where do we stand with fuzzy project scheduling?

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Where Do We Stand with Fuzzy Project Scheduling?Pierre Bonnal1; Didier Gourc2; and Germain Lacoste3

Abstract: Fuzzy project scheduling has interested several researchers in the past two decades; about 20 articles have been wissue. Contrary to stochastic project-scheduling approaches that are used by many project schedulers, and even if the axiomatto the theory of probabilities is not always compatible with decision-making situations, fuzzy project-scheduling approaches thasuited to these situations have been kept in the academic sphere. This paper starts by recalling the differences one can obsuncertainty and imprecision. Then most of the published research works that have been done in this field are summarizedframework for addressing the resource-constrained fuzzy project-scheduling problem is proposed. This framework uses tempotic descriptors, which might become very interesting features to the project-scheduling practitioners.

DOI: 10.1061/~ASCE!0733-9364~2004!130:1~114!

CE Database subject headings: Scheduling; Project management; Fuzzy sets.

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Introduction‘‘Simple things are wrong, while complex ones are unusabFrench poet Paul Valery said one century ago. Project scheare facing a similar dilemma when trying to find the most appriate tool for scheduling their projects. Project-scheduling mels that are simple can sometimes be misleading. Models thtoo sophisticated, because of their low reactivity, can be use

Critical-path method~CPM! is the simplest method availabfor modeling the execution of a project. If the complexity ofproject is low, one can believe that such an approach masufficient. For more complex projects, such as large-scale intrial or construction projects, additional features such as a wvariety of precedence constraints including start–start, sfinish, finish–finish, and with or without a lag; temporal cstraints such as milestones; as well as resource constraintsbe considered.

Among these additional features, uncertainty is an issue taddressed in one of the first publications related to the proscheduling problem—the Malcolm et al. paper on the progevaluation and review technique~PERT! method ~Malcolm1959!. Several interesting papers have been issued followingprimary paper, and among them, a paper proposing the uMonte-Carlo simulations by Van Slyke~1963!, also papers extending the stochastic scheduling problem to branching~the so-called generalized project-scheduling problem! ~Eisner 1962; Elmaghraby 1964; Elmaghraby 1966; Moore and Clayton 1Pritsker 1968!, and a few others aiming to criticize the PE

1Cost and Schedule Manager, LHC Project, CECH1211 Geneva 23, Switzerland. E-mail: [email protected]

2Assistant Professor, E´ cole des Mines d’Albi-Carmaux, 81000 AlbFrance. E-mail: [email protected]

3Professor and Chair, E´ cole nationale d’inge´nieurs, 65000 TarbeFrance. E-mail: [email protected]

Note. Discussion open until July 1, 2004. Separate discussionsbe submitted for individual papers. To extend the closing date bymonth, a written request must be filed with the ASCE Managing EdThe manuscript for this paper was submitted for review and pospublication on February 26, 2002; approved on October 4, 2002.paper is part of theJournal of Construction Engineering and Management, Vol. 130, No. 1, February 1, 2004. ©ASCE, ISSN 0733-93
2004/1-114–123/$18.00.
114 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT

J. Constr. Eng. Manage. 2

ll

method ~Kotiah and Wallace 1973; McCrimmon and Ray1964; Parks and Ramsing 1969!.

Additional criticisms came from some writers interested inimplementation of the fuzzy-set and the possibility theoriesolve practical decision-making problems. They argued, anddo, that the project-scheduling problem is not a domain thatthe axiomatic associated with the probability theory. Fuzzpossibilistic approaches are much more appropriate.

The first paper addressing the project-scheduling problema fuzzy point of view was by Chanas and Kamburowski~1981!and was published in the early 1980s. In the last 20 years,20 papers have been published on that issue. Most of thevery theoretical and cannot be implemented in real-life situat

We believe that these scheduling approaches are menough and that, through some adaptations, project-managpractitioners can benefit from them. The aim of this papersum up the knowledge acquired on fuzzy/possibilistic proscheduling over the years and to propose a practical methadd to the project-scheduler toolbox.

