research article developing a mathematical model for...

Research ArticleDeveloping a Mathematical Model for Schedulingand Determining Success Probability of Research ProjectsConsidering Complex-Fuzzy Networks

Gholamreza Norouzi Mehdi Heydari Siamak Noori and Morteza Bagherpour

Department of Industrial Engineering Iran University of Science and Technology Narmak Tehran 1684613114 Iran

Correspondence should be addressed to Gholamreza Norouzi peyghogmailcom

Received 23 January 2015 Revised 25 June 2015 Accepted 25 June 2015

Academic Editor Ching-Jong Liao

Copyright copy 2015 Gholamreza Norouzi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In projectmanagement context timemanagement is one of themost important factors affecting project successThis paper proposesa newmethod to solve research project scheduling problems (RPSP) containing Fuzzy Graphical Evaluation and Review Technique(FGERT) networks Through the deliverables of this method a proper estimation of project completion time (PCT) and successprobability can be achieved So algorithms were developed to cover all features of the problem based on three main parametersldquoduration occurrence probability and success probabilityrdquo These developed algorithms were known as PR-FGERT (Parallel andReversible-Fuzzy GERT networks) The main provided framework includes simplifying the network of project and taking regularsteps to determine PCT and success probability Simplifications include (1) equivalent making of parallel and series branches infuzzy network considering the concepts of probabilistic nodes (2) equivalent making of delay or reversible-to-itself branches andimpact of changing the parameters of time and probability based on removing related branches (3) equivalent making of simpleand complex loops and (4) an algorithm that was provided to resolve no-loop fuzzy network after equivalent making Finallythe performance of models was compared with existing methods The results showed proper and real performance of models incomparison with existing methods

1 Introduction

Research and RampD projects are often conducted initially todesign andmanufacture a product with certain capabilities Amajor part of projects is conducted in research organizationsand institutes especially military ones These projects havecertain features including [1 2] (1) uncertainty in definingthe activities (2) unclear and uncertain results of activities(3) uncertainty in time and cost of activities (4) noniterativeactivities (5) loops and iterations to previous activities and(6) parallel paths Due to such features a certain techniqueis needed to deal with such projects under determined timeand cost effectively On the other hand backwardness andlongtime of undertaking project would lead into huge costsnot forecasted in initial budget plan and organizations mayface numerous problems

In practice the projects have been implemented throughan uncertain environment that ambiguity is one of the major

features of such environments Uncertainty may be con-sidered as a property of the system which is an indicativedefect in human knowledge towards a system and its state ofprogression [3] Introduction of uncertainty topic for the firsttime entitled probability and it was attributed to AristotleMeanwhile there are various types of uncertainty in the realworld including fuzzy and random and being uncertain thatincludes the former two items [4] Alongwith randomizationfuzziness is introduced as a fundamental type of subjectiveuncertainty [5]

In the early 20th century Henry Gant and FrederickTaylor introduced Gantt chart which showed start and endof projects However overall timescale of the projects basedon precedence relationships and analyses was not possibleby this technique Therefore network designing in projectcontrol and design to remove raised problems and providinga more comprehensive technique was developed by a groupof operation research (OR) scientists in 1950 and different

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 809216 15 pageshttpdxdoiorg1011552015809216

2 Journal of Applied Mathematics

methods that is CPM (Critical Path Method) and PERT(Program Evaluation and Review Technique) were intro-duced and developed [6]

Project management in particular project planning is acritical factor in the success or failure of new product devel-opment (NPD) [7] and research and development (RampD)projects [8] Limited facilities of CPM and PERT methodsfor modeling of projects having complex and uncertainnetworks [6] caused the great scientists such as Pritsker andWhitehouse [9] and Pritsker and Happ [10] to introducethe probability GERT (Graphical Evaluation and ReviewTechnique) which is amethod consisted of flow graph theorymoment generation functions and PERT to solve problemswith probability activities Pritsker [11] developed GERTmethod and used it for different nodes and introduced useof simulation in that method Of course it is pointed outthat Pritsker and Whitehouse [9] used Maisonrsquos role whichhad been introduced in graph theory to solve probabilitynetworks The pointed out probability methods are all basedon probable distributions such as Beta (120573) or normal dis-tribution which are used for estimation of project activitiesperiod Thus to use probable distributions random samplesrepeatability and statistical deduction will be required

