New Fuzzy GERT Method for Research Projects Scheduling M. H. Karimi Gavareshki

Department of Industrial Engineering and Management, Malek Ashtar University of Technology, Tehran, Iran

Email: mh-karimi@cic.aut.ac.ir

Abstract-The aim of this paper is to present a new applicable and suitable technique for research projects scheduling. Research projects have special characteristics that separate them from other projects. These include uncertainty in definition, sequence and duration time of activities; uncertain results; having loops and cycles; and no repetitive activities, So there is B need for special tools and techniques for scheduling these projects. Conventional scheduling techniques are not able to model research projects or have a lot o f errors. Fuzzy GERT networks using capabilities of GERT and FUZZY, is powerful tools far scheduling research projects. This paper presents B new method far solving fuzzy gert nehvorks. I n this method, activity duration time and loops trapezoidal fuzzy numbers states repetition number, and output activities from nodes of network belong to a furzy set. In this method, the Iimt time gert networks Computstioos are done based an evaluating nodes. Process outputs are scheduled network and project fuzzy completion time. These outputs are fuzzy numbers and can be analyzed by (I - cuts. This method is more applicable and needs less computation than existing fuzzy and probability gert methods. Besides, this method can get computerized and up to date easily. This method can be also applied for cost estimstons in research projects. For checking validation, we compared this method with existence fuzzy and probability gert methods, the results were very near to each other. We also applied it in an electronic institute. The result^ were more realistic than the results of their pervious method (CPM).

Keywordr - research project, scheduling, fuzzy GERT, fuuy numbers

1. INTRODUCTION

Research projects have special characteristics that separate them from other projects. These include uncertainty in definition, sequence and duration time of activities; uncertain results; having loops and cycles; and no repetitive activities, So there is a need for special tools and techniques for scheduling these projects. So far different methods such as heuristic techniques (resource assignment), CANT chart, network analysis techniques, mathematical methods, simulation for solving problems of project scheduling have been presented but totally network analysis techniques are effective for solving complicated problem of project scheduling. Totally project scheduling network techniques are containing conventional deterministic and stochastic network and fuzzy networks techniques. Deterministic network techniques (CPM) that using them for simplicity is very comertial can not consider uncertainty of definition and time activities and cycles existence. In this cases

Deterministic network techniques (CPM) have a lot of error and that is bener being used other techniques. Results of stochastic network techniques (as GERT, PERT) are better than deterministic techniques but inexistence of analytic solution techniques in GERT network with composition different nodes are being of using difficulty this technique in R & D project scheduling. in fuzzy network techniques, project network parameters and relation between them are fuzzy. Reasons of using fuzzy in projects scheduling contains: uncertainly in activities, sequence and definition, uncertainty in time estimate of project activities, subjective view of expert, less necessity fuzzy procedures to information than probability methods, less computation of fuzzy methods than probability methods. These techniques with a view of using kind were divided to three groups: fuzzy time, network with fuzzy parameters and fuzzy GERT. In technique with fuzzy time (fuzzy CPM), only time parameter of project network is being f u y and another parameters of project network means activities sequence and definition are certain and determinstic. Prade [2]and CHANAS[I] were first person that used fuzzy in project scheduling and in their maters only time parameter considered furzy. Later on many atticles was published about using of fuzzy in scheduling that has been more complete gradually[3],[4],[5][6][7]. This technique can’t demonstrate uncertainty of activities sequence and definition and existence of loops in R & d scheduling. In network technique with fuzzy parameters, activities definition, sequence and time parameters can be fuzzy hut in this techniques can’t consider cycles, mean while, suitable method haven’t been presented for solving them. Only method have been presented in 1989 by Maris[S]. Fuzzy GERT network were the same probability GERT network that fuzzy parameters substitute probability parameters. This network using capabilities of GERT networks and fuzzy set are powerfull tools for R & d scheduling. This technique with consideration more charactristices of R & d projects are giving more actual results than mentioned techniques. Up to now two method fuzzy GERT have been presented (Cheng method and Itakura method).Cheng method[lO],[l I] have investigated special kind of fuzzy GERT networks (exclusive or nodes) that can’t use for R & d scheduling projects. Fuzzy GERT method (Itakura and Nishikava) [9]can use R & d project scheduling, but it has some restriction and complexity problem. In attention to existent methods problems, and to target having applicabel method and computerized capability, new method have been planed for solving this network.