Uncertainty versus Imprecision

Some definitions of terms associated with the notions of utainty and imprecision are given in this section. From DuboisPrade~1988! and from Klir and Folger~1988! we can derive thfollowing:• Uncertainty usually refers to the random nature of a result

term is of a probabilistic nature, and• Imprecision refers to the incompletely defined nature of a

sult; imprecision has a deterministic nature.A pragmatic approach to the notion of information consist

considering it as a predicate, a truth, to which a balancing qfier is attached. Several qualifiers come to mind including pable, possible, necessary, plausible, credible, and so on.

Probable

This qualifier gets its meaning when used in the perspective oprobability theory. If we consider that this theory refers to
measurement of the frequency of event occurrences, then the
© ASCE / JANUARY/FEBRUARY 2004

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probable has a pure arithmetical meaning. In certain caseprobability theory refers to the feeling of uncertainty, i.e.,price that someone may pay if the proposal he/she believestrue is really false. In such situations, the probable has asubjective meaning. In the project-management domain, ththe latter meaning that is usually considered, probable acduration for instance.

Possible and Necessary

Aristotle already mentioned the duality that exists between ttwo terms. The possible expresses the difficulty associatedthe realization of an event, while the necessary refers to thegation to have an event realized. If an event is necessarymeans its contrary is impossible. One can locate the prosomewhere in between the possible and the necessary.

Plausible and Credible

Everything that comes from a corpus of knowledge is said tcredible. Everything that does not is said to be plausible.

The probability theory is a proven mathematical theory.though it has been in existence for several centuries, its axiois quite recent as shown in Kolmogorov’s work in the 1930s.theory has many axioms, but they are clear. Its global consisis uncontested. Its implementation to a wide range of domaingiven valuable results. However, because it is a particularlyomatic theory, several limitations can be observed when apto other fields such as project management.

Prior to any calculation, two of its axioms require that allpossible discrete events are identified, and that the sumthese probabilities equals one. When applied to decision-mprocesses, these two conditions are rarely satisfied. By definthe knowledge someone can have of the future is vague. Tfore, it is extremely difficult to identify all the possible events tmay occur.

The distribution law of a random variable gives the frequeof occurrence of each of its possible values. Math textbookfull of distribution laws, but their use supposes that enoughhas been gathered to let the analyst check if the envisionedfit the data. Unfortunately, the project-management analystnot have enough information to perform this prerequisite. Thdue to the fact that, by definition, a project is a collectionnonrepetitive activities. Even if the closing-out phases arerectly performed, the collected data does not have the qurequired for ensuring a good fit of distribution laws.

Project activities are usually not fair dice because project magers obviously have some power to steer the execution ofprojects. With this last argument, one has three irrefutable refor concluding on the inappropriateness of the probability thwhen used in the project-management field, and especially foresolution of the project-scheduling problem.

Cantor and Dedekind launched the basis of the set theothe middle of the 19th century. It was quite a controversial mematical theory at that time. That is not the case anymore.theory’s principle is that all of the elements of the universe casorted out in sets, and every set can be considered as aWhen this theory is applied to real-life problems, it is sometidifficult to decide whether an element belongs to a set or nothe 1920s, the Polish logician Łukasiewicz and the Romamathematician Moisil, proposed a multivalent logic that hanthe notion of doubt to deal with this. If 0 is false and 1 is tr
then the notion of uncertainty can be associated with 1/2 for in-
JOURNAL OF CONSTRUCTION ENGINEERIN


.

stance. One had to wait until the 1960s and for Zadeh’s wohave a complete algebra for treating vagueness and fuzzinesfuzzy-set and possibility theories have much less axiomatictations as compared to the probability theory. For instanceevent can be possible, so does its contrary. Because this thecompatible with the limitations given above, we are convinthat project schedulers may benefit from using it. This is parlarly the case for planning and scheduling large-scale industrconstruction projects.

