapplied points of view.

The newly developed resource constrained scheduling model for projects under random disturbancesand with alternative structure is a future extension of our previous publications[4,6], in which activityrelated resources with fixed and variable capacities are imbedded in a PERT type network model withoutalternative branchings.

We will henceforth consider an activity-on-arc network projectG(N, A) of the CAAN type[2,3], wherethe set of alternative nodes is subdivided into subsets:

• ¯N ⊂ N: alternative nodes with stochastic branchings;• N ⊂ N: alternative deterministic nodes (decision nodes).

We have chosen the CAAN model since within the recent decade it has been used in various main typesof alternative network projects[2,3,5].

Each activity(i, j) ∈ A ⊂ G(N,A) requires renewable resources of various types with fixed orvariable capacities. In order to simplify the problem we will consider the case of fixed capacities, althoughintroducing variable capacities results only in additional technical difficulties. Each type of resources isin limited supply with a resource limit that is fixed at the same level throughout the project duration. Theduration of each activity is a random variable with given density function.

The problem is to determine starting time valuesSij for each activity (i, j) which will be actually realizedin the course of the project’s development. Note that due to the project’s alternative structure, not all theactivities entering the project will be realized. ValuesSij are not calculated in advance and are randomvariables conditioned on the model’s future decision. The model’s objective is to minimize the expectedproject’s duration. Such an objective is mostly used in project management (see, e.g.[1,10,12,14], etc.),and the problem of decreasing the project duration is considered as one of the most important goals, espe-cially for projects under random disturbances[7,9,10,12]. The suggested heuristic algorithms is performedin real time via simulation. Decision-making in the course of monitoring the project is carried out:

• at alternative deterministic decision nodes to single out all alternative sub-networks (the so-called jointvariants) in order to choose the one with the minimal average duration;

• at the project’s essential moments when at least one activity is ready to be operated but the availableamount of resources is limited. A competition among those activities is carried out to determine the


subset of activities, which have to be operated first and can be supplied by available resources. Such acompetition is realized by a combination of a knapsack resource reallocation model and a subsidiarysimulation algorithm.

Note that those essential moments are as follows:

• when one of activities (i, j) is finished and additional resources become available, or• when a certain event (node)i is realized and all activities leaving that node are ready to be processed.

Since a joint variant of a CAAN model is a GERT type sub-network with probabilistic outcomes in keyevents, the problem’s solution is based on developing a resource constrained scheduling model for GERTprojects. The corresponding algorithm is, in essence, the backbone of the general resource constrainedmodel, and a future development of the models outlined in our previous publications. Thus, developingthe resource-constrained project scheduling model for networks with purely stochastic alternatives is themain contribution of this paper.

The structure of the paper is as follows: inSection 2, the description of the CAAN model is outlined.Section 3presents the general problem of resource constrained project scheduling for a CAAN model isoutlined. InSection 4, the description of the heuristic algorithm is presented. InSection 5the subsidiaryalgorithm for the GERT type network model is outlined.Section 6describes a numerical example, whileSection 7presents conclusions and future research.

2. Description of the CAAN model

A CAAN model is a finite, connected, oriented, activity-on-arc networkG(N, A) with the followingproperties[2]:

(A) Network G(N, A) has one source node and not less than two sink nodesn′.(B) The set of nodes of networkG(N, A) includes four types of nodes:

• type 1 (x): with the logical “and” receiver;• type 2 (α): with the logical “and” receiver and the “exclusive or” emitter;• type 3 (β): with the “exclusive or” receiver and the “must follow” emitter;• type 4 (γ): with the “exclusive or” receiver and the “exclusive or” emitter.

(C) The set of alternative nodes (types 2 and 4) for a CAAN model is subdivided into subsets:

• ¯N ⊂ N: alternative nodes{ ¯α}with stochastic branching;• N ⊂ N: alternative deterministic (decision) nodesα.