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0-7803-8519-5/04/$20.00 @ 2004 IEEE

Kind ofside

EXCLUSIVE- OR

INCLUSIVE- OR

AND

Kind ofside

deterministic

Different COI

Symbol Property Doing each input activities causes releasing node. Also only One input activiry must he done.

Doing each input activities E ~ U I C S

releasing node. Releasing time of node is equal to minimum of ending time of input activities.

Doing all input activities is

Releasing time of node is equal to maximum of ending time of input activities.

4

a for releaSing node.

I I I TABLE 2

node must be done.

are belonging to a fuzzy set. Each activity has a membership degree.

I I )inations of nodes:

In this networks, fuzzy branche is replaced probability branche. Each branch is characterized with membership degree ( w - ~ ) and fuzzy duration time( tj.j ). tj.j

- - - f - , P!.j

K3 Firlire 1 -fiiim hrmch

Loops are activities that are repeated for one or more times. Each loop is characterized with membership degree or occurrence possibility( ,UL )and f u u y repetition number( r, , ).

- 7

- 2 J

Fiewe 2 -fuzzy l m o

4ssumption: In this method, we consider following assumption for simplicity and applicability, but this method easily can be generalized I- Duration time of activities is represented by triangular or trapezoidal fuzzy numbers. 2- Repetitions number of loops is represented by triangular or trapezoidal fuzzy numbers. 3- Membership degree of activities and loops is a number between o, I . 4- occurrence possibility of loops for different repetitions is equal. 5- We used fallowing fuzzy relations in ow

A, = [ ~ : . ' , n : " ' ] = [ ( b - a ) a + a + ( b - c ) a + c ] a ~ [ O , i ] (6)

: ifh node : activiry I-j

,U,-j : memhsmhip degree ofactivity 1-j

f - j : h y dmtion time of activity I-j -

loop n to I ,UL, : membership degree ofloop L "i I r, : Fuzzy repetition number

pi : preeedencses nodes(activities)

fuzzy ending lime of activily i-j fi;.,: : membership degree of ending time

K,-,

' Initial release time of node S F < '

MTi - . average time of node

' membership degree of node p ui,

Ill. METHODOLOGY In attention to existent methods problems, and to

target having applicabel method and computerized capability, new method have been planed for solving this networks. For first time in this method, GERT network computations like fuzzy CPM method ( forwarding computation). is performed based on nodes . In this method, nodes were evaluated from start node to end node. Nodes evaluating was doing based on input and output

activities to every node. initial release time of node (ST. ) is time that node with attention to input activities to node was releasing. If output side of node has been loop, Namely have existed r e m possibility, loop time would

-

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have been increased to release time of node. Therefore, we were defining for every node another parameter under the title of average time of node ( ~ ) that was being

indicative of average possible times of being released nodes. Steps of algorithm are :

MT.

1- For start node set

ME,*,, = SF,<,, = (O,O,O) (7) 2- For each node from sfan to end Comput average time of

Figure3- EXCLUSIVE-OR NODE

2-1 Computing ending time and membership degree of precedence activities(far input activities to node):

= MT 8 T." Vi E P, (8)

PR." = m4P<.n >PM 1 Vi E P" (9) -

2-2 Computing initial release time ofnade (ST )

initial release time of node is calculated based on kind of

input side

a) EXCLUSIVE-OR NODE

normalization of membership degree

3- Computing average time of node ( - ) MT i

4- Computing project completion time With evaluating network nodes from first node to end node, project network is scheduled and project completion time is obtained that is equal with average time of end node of project network.

I

T,, = M T d Since our input parameters are trapezoidal and triangular fuzzy numbers, project completion time and also nodes will be trapezoidal and triangular fuzzy number. Now with using of a-cuts operation and geometrical center of trapezoidal and triangular fuzzy numher(defuzzification) can analyze result of scheduling . if project completion time have been triangular fuuy number (a,b,c), a can be considered as risk level and project manager can compute and analyze time arithmetic of project completion at different risk levels(6). Also can get project completion time average with computing of geometrical center of trapezoidal and triangular fuzzy numher(deiiuzification) that is a certain number(5).

t

FigureC a-Cut

Nat.l:This method can be developed for cost estimation in R&D projects. If cost of project is based an activity cost (Activity base costing) .in this method cost of activity can be replaced by duration time.