Deterministic Activity Network Calculations

A project can be broken down into activities that must beformed in a determined sequence. An acyclic-directed grausually used to model this sequence of activities. The CPMproach associates activities to vertices and precedence consto nodes. Another approach makes the contrary, and is prebecause it is easier to handle. Let us callA5$a1 ,...,an% theproject activities andU5@ujk# the precedence matrix madefollows: ujk51 if ak is an immediate successor ofaj , and ujk

50 otherwise. Let us callG j,A the subset of the activities thare immediate successors ofaj , and G j

21,A the subset of thactivities that are immediate predecessors ofaj . It shall be mentioned that all these subsets derives fromU. The resolution of thso-called project-scheduling problem, i.e., of the graphG5(A,U) supposes that all activity duration is known. To simpthe understanding of the calculation procedure, two zero-duractivities are added:aa andav . aa precedes all the activities fwhich G j

215B; aa is called the initial activity of the networand its start date is known. Theav succeeds all the activities fwhich G j5B and av is called the terminal activity. The aimthe exercise is to determine the minimum makespan forproject and the earliest start datest j and the latest finish datest j

of each activity. Thet j are computed throughout the so-caforward calculation at the end of which the minimum makespobtained;tv2ta . Then tv and tv are made equal in ordercompute thet j throughout the so-called backward calculatTotal and free floats can be computed when all the dates aated to the activities are known.

TFj5t j2t j1dj and FF j5minkPG j$tk%2t j1dj

Calculations are performed fromaa to av ~forward! and fromav to aa ~backward!. An activity can be scheduled if its immeate predecessors~forward! or successors~backward! are alreadcalculated.

t j5maxkPGj21$tk1dk% if G j

21ÞB and t j5ta otherwise

t j5minkPG j$tk2dk% if G jÞB and t j5tv otherwise

In the scheduling of real-life projects, schedulers often neconsider temporal constraints~fixed milestones which correspoto external events! and a wider range of precedence constraincluding finish–start, start–start, start–finish, and finish–finThe introduction of these additional features in the calculaprocedure given previously is quite straightforward.

The scheduler may also need to consider resource constThe problem becomes more difficult to resolve as no exacttion can be found in a polynomial computational time. Thiwhy several attempts were made, and are still being made, theuristics and metaheuristics that give solutions close to theone. Several domains of the applied mathematics including b
light-beam, and taboo search, as well as bound and branch, ge-
G AND MANAGEMENT © ASCE / JANUARY/FEBRUARY 2004 / 115

004.130:114-123.

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netic algorithms, and simulated annealing have been investigThe use of priority rules is undoubtedly the simpler approachcan implement to resolve the resource-constrained proscheduling problem~RCPSP!. Assuming that the remainderthis paper treats fuzzy project scheduling, this heuristic metology gives good enough results~Alvares-Valdes and Tama1989; Davis 1973; Davis and Patterson 1973!. As a prerequisite tthe RCPSP, a calculation of the activity network as per thecedure is given above, i.e., the resource constraints are omThey are then added in the following way:(aj PPr j

l <Rl , ; l ,whereP is the set of the activities in progress at a given timer j

l

is the amount of thel th resource needed for performing activaj , andRl is the availability of thel th resource over the givetime period.

The RCPSP calculations are similar to the ones just gBecause of limited resources, it may be possible for just a sof the eligible activities to be scheduled. In such situations,ority rules are used to make a selection among all the eliactivities. Alvares-Valdes and Tamarit~1989! showed through aactivity network benchmarking that the combinations ofgreatest-rank positional weight~GRPW! added to the latest statime or GRPW added to the most total successors lead to afactory optimization.

Therefore, we can observe that a limited number of arithmcal operations are required for resolving the project-schedproblem, and the RCPSP including sum, difference, maximminimum, and ranking.

Fig. 1. Fuzzy versus

Fig. 2. Trapez



.