When fuzzy theory was introduced and developed byZadeh [12] scientistsrsquo and researchersrsquo attention in the fieldof network and science engineering was gradually caughtby the issue that they will be able to solve the problemof uncertainty in problems and also expressed problemsby a fuzzy approach Thus for the first time Chanas andKamburowski [13] introduced a new technique called FPERT(Fuzzy ProgramEvaluation and ReviewTechnique) in whichby using fuzzy theory the estimation of activities time andproject completion time had been shown by triangular fuzzynumbers Gazdik [14] presented another technique based onfuzzy sets and graph theory called FNET (fuzzy network)This is for estimation of activities time by use of algebraicoperators and by deductive method in the projects withoutdata Itakura and Nishikawa [15] applied fuzzy concepts inGERT networks for the first time and by replacing fuzzyparameters for probability parameters presented FGERT(FuzzyGraphical Evaluation andReviewTechnique)methodof solution is similar to probability GERT but it is differentbecause fuzzy theory was used to review GERT networksMcCahon [16] presented FPNA (Fuzzy Project NetworkAnalysis) technique to identify critical path and obtain totalfloat value and probability time for project completion Tenyears after presentation of his first measures on fuzzy GERTCheng [17 18] presented another method of solution Ofcourse this method was presented to solve problems of seriessystems reliability in which merely Exclusive-or nodes weredealt with Basis of his method is the same probable GERTbut it is only different in using fuzzy parameters instead ofprobable functions Nasution [19] dealt with developmentof FNET technique He solved fuzzy networks with themulticritical paths approach by using interactive differentialfuzzymethod in the network reversible computations Changet al [20] considered projects of large size planning networksWhen such projects are under uncertain conditions thenthe techniques presented to solve timed networks become

complicated and sometimes unsolvableThus they presenteda technique by merging comparison and composite methodsand also by using fuzzy Delphi which was efficient forsolving problems with the foregoing features Shipley et al[21] presented a technique called Belief in Fuzzy Probabilitiesof Estimated Time (BFPET) which is based on the fuzzylogic belief functions and fuzzy probability distributions anddevelopment principle Wang [22ndash24] dealt with develop-ment of fuzzy sets approach for planning product develop-ment projects of limited and imprecise data The approachhe adopted was in line with the works of Buckley [25] andTatish that is use of trapezoidal fuzzy numbers and theircomputational procedure Chanas and Zielinski [26] Chenand Huang [27] Chen and Hsueh [28] Fortin et al [29] andKe and Liu [30] dealt with the issue of critical path in thenetwork of projects scheduling problem that contains fuzzyduration times of activities All these studies represent activityduration times described by means of fuzzy sets howeverwithout any component of time dependency Huang et al[31] have studied the project scheduling problem in CPMnetworks with activity duration times that are both fuzzy andtime dependent

Some new analytical methods for determining the com-pletion time of GERT-type networks have been proposed byShibanov [32] Hashemin and Fatemi Ghomi [33] The mainproblem of these methods is high complexity of relationsand computations in a network without loops Kurihara andNishiuchi [34] solved probability GERT networks by MonteCarlo simulation and by this method they could analyzespecific kind of nodes present in GERT networks withoutloops (with loops [35]) Gavareshki [1] and Lachmayer et al[2 36 37] presented a new method to solve fuzzy GERTnetworks whose basis goes back to the method for solvingCPM networks that is moving forward and implementationof fuzzy literature and inference of the nodes Hashemin [38]provides an analytical procedure for this kind of networksby simplifying GERT networks Shi and Blomquist [39] dealtwith using matrix structures for solving time schedulingnetworks By combination of fuzzy logic and DSM (Depen-dency Structure Matrix) he could introduce a new approachfor solving problems on time scheduling of projects withuncertainty in periods of activities and uncertainty in over-lapping the activities Martınez Leon et al [40 41] presentan analytical framework for effective management of projectswith uncertain iterations in probable GERT networks Theframework is based on the Design Structure Matrix As theintroduced approachwas new it is capable of being developedin the fuzzy networks and uncertainty in the definition ofactivities

Usual techniques are not able to estimate project com-pletion time (PCT) of the projects that are executed for thefirst time or projects having computational problems [8 4243] Therefore only networking techniques with particulardefinitions on parallel nods and branches may be regressedand can create loops in fuzzy environment which provideproper timescale for these projects by using the capabilitiesof fuzzy sets and networks adapted to the attributes andfeatures of these projects Concerning the bottlenecks ofthese techniques a new method is developed to resolve

Journal of Applied Mathematics 3

the problems of these networks and to understand thembetter Thus the objectives of this research are