To analyze the CAAN model a special networkG∗(N∗, A∗) is used which is called the outcome graph[2,3]. The latter is designated as

G∗(N∗, A∗) ⊂ G(N,A), (1)

N∗ = N\{x} ≡ n0 ∪ {n′} ∪ {α} ∪ {β} ∪ {γ}, (2)

(i, j) ⊂ G∗(N∗, A∗) ⇒ Gij ⊂ G(N,A), (3)

whereGij is a fragment entering the initial graphG(N, A). Note that for the CAAN model fragmentsGij

do not intersect and thus bothG(N, A) andG∗(N∗, A∗) are fully divisible networks.


A very important conception of the CAAN model’s activity is the activity’s direction. If (i, j) leavesnodei andi is a decision node, all activities leavingi are indexed clockwise ashi = 1,2, . . . , ni, whereni is the number of activities leavingi [2].

After determining the outcome graphG∗(N∗, A∗) the problem is to single out from it the so-calledjoint variants. A joint variant is a PERT or a GERT type sub-network which can be chosen by theproject management in order to realize the project. Every joint variant is defined by a choice of certaindirections in some of the decision nodes of the outcome graph the (so-called non-contradictory nodes[2]). A set which indicates the set ofα-nodes and their directions, by set{α1, hi1, α2, hi2, . . . , αr, hir}and, thus, completely defines the joint variant, is called anadmissible plan [2]. The set of joint variantsis in one-to-one correspondence with the set of admissible plans.

Thus, to sum up, a joint variant is a sub-network which can be extracted from the outcome graphby fixing non-contradictory directions and excluding unfixed directions. A detailed description of thealgorithm to determine all joint variants from the initial CAAN model is outlined in[2]. Therefore, whendescribing the newly developed resource constrained model for the CAAN network, we will make, whennecessary, appropriate references. Call henceforth AJV the algorithm for determining joint variants.

3. The problem

The general resource constrained scheduling problem for a CAAN type modelG(N, A) is to minimizethe expected project’s duration

Min E{T(G|S∗ij, J

optrt )} (4)

s.t.

Rmaxk (t|S∗

ij, Joptrt ) ≤ Rk(t) ∀t ≥ 0, 1 ≤ k ≤ n. (5)

Problem (4–5) is a complicated stochastic optimization model for projects with an alternative structureand topology. The problem cannot be solved in the general case and allows only a heuristic solution.

The general idea of the heuristic algorithm is as follows. Decision-making is carried out in real time,at any routine essential momentt (decision point), either when one of the activities (i, j)∗ is finishedand additional resourcesrijk, 1 ≤ k ≤ n, become available, or when a certain non-alternative nodeiis realized and all activities leaving that node are ready to be processed, or when a decision node (anode with deterministic alternative outcomes of typeα andγ) is reached. In the latter case, by usingalgorithm AJV[2], all joint variantsJrt, 1 ≤ r ≤ mt, are singled out and later on examined, to determinethe optimal joint variant with the minimal expected duration. The procedure of determining the averageduration of a joint variant (which is, in essence, a PERT or a GERT type network with purely stochasticalternative outcomes at certain nodes) is carried out by a newly developed resource constrained GERTproject scheduling algorithm (RCGPS), which will be outlined below.

After examining all the joint variantsJrt, 1 ≤ r ≤ mt, an optimal joint variantJ∗t is chosen and future

monitoring centers on realizing resource constrained scheduling for pre-given total available resourcesRk(t), 1 ≤ k ≤ n.

If a routine essential moment is a node withstochastic alternative outcomes, the latter are simulatedaccording to their outcome probabilities (in real time projects as well); the simulated activity thus obtainsvia simulation the corresponding duration.


If an essential moment centers on determining a subset of activities (from a set of activities ready to beprocessed and waiting to be supplied by resources), a competition among the activities has to be arranged.For the case of a PERT network, the corresponding algorithm (call it henceforth RCPPS) is outlined in[4,6]. The general idea of the RCPPS algorithm is to reallocate resources among the project’s competitiveactivities on the basis of priority levels assigned to those activities. Those priority levels are the activities’contributions to the project’s average duration. They depend both on the activity’s average duration andon the probability to be on the critical path in the course of the project’s realization. Those probabilityvalues are also obtained via simulation.