Some Definitions From Fuzzy-Set and PossibilityTheories

Most of the results given hereafter can be found in most otextbooks addressing fuzzy arithmetic or fuzzy logic~Dubois andPrade 1988; Klir and Folger 1988!. In the crisp-set theory, aelementx does or does not belong to a setX. The membershifunction mX(x) equals 1 if xPX, and zero otherwise, i.emX(x)P$0,1%. In the fuzzy-set theory, an element may moreless belong to a set:mX(x)P@0,1#. Fuzzy numbers and fuzintervals are fuzzy sets over the real lineR. Some characteristiof fuzzy vs. crisp numbers and intervals are shown in Fig. 1

Throughout this paper, and for computational efficiency,trapezoidal notation is used. A fuzzy interval is defined fromcrisp numbers xL0 , xL1 , xR1 , and xR0 as follows: x5^xL0 ,xL1 ,xR1 ,xR0& ~Fig. 2!. Additional definitions such aa-cuts are given in Fig. 2.

Addition and Subtraction

The most often used formulas for adding or subtracting fintervals x5^xL0 ,xL1 ,xR1 ,xR0& and y5^yL0 ,yL1 ,yR1 ,yR0& arethe following:

x1 y5^xL01yL0 ,xL11yL1 ,xR11yR1 ,xR01yR0&

x2 y5^xL02yR0 ,xL12yR1 ,xR12yL1 ,xR02yL0&

numbers and intervals

notation;a-cuts

crisp

oidal


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These two formulas are known to be pessimistic. To avoidinflation of the imprecision, the following operators were pposed:

x1 y5^xL11yL12@~xL12xL0!q1~yL12yL0!q#1/q,xL11yL1 ,

xR11yR1 ,xR11yR11@~xR02xR1!q1~yR02yR1!q#1/q&

x2 y5^xL12yR12@~xL12xL0!q1~yR02yR1!q#1/q,xL12yR1 ,

xR12yL1 ,xR12yL11@~xR02xR1!q1~yL12yL0!q#1/q&

For q.1, the two operators give more optimistic results. In sways, they average the imprecision. We are convinced thadecision-making context, the imprecision of a result~a sum or adifference for instance! shall have at least the imprecision ofmost imprecise of its operands. This is why we are proposinfollowing alternative operators. Letz5^zL0 ,zL1 ,zR1 ,zR0& be theresult of x1 y and w5^wL0 ,wL1 ,wR1 ,wR0& the result ofx2 y.

zL05zL12max$xL12xL0 ,yL12yL0%3c2

zL151

2~xL11xR11yL11yR1!2max$xR12xL1 ,yR12yL1%3c1

ZR151


zR05zR11max$xR02xR1 ,yR02yR1%3c2

wL05wL12max$xL12xL0 ,yR02yR1%3c2

wL151


wR151


wR05wR11max$xR02xR1 ,yL12yL0%3c2

Interesting results can be obtained withc151/2 andc251.

Maximum and Minimum

The most often used formulas for fuzzy maximum and minimoperations are the following:

max$x,y%5@max~xL0 ,yL0!;max~xL1 ,yL1!;max~xR1 ,yR1!

max~xR0 ,yR0!#

min$x,y%5@min~xL0 ,yL0!;min~xL1 ,yL1!;min~xR1 ,yR1!

min~xR0 ,yR0!#

These two operators also give quite pessimistic results, butoperands are fuzzy intervals with similar imprecision ratios,precision can be kept within acceptable limits.

Ranking

Let x5^xL0 ,xL1 ,xR1 ,xR0& and y5^yL0 ,yL1 ,yR1 ,yR0& be twofuzzy intervals. IfxL0<yL0 , xL1<yL1 , xR1<yR1 andxR0<yR0 ,the ranking ofx and y is straightforwardy is said to be stronglgreater thanx. If one or two of these four inequalities is/aretrue, the strong comparison rule must be abandoned to the atage of the so-called weak comparison rule~WCR!. Several pro
posals have been made to address this issue including WCR using


-

area compensation, barycentre comparison, measures of poties, and of necessities. The WCR that uses the middle ofa-cuts,proposed by Dubois, Fargier, and Fortemps~2000! seems to bthe most efficient one. It also has the advantage of the lineThis method associates to each of the fuzzy intervals to bepared, a crisp quantity that is obtained from the middle ofsegments generated bya-cuts ~Fig. 3!.

x°F~ x!51

2 E0

1

~aL1aR!da

Also, when the fuzzy intervals to compare are trapezoidal ivals

F~ x!51

4~xL01xL11xR11xR0!