IV. validity with practical and theoretical example For testing validity new method,we compare it with

existent methods. For this was selected some theontical and practical examples and have solved with new method and existent methods of probability GERT and fuzzy GERT and have compared results together that result were near to each other. In addition to this we exploit new method of fuzzy GERT at an research center that scheduling result with new method was nearer to reality in

c) AND NODE comparison to conventional method CPM at research center.

&GP"

b) MCLUSIVE- OR NODE

ST, = mKx cfil. 1 i'P"

- (13)

= min b,-. 1 (14) i'P"

a) Comparing with probability GERT method S E =&JE.. } (15) iSP"

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In this section we consider example of probability GERT network that is connection to R & d project and have been acquired from [I21 source with a few changes in it's data. For solving of this example with new method of fuzzy GERT, we approximate input parameters of probability network to fuzzy data and solved problem. project completion time obtained by two method is following : fuzzy time of project completion

T,, = MT, = (1 1.4,11.8,20.9) - - probability time of project completion . _

6(s)=4.1 E(s) = aM~~(s ) ls=o = 11.74

nrohahlitv

fuzzy

0 4.4 11.7 20.9 24 11.8

Figure 5- fizzy and probanlity time

As was being obsewed scheduling result of two method have nearly been equal , but few difference will be natural , because data weren't equal and have been approximated. h)Comparing with ITAKURA method

We solved problem of ltakura [9] with changing data to triangular fuzzy numbers that results is presented below figure.

~~~

new methid ~

I , IS ,I ,. ,, ,. ,> ,I I. m 2, n

t h e L - ~ _ _

Figure 6- fumy time of itakura and new method

- I

T,, =MT, =(11.8,15.6,19.8)~(12,16,20)

c) Scheduling of sample R & d project: In this section one of R & d project as making a plant

that have been scheduling by conventional method (CPM)

have scheduled with new method of fuzzy GERT and were comparing. For scheduling of this project with new method of fuzzy GERT, were doing / process below serialization:

1) qualitative description and drawing of GERT network and appointment necessary parameters of network. project GERT network was getting with understanding of information from project manager that have been showing in figure8 andtahel3. 2)solving project fuzzy GERT network that is evaluating nodes (part of computaions have been shown in tahel4). 2) Computing of project completion time :With

evaluating end node also project completion time is getting( figure7).

1 7 5 ion

Figure 7- fuzzy completion time ofproject

Figure 8- gen network of project

Tabel 3- parameters of fuzzy serf network

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Tabel4- evaluating nodes of fuzzy gen network

Reality time of project ending is 12 month. Result of project furzy completion time with CPM method is 3.5 month. result of project fuzzy completion time with new method of fuzzy gert is equal to (4.5, 6.5,IOS)month . As was being observed result of new method have near to reality ending project rather than CPM method.

V. CONCLUSION Fuzzy gert network with regarding to gert capabilities (using of logical nodes and branches and loops)in modeling of research projects and fuzzy ability for uncertainty of project parameters(time , activity definition and sequence)are suitable especially for R&D projects scheduling. Also fuzzy gert can be used in other content such as inventory control, reliability and etc. Fuzzy GERT network is the same probability GERT network that fuzzy parameters have replaced probability parameters and were composed from 3 part: logical nodes, fuzzy branches and loops. For first time in this method, GERT network computations is performed based nodes and resembling of fuzzy CPM method ( forwarding computation). In this method, nodes were evaluated from start node to end node. Nodes evaluating was doing based on input and output activities to every node. Process outputs are scheduled network and project fuzzy completion time. These outputs are fuuy numbers and can he analyzed by a - cuts. This method is more applicable and needs less computation than existing fuzzy and probability gert methods. Besides, this method can get computerized and up to date easily. This method can be also applied for cost estimations in research projects. For checking validation, we compared this method with existence fuzzy and probability gert methods, the results were very near to each other. We also applied it in an electronic institute. The results were more realistic than the results of their pervious method (CPM). Future development can be include generalizing method for accounting different fuzzy numbers and t-norms and s- norms and fuzzy ranking method and defuzzification and find best way for those. Also participating probability in this method with fuzzy and accounting fuzzy interval for estimation membership degree can he future development.

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