The outlined below algorithm RCGPS is a modification of the RCPPS algorithm since stochasticalternative outcomes have to be taken into account.

After singling out the subset and supplying the later by available resources, activities begun to beprocessed. A new routine essential moment is determined, etc. until the project is accomplished.

Note, in conclusion, that in the course of developing a real project there may be changes in the parametersof some activities, e.g. probability density functions of the activities’ durations, outcome probabilities,etc. since activity networks are revised over time. In such a case the problem of determining all the jointvariantsJrt has to be resolved at each sequentially encountered decision node at momentt, since revisinga project may result in changing its optimal joint variantJ

optt . If the network does not undergo revision

the problem has to be solved only once, att = 0.

4. The general resource constrained scheduling heuristic algorithm for a CAAN type model

The outlined below algorithm incorporates two currently developed algorithms, namely:

• the algorithm of determining all joint variants from the initial CAAN model (algorithm AJV[2]);• the resource-constrained project scheduling algorithm RCPPS for non-alternative PERT networks for

cases of fixed and variable resource capacities[4,6].

In the course of developing the procedure of the general algorithm, we will refer the reader to thosereferences and will give only a short description of both algorithms.

The enlarged step-by-step procedure of the heuristic algorithm is as follows:

Step 1. The routine essential momentt of the project’s progress is determined at the beginning of theproject’s realization. An essential moment may occur:

(A) at a decision node (α) with alternative deterministic outcomes (ofα andγ types);(B) at an alternative node (¯α) with stochastic outcomes (ofα andγ types);(C) at a non-alternative node (j) (of x andβ types);(D) at the moment a certain activity (i, j) is finished, but eventj is not realized as yet.

In case A apply the next step; in case B go to step 8; in both cases C and D go to step 11.Step 2. Determine the remaining network projectGt, t ≥ 0. Note that

Gt ≡ G(N,A)\{(i, j)∗t }\{(i, j)∗∗t }, (6)

where{(i, j)∗t } denotes the set of activities which have been already processed till momentt, and{(i, j)∗∗

t } denotes the set of activities which have not been realized and, due to the alternativestructure ofG(N, A) and prior decision-making, will not be realized in the future.


Step 3. Apply algorithm AJV[2] to single out all the joint variantsJrt from the sub-networkGt. Tocarry out the algorithm one has to realize sequentially four sub-algorithms as follows[2]:• determining theα-frame for the outcome graph;• determining the maximal path in the outcome graph;• determining the admissible plans;• determining the joint variants which correspond to admissible plans.Let the determined joint variants beJrt, 1 ≤ r ≤ mt (see Nomenclature).

Step 4. For each joint variant,Jrt determine its average durationTrt by using the resource constrainedscheduling model either for GERT or PERT projects. The total pre-given available resourcecapacities areRk, 1 ≤ k ≤ n (see Nomenclature). For GERT projects, the newly developedresource constrained project scheduling algorithm RCGPS to determine the project’s averageduration is outlined in the next section. For PERT projects, the corresponding algorithm isoutlined in[4,6] and enters the RCGPS as a basic part.

Step 5. Choose the joint variantJξt with the minimal average duration, i.e.

Tξt = Min1≤r≤mt

Trt (7)

Thus, joint variantTξt is considered as an optimal one,Joptt .

Step 6. Choose the outcome direction (activity) leaving nodeαwhich corresponds to the chosen optimaljoint variantJopt

t . Let it be(α, j).Step 7. Cancel all other alternative outcome activities leaving nodeα.Step 8. Determine all the nodesi ⊂ Gt with no activitiesentering those nodes. If such nodes exist,

cancel them together with all activitiesleaving those nodes. Proceed realizing step 8, until onlynodes with a receiver (except the source node) will remain. Go to step 11.