Fuzzy ÕPossibilistic Project-Scheduling Problem

Over the last 20 years, two dozen articles or so tried to adthe fuzzy project-scheduling problem. Three different axesbeen investigated so far~Galvagnon 2000!.• The membership level of an arc to a graph has been co

ered. In an activity-on-arrow formulation, it is the existencan activity which is considered, while in a activity-on-noformulation, it is the existence of a precedence constraint.few papers address this issue. There is only one by DuboiPrade~1978! and a second by Klein~1988!.

• The project-scheduling problem can also be seen as astraint satisfaction problem. In that case, the aim is to calca satisfaction level of an activity network~Fargier 1994!.

• The imprecision associated with the quantitative informawhich characterizes an activity network can also be lookefor instance, the activity duration, the work load required,resource availability, fixed dates and soon.

The latter axis is by far the one that has been investigatemost. The most significant contributions are numerous; thebriefly described hereafter. The chronological order of publicais used.

Chanas and Kamburowski

The two writers are the very first who processed the proscheduling problem in a fuzzy environment~Chanas and Kambrowski 1981!. The criticisms made by several writers~Kotiah andWallace 1973; McCrimmon and Rayvec 1964; Parks and Rsing 1969! toward the PERT probabilistic approach motivatedpublication of this paper. Chanas and Kamburowski featurmethodology for addressing the fuzzy project-scheduling prothat slices the activity duration given as fuzzy intervals, ina-cuts.

Fig. 3. Comparison rule that uses middle ofa-cuts

The project-scheduling problem is then solved for eacha-cut


004.130:114-123.

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level. Hence, it is crisp quantities that are handled. A recombtion of all these calculation results gives a fuzzy solution toproject-scheduling problem.

Dubois and Prade

The proposal made by these two writers~Dubois and Prade 198!is certainly the most significant. It is also the most straighward. They have also considered that, by definition, activityration is imprecise, and that this information must be estimusing fuzzy quantities. Further on, they used trapezoidal intefor modeling fuzzy duration. The methodologies used for solthe project-scheduling problem and the RCPSP are very simithe ones recalled above. They differ by the fact that addisubtraction, maximum, minimum, and ranking have beenplaced by their fuzzy equivalents. Gazdik’s~1983! and Wang’s~1999! papers feature methodologies similar to that of DuboisPrade.

Lootsma

After comparing the benefits of possibilistic schedulingproaches with respect to probabilistic ones, and after recallininconsistency of building a probabilistic activity netwoLootsma ~1989! proposed a comparative analysis of themethodologies through an example. From it, he observed thtotal project makespan is shorter when the possibilistic appris used. The dispersion over the project termination date isgreater.

Nasution

In their approaches, none of the writers cited previously mtioned the phenomenon that may occur at the backward cation of the activity network. Nasution~1994! addresses it througan example that is recalled hereafter. LetG be a project made otwo activitiesa12 and a23, which can be performed in parallBecause the writer prefers an activity-on-arrow formulatiodummy activitya13 is introduced. Lett j and t j be the early anlate date associated with the eventej . Exceptt 1 , which is a crispdate t 15t15^0,0,0&, all dates and duration of this microprojeare expressed with triangular fuzzy numbers. Let us say thad13

.d23, then t 35max$t11d13,t11d23%5d23. e3 is the termina

Fig. 4. Estimating fu

event, then prior to the backward calculation of the activity net-



work: t35 t 3 and t25 t32d135d132d23. If d135^xL0 ,x1 ,xR0&and d235^yL0 ,y1 ,yR0&: t25^xL02yR0 ,x12y1 ,xR02yL0&. Also,becausea13 is a dummy activityt15 t2 . The expression showthat, even ifx1 andy1 are positive numbers,t1 can be a negativfuzzy number. This phenomenon, also mentioned by Galvaet al.~2000!, comes from the fact thatx1 y2 yÞ x, x andy beingtwo fuzzy numbers. Two proposals have been made to avoidifficulty. Nasution proposes to replace~in the backward calculation! the ordinary fuzzy subtraction operator by another oncalled interactive fuzzy subtractor that makesx1 y2 y5 x. Hapkeand Słovinski ~1996a! propose to use the optimistic fuzzy opetor ~see previous section!.