Step 9. Applying this step means that we have reached an alternative node¯α with stochastic outcomesand corresponding probabilities. Simulate the set of full events in order to obtain the outcomeactivity. Let it be( ¯α, j).

Step 10. Cancel all other non-simulated outcome activities leaving node¯α. Go to step 8.Step 11. Applying this step means that there may be activities ready to be processed at momentt, e.g.

• activity (α, j) (step 6);• activity ( ¯α, j) (step 9);• activity leaving nodej (case C, step 1), etc. At step 11 in case D (see step 1), return the utilized

resourcesrijk, 1 ≤ k ≤ n, to the project management store.Step 12. Determine the set of activities(i1, j1), . . . , (iq, jq), q ≥ 1, which are ready to be processed at

momentt, together with all available resourcesRk(t), 1 ≤ k ≤ n.Step 13. If all activities(iv, jv), 1 ≤ v ≤ q, can be supplied by available resources, the needed resources

are fed-in and activities{(iv, jv)} begin to be operated at momentt, i.e.Sivjv = t, 1 ≤ v ≤ q. Ifthere is a lack of available resources, go to step 15.

Step 14. Simulate (according to the density function) the durations of all activities which have beensupplied with resources and started to be realized at momentt. Go to step 1 to determine thenext routine essential moment.

Step 15. Applying this step means that, due to limited amount of resources, a competition among activities(iv, jv), 1 ≤ v ≤ q, has to be arranged in order to single out the subset of activities which canbe supplied with resources and can start to be operated at momentt.


The competition is realized by solving a knapsack resource reallocation problem to maximize the totalcontribution of the chosen activities to the average project’s duration. For each activity under competition,its contribution is the product of the average duration of the activity and its probability of being on thecritical path. Those probability values are calculated via simulation.

Since monitoring the CAAN type project results in monitoring a joint variant, i.e. a GERT type project,the algorithm outlined in[4,6] needs modification. The newly developed resource constrained projectscheduling algorithm for a GERT network model will be outlined in the next section (algorithm RCGPS).

After applying algorithm RCGPS and determining the subset of chosen activities go to step 14.The general algorithm terminates when the project will reach its target, i.e. when the remaining graph

Gt becomes an empty set.

5. Resource constrained project scheduling algorithm for GERT models (RCGPS)

As outlined above, the RCGPS algorithm is a future development of the resource constrained projectscheduling algorithm for PERT projects[4,6].

It is assumed that the project’s network isproperly enumerated, i.e. for all activities (i, j) entering thegraphG(N, A) relationi < j holds. The enlarged step-by-step procedure of the algorithm is as follows:

Step 1. Similar to the general algorithm inSection 4, the routine essential momentt is determined(for the monitored optimal joint variantJopt

t ). An essential moment occurs:• at any alternative node (¯α) with stochastic outcomes;• at any non-alternative node (j);• at the moment a certain activity (i, j) is finished, but eventj is not realized as yet.

Step 2. The remaining monitored network projectGt for the previously chosen joint variantJoptt

is determined. In case A (see step 1) apply the next step. In cases B or C go to step 6.Step 3. Similar to step 9 of the general algorithm, simulate the corresponding probabilistic out-

come activity( ¯α, j).Step 4. Is similar to step 10 of the general algorithm (seeSection 4), and results in canceling all

non-simulated alternative stochastic outcomes leaving node¯α.Step 5. Is similar to step 8 of the general algorithm.Step 6. Is similar to step 11 of the general algorithm and results in returning the utilized resources

rijk, 1 ≤ k ≤ n, in case C (see step 1) to the project management store.Steps 7 and 8. Steps 7 and 8 are similar to steps 13 and 14 of the general algorithm, with one exception:

in the case of lack of resources step 10 is applied.Step 9. In order to arrange the competition among the activities(iv, jv), 1 ≤ v ≤ q, sub-network

Gt has to be simulated in order to be transformed to a PERT network. The simulationalgorithm at step 9 comprises the following operations:(i) From the set of stochastic alternative¯α-nodes entering the remaining network project

Gt before realizing step 9, determine the node with the minimal number (call ithenceforth¯αmin). If the set of those nodes is empty, go to step 10. Otherwise apply(ii).