It should also be mentioned that the core of Nasution pconsists of symbolic calculations, with perhaps some intefrom the academic point of view, but due to their complexity, wvery few interests to the practitioner.

Geidel

To respect the chronological order, we have to mention Geworks ~Geidel 1988, 1989!. This writer proposes the use of msures of possibility and necessity for a better ranking of activ

Lesmann et al.

This approach~Lessmann 1994! is similar to the one of Duboand Prade. They also noted the phenomenon depicted prevTo address the latter, they made two proposals. They arguein most construction projects, the project due date is specifithe contract. Hence, backward calculations can be initiated frcrisp date. Otherwise, because they preferred to use polynfuzzy numbers, they suggested transferring the earliest dateterminal milestone to the latest date as follows:tv5tv5x1 as-suming tv5^xL0 ,...,x1 ,...,xR0&. A methodology for estimatinfuzzy numbers is also given~Fig. 4!.

Hapke and Słovin´ ski

Most of the papers addressing the fuzzy project-schedulinglem are from Hapke and Słovin´ski together with few othe~Hapke and Słovin´ski 1993, 1996a,b; Hapke et al. 1994, 19971998; Hapke 1995!. Their research involves the RCPSP w

ration~Lesmann et al.!
zzy du
fuzzy data. To solve this problem, they investigated the use of


004.130:114-123.

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several heuristics and metaheuristics such as the Pareto-simannealing, the light beam search and so on.

Galvagnon et al.

Their works were motivated by the fact that no satisfactoryposal had been made to tackle the backward calculation proin a fuzzy project-scheduling environment. Even though tframework addresses this more accurately, no solution cafound in a polynomial computational time unless the activitywork is modeled as a series-parallel graph, which is quitestrictive simplification~Galvagnon et al. 2000; Galvagnon 200!.Despite the analytical rigor and interest of this proposal, it ilow interest to the project practitioner.

Chanas and Zielin´ ski

The two most recent papers by these writers are of great in

Fig. 5. Linguistic

especially because no one took care of the problem associated



dwith the determination of critical paths in a fuzzy environmbefore. In their first paper on this issue~Chanas and Zielin´ski2002!, they demonstrated that• The problem that consists in finding a critical path~and dem

onstrating its criticality! is an easy problem that can be solin a polynomial computational time.

• The problem of searching for all of the critical paths ofactivity network can only be solved in an exponential comtational time.

In their second paper, Chanas and Zielin´ski ~2001! propose twoalgorithms for determining the so-calledf-criticality index of apath.

Temporal Linguistic Descriptors

Linguistic descriptors are certainly among the very interesfeatures provided by the fuzzy-set theory. The relationship

riptors of duration
desc
can find between everyday speaking for describing things and


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situations and their mathematical equivalent is through thecalled linguistic descriptors~LD!. We are making some proposto associate trapezoidal fuzzy intervals to temporal LDs, i.e.,of dates and LDs of duration~Figs. 5 and 6!.

The benefits of gathering information in a fuzzy projescheduling environment are straightforward. For instance, whproject planner asks an engineer for the time he needs to comthe writing of a technical specification, an answer ofabout 1 weedoes certainly not mean exactly 5 working days or 40 worhours. If nothing happens, the specification can be issued witdays. On the contrary, if the completion of the specificationcomes slightly more difficult than foreseen, then its durationequal 6 or 7 days. This is just a matter of precision~or of impre-cision! of the estimation. In hindsight, the engineer considersin both cases the precision he has given is accurate enoughpurpose.

One can also apply modifiers to this imprecise duration.A bigweekis slightly longer than1 week, and approximately 1 monthis

Fig. 6. Linguist

more imprecise than1 month~Figs. 5 and 6!.