(ii) Simulate the probability outcomes leaving node¯αmin (similar to step 3).(iii) Cancel all non-chosen outcome activities leaving node¯αmin.


(iv) Determine all the nodes (alternative and non-alternative) with no activities enteringthose nodes. If such nodes exists, cancel them together with all activities leaving thosenodes. Proceed realizing subset (iv) until only nodes with receivers will remain. Goto (i).

Step 10. Realizing that step means that we have simulatedall non-contradictory alternative stochas-tic nodes (this is provided by introducing proper enumeration) and only nodes ofx-typeremain. Thus, simulating a GERT network at step 9 results in obtaining a PERT network.

Simulate the durations of all remaining activities according to their density functions.Step 11. Determine the critical path of the simulated network.Step 12. Repeat the procedure of steps 9–11 M times in order to obtain representative statistics.Step 13. Calculate the frequency of each activity(iv, jv), 1 ≤ v ≤ q, to be on the critical path. Call

them henceforth(iv, jv).Step 14. In accordance with[4], determine the subset of chosen activities by solving a zero-one

programming problem: determine integer valuesξivjv , 1 ≤ v ≤ q, to maximize the objec-tive

Max{ξivjv }

{q∑

v=1

[ξivjvp(iv, jv)µivjv ]

}(8)

s.t.

q∑v=1

(ξIvJvrIvJvK) ≤ Rk(t), 1 ≤ k ≤ n, (9)

where

ξivjv ={

0, if activity (iv, jv)will not obtain resources;1, otherwise.

Note that solving problems (8) and (9) results in realizing a heuristic approach to decreasethe project’s duration as much as possible[4]. Models (8) and (9) are, in essence, thebackbone of the RCPPS algorithm, which is outlined in[4,6] and has been successfullyapplied to many medium-size PERT projects[6].

Go to step 8 in order to simulate the durations of the chosen activities and, later on, todetermine the next routine essential momentt.

Note that simulating activity durations at step 10 is an auxiliary procedure (in orderto determine probabilitiesp(iv, jv) for problems (8) and (9) while simulating activitydurations at step 8 is anactual activity realization.

The outlined above algorithm RCGPS is performed in real time: namely, all the activities can be operatedonly after obtaining necessary resources. However if we want to evaluate the average project’s durationTrt for the set of joint variantsJrt (see step 4 of the general algorithm), we can obtain a representativestatistics by simulating each joint variantJrt many times to determine its average duration. The numberof simulation runs can be obtained from the classical sampling theory[13].


6. Numerical example

In order to verify the efficiency of the developed algorithm, extensive experimentation have beenundertaken. A GERT project with constrained renewable resources is presented inTable 1. The projectrequires resources of one type with resource limit valueR1 = 50. The initial given data for each activity(i, j) entering the GERT model is as follows:i; j; aij; bij; rijl; pij, wherepij denotes the probability ofrealizing activity (i, j). Thus,Pij = 1 means that nodei is ofx-type, while 0< pij < 1 denotes a stochasticalternative outcome, i.e.i ≡ ¯α.

Three alternative distributions are considered:

1. tij has a normal distribution in the interval [aij, bij] with the meanµij = 0.5(aij + bij) and the varianceVij = (1/36)(bij − aij)

2;2. tij has a uniform distribution in the interval [aij, bij];3. tij has a beta distribution with the density function

Pij(x) = 12

(bij − aij)4(x − aij)(bij − x)2. (10)

The computer program is written in the PASCAL 7 programming language. In order to check thenewly developed RCJPS algorithm, 100 simulation runs were undertaken. The histograms for the threeconsidered density functions are presented onFigs. 1–3.