Unfortunately, the multiplication of two trapezoidal fuznumbers or intervals is a problem that is not as trivial as thethat consist of adding or subtracting such numbers. The prof trapezoidal fuzzy interval is not a trapezoidal interval! Forremainder of this paper, the multiplication of crisp numbertrapezoidal fuzzy numbers is sufficient. For instance

2 weeks5231 week523^3,4,6,7&5^6,8,12,14&

The multiplication is also required when resources arevolved. We limit ourselves to crisp resources allocation and aability. It should be noted that Galvagnon~2000! has made proposals for using fuzzy intervals for describing workloadsresource-constrained project-scheduling environment.

Fuzzy Activity-Network Calculations

The methodology proposed here for resolving the resoconstrained fuzzy project-scheduling problem is made of se

criptors of dates
ic des
phases.


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Phase 1

The transformation of the activity-network information gathewith temporal linguistic descriptors into trapezoidal fuzzy invals is a prerequisite. This first phase straightforwardly asswhat has been said previously.

Phase 2

The second phase aims to identify critical activities. Becaussearch of critical paths may sometimes be a problem that cbe solved in a polynomial computational time, the following pcedure is proposed to approximate criticality indices of activi

The fuzzy activity network is transformed in a crisp netwby considering the most optimistic duration of each activity,dj5dL0 j . This optimistic network is then calculated~forward andbackward processes!. At the end of this first treatment, critical anoncritical activities can be distinguished. Let us callC the set othe possible critical activities. These calculations on crisp actnetworks are reiterated, considering the most optimistic dur(dj5dL0 j) for the activities that belong to the setC and the mospessimistic duration (dj5dR0 j) for those that are not in this sAt the end of each iteration, newly critical activities are appento C. The process is stopped when the cardinality ofC has become constant.

This algorithm is bound to be processed in a polynomial cputational time because the number of iterations is obviouslythan the number of the activities of the network. Let us calln thetotal number of iterations, andpj the number of iterations at thend of which the activityaj is found critical. The criticality indeCI j can be appraised as follows:CI j5pi /n. If CI j51, thismeans that activityaj is always critical, as per the global impcision associated with the activity network. If 0,CI j,1, the activity aj can be critical in some configurations, and ifCI j50, onecan conclude that there is no way to have the activityaj critical.One of the interesting features of these criticality indices isthey can be ranked.

Phase 3

Up to that phase, resources have not been taken into accoschedule calculations. The procedure for calculating a resoconstrained fuzzy activity network is similar to the RCPSP u

Table 1. Input Data to Numerical Example

Activity ID Temporal LD G21 G

a — — A, IA 3 weeks a B, EB About 1 month A CC About 1 month B DD 2 weeks C, F LE 3 big weeks A FF About 3 weeks E D, GG About 3 weeks F HH About 2 months G v

I About 3 weeks a JJ 2 small weeks I KK 2 big weeks J LL About 6 weeks D, K v

v — L, H —

priority rules with the following differences:



• Crisp arithmetic operations are replaced by their fuzzy eqlents,

• The ranking of eligible activities is made with respect tolate start time (t j2d j). The CIs are then used for discrimining equalities. Other ranking combinations can be envisaand

• To avoid imprecision propagation, we propose thatt v is trans-ferred totv as follows:

tv5 K tL11tR1

2,tL11tR1

2,tL11tR1

2,tL11tR1

2 LPhase 4

It is of interest to any project practitioner to know the critactivities of a project. The critical activities ofC that have beeidentified in Phase 2 of the present scheduling methodologynot reflect the results of Phase 3 calculations. Hence, it is wrelaunch a process similar to the one of Phase 2. In order to uidentical algorithm, pseudo-precedence constraints can bebetween activities that share identical resources.

Phase 5

Despite the fact that attenuated arithmetical operators areused all along this process, the transformation of the calcudates (t j and t j) into temporal LDs is not straightforward. Wpropose to use the means (t j), associated with an imprecisirange (Dt j) to make the following transformations:

t j51

4~ tL01tL11tR11tR0! j and t j5

1

4~tL01tL11tR11tR0! j

Dt j51

6~ tL012tL122tR12tR0! j and

Dt j51

6~tL012tL122tR12tR0! j

If Dt j,5 days (Dt j,5 days!, the datest j ( t j) can be expressed in weeks. One can easily find the week number assoto t j ( t j), and these figures serve as a basis for the correspotemporal LDs. For instance, ift j5November6, 2005, then,week

˜

Resources dL0 dL1 dR1 dR0

— 0.0 0.0 0.0 0.0R1 9.0 12.0 18.0 21.R2 10.0 15.0 20.0 25.R3 10.0 15.0 20.0 25.R3 6.0 8.0 12.0 14.R2 12.0 15.0 18.0 24.

R1,R3 3.0 7.5 22.5 37.R2 3.0 7.5 22.5 37.R2 20.0 30.0 10.0 50.R1 3.0 7.5 22.5 37.R1 4.0 8.0 10.0 12.R1 8.0 10.0 12.0 16.R1 6.0 15.0 45.0 54.— 0.0 0.0 0.0 0.0

No. 45can be used to expresst j .


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If Dt j>5 days (Dt j>5 days!, the datest j ( t j) should beexpressed in months. Similarly, the day of the month,qj , caneasily be found for each datet j ( t j). In the present methodologfive cases may be distinguished.

if qjP@1,6# then t j5beginning of month

if qjP@7,12# then t j5first half of month

if qjP@13,19# then t j5middle of month

if qjP@20,25# then t j5second half of month

if qjP@26,31# then t j5end of month

For dates that have an imprecision range lower than 5i.e., expressed in weeks, it appears useless to state this inftion. If Dt j>5 days orDt j>5 days, this information is of inteest. We propose these imprecision ranges in weeks, andthem to the closest integer. Let us use againt j5November6,

2005. If Dt j521.6 days for instance, thent j5beginningNovember 2005,64 weeks.

Numerical Example

A network made of 12 activities has been used as a numexample. Table 1 gives information on input data, such as acduration ~expressed in the form of temporal LDs!, predecessoand successors, and the corresponding trapezoidal intervalciated with the activity duration~Fig. 5!. The results of the computations are in Table 2.

Conclusions

After about 20 years of maturation in the academic spherebelieve that fuzzy project scheduling is mature enough to bein real-world projects. If classical deterministic approachesas CPM remain the most valuable for small projects, large-industrial or construction ones may benefit from fuzzy projscheduling approaches. The methodologies one can find iliterature on this issue are quite theoretical and difficult to im

Table 2. Results of Computations

Activity ID

Criticality index Earliest da

Withoutresources

Resourcesconstrained tD tF

a 1 1 Week 0 WeeA 0.7 1 Week 0 WeeB 0.3 0.3 Week 9 WeeC 0.3 0.3 Week 13 WeeD 0.3 0.3 Week 18 WeeE 0.3 0.7 Week 4 WeF 0.3 0.3 1st half February 1st haG 0.3 0.3 1st half March EndH 0.3 0.3 End March Mid-I 0.3 0.3 2nd half January Mid-FJ 0.3 0.3 Mid-February End FK 0.3 0.3 1st half March EndL 0.7 0.7 Mid-April End Mv 1 1 End May End

ment. The approach we propose in this paper is pragmatic enough



-

-

to be understood by project-scheduling practitioners, and thbe implemented in real-world projects. Among the difficultiescould face, the gathering of imprecise data in the form of funumbers or intervals could constitute a restraint to the impletation of fuzzy project-scheduling approaches. We are nowvinced that this obstacle is removed thanks to the use of temLDs. This paper aims at making the link between researcherpractitioners. Feedbacks from field implementations are nowquired for making research on this issue advance. One oissues pending is the tuning of temporal LD definitions.

Acknowledgments

The writers’ thanks go out to all those who helped us collecthe literature that has been published on the fuzzy proscheduling problem especially Didier Dubois, Vincent Galvnon, Joachim Geidel, and Professor Michael Oberguggenbe

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