Using the RCJPS algorithm results in the following project’s average durationsT :

(a) Normal distribution:T = 517.4.(b) Uniform distribution:T = 533.4.(c) Beta-distribution:T = 511.6.

The following conclusions can be drown by analyzing the histograms:

1. Introducing the beta distribution results in projects with shorter durations in comparison to the normaland uniform distributions.

2. Introducing the normal distribution results in projects with shorter durations in comparison to theuniform distribution. Thus, the latter can be regarded as the least efficient distribution.

Fig. 1. Final histogram of the normal distribution.


Table 1The GERT project’s initial data

No. i j aij bij rijl pij

1 1 2 40 60 16 1.0002 1 4 35 70 15 1.0003 1 5 25 35 18 1.0004 1 6 30 45 19 1.0005 1 7 26 33 10 1.0006 2 3 9 15 18 1.0007 2 15 38 50 24 1.0008 3 14 10 18 25 1.0009 3 15 16 24 16 1.000

10 4 9 30 38 19 1.00011 4 14 18 22 20 1.00012 5 13 25 32 18 1.00013 6 9 31 45 15 1.00014 7 8 58 78 16 1.00015 8 11 35 45 17 1.00016 8 12 25 35 19 1.00017 9 10 35 60 21 1.00018 9 11 30 50 24 1.00019 10 21 35 42 13 1.00020 11 19 20 30 16 1.00021 11 20 14 21 12 1.00022 11 21 15 20 14 1.00023 12 19 30 42 16 1.00024 13 17 28 40 15 1.00025 13 18 22 28 13 1.00026 14 16 20 35 14 1.00027 15 16 16 24 18 1.00028 15 17 15 22 22 1.00029 16 22 13 18 10 1.00030 17 22 27 38 18 1.00031 17 23 35 55 16 1.00032 18 20 20 30 17 1.00033 19 23 25 37 19 1.00034 20 22 17 38 20 1.00035 21 23 38 55 15 1.00036 22 23 12 22 24 1.00037 23 24 40 60 30 0.33338 23 25 50 70 20 0.33339 23 26 30 50 25 0.33340 24 27 30 40 30 0.50041 24 28 30 40 30 0.50042 25 29 20 30 25 0.50043 25 30 20 30 25 0.50044 26 31 40 50 25 0.50045 26 32 40 50 25 0.500


Fig. 2. Final histogram of the uniform distribution.

Fig. 3. Final histogram of the beta-distribution.

3. The variance obtained for the project’s duration by using the normal distribution is the smallest one,while the variance obtained by using the beta-distribution is the largest.

7. Conclusions and future research

1. The heuristic algorithm presented here is, probably, the first one developed in the area of resourceconstrained project scheduling for alternative stochastic network projects. It can be successfully usedfor monitoring complicated medium-size projects with alternative structure and topology, and withlimited activity related renewable resources. The algorithm can be used for CAAN models whichcover a broad spectrum of alternative stochastic networks.

2. Since a CAAN model is structured from sub-networks of GERT type, the developed resource con-strained project scheduling algorithm is based on multiple realization of a standardized resource con-strained algorithm for GERT models. Such a basic algorithm is easy to apply and can be implementedon a PC. The algorithm can be used for any probability distribution of activity durations.

3. The algorithm is performed in real time and adopts a wide range of revisions, alterations, etc. overtime, in the course of the project’s realization.


4. Future research can be undertaken in several directions:• to develop a more complicated resource constrained project scheduling algorithm with activity

related resources of variable capacities[6];• to implement a more universal alternative activity network, e.g. the non-divisible generalized alter-

native activity network (the GAAN model[5]), which comprises CAAN, VERT and GERT models.The GAAN model can be used for any type of alternative network projects, including highly com-plicated R&D projects where decision-making has to be introduced with incomplete or inadequateinformation about the alternatives.

Acknowledgements

This research has been partially supported by the Paul Ivanier Center of Robotics and ProductionManagement, Ben-Gurion University of the Negev, Israel.

References

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resource constrained scheduling simulation model for alternative stochastic network projects

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