(1) to apply three parameters duration occurrence prob-ability and success probability for each of the projectactivities as well as use of trapezoidal fuzzy numbersto present duration and probabilities to occurrenceand success of activities

(2) to present methods for equivalent making of paralleland series branches in fuzzy networks consideringthe meanings and concepts of deterministic andprobabilistic nodes

(3) to provide a new mathematical model for removingdelay or reversible-to-itself branches in fuzzy net-works

(4) to provide a new mathematical model for removingsimple and subindependent complex loops in fuzzynetworks based on transformation to reversible-to-itself branches

(5) to provide an algorithm for estimating of PCT andscheduling of the simplified (no loops) fuzzy net-works based on equivalent making of parallel andseries branches

(6) to provide a new mathematical model for estimatingof the probability of project success

In Section 2 features of studied problem (PR-FGERT) aredescribed In Section 3 assumptions and parameters will bedealt with which we come across in the process of researchstages In Section 4 the analytical approach and proposedalgorithm are described In Section 5 a numerical case studyis given so that the algorithm efficiency is demonstratedIn Section 6 the proposed algorithm and solution modelvalidation have been presented Finally the conclusion andfuture research are presented in Section 7

2 Problem Definition and Features of StudiedProblem (PR-FGERT)

As mentioned in literature review the closest and the mostcompatible type of network to display RampD projects is GERTnetwork Due to limitation of accessing time information oneach activity we used times of trapezoidal fuzzy In most ofthe engineering applications trapezoidal fuzzy numbers areused as they are simple to represent are easy to understandand have a linear membership function so that arithmeticcomputations can be performed easily [44] These networkshave capabilities such as different types of nodes parallelbranches between two nodes (various ways to do an activityandor transition from one node to another one Table 3) theexistence of delay or reversible-to-itself branches as well assimple and mixed loops precedence relationships to severalnods and the existence of several end nodes to describe andto convert a project to a model properly

The main assumption of the research is the existence ofthree parameters (1199012

119894119895 1199011119894119895 119894119895) for each activity where

119894119895is

duration of doing the activities as a trapezoidal fuzzy number

1199011119894119895is occurrence probability and 1199012

119894119895is the probability of

ending an activity successfully (this value is independent ofother activities for different branches and activities)

Concerning GERT networksrsquo features with fuzzy timesas well as presented assumptions and parameters the first-time-executed project can bemodeled easily To resolve thesenetworks so that they have high performance in terms ofunderstanding deliverables and functionality an algorithmis provided to resolve the probability and fuzzy of thesenetworks by using an innovative technique It also provides anestimation of PCT along with the probability of its successfulending

3 Model Assumptions and Parameters

The assumptions and parameters used in this paper are asfollows

31 The Modeling Procedure Has Been Done according to theFollowing Assumptions

(i) Activity of the network has a single source and a singlesink node If several initial nodes and some end nodesexist they should be connected to a dummy node

(ii) Input side of a node was applied of And Exclusive-or and Inclusive-or types output side of a node wasapplied of deterministic and probabilistic types

(iii) Uncertainty in estimating duration of activities is ofpositive trapezoidal fuzzy number

(iv) Uncertainty in occurrence of activities is consideredof probable type

(v) Uncertainty in success of activities is considered ofprobable type

(vi) In the project activities network loops are considered(vii) Maximum number of parallel branches between two

nodes is three(viii) The number of loops iteration is uncertain(ix) Establishing the law of independence between exist-

ing loops in the network

32 Parameters The parameters are as follows

119883(119894-119895) output activity from node 119894 and input to node

119895119885119884119894 output set of activities from node 119894

(119894-119895) fuzzy duration of doing activity119883

(119894-119895)119889119905(119894-119895) defuzzy duration of doing activity119883

(119894-119895)

1199011(119894-119895) occurrence probability of activity119883(119894-119895)

1199012(119894-119895) success probability of activity119883(119894-119895)

ln(119900-119898) output loop from node (119898) and input to node

(119900)ln119894(119900-119898) loop of level 119894 output from node (119898) and input

to node (119900)


1199011ln(119900-119898) occurrence probability of loop ln(119900-119898)

1199012ln(119900-119898) success probability of loop ln(119900-119898)

119883ln(119900-119898) set of activities in loop ln(119900-119898)

ln(119900-119898) time value of loop unit ln(119900-119898)

1015840

(119894-119895) fuzzy duration of activity 119883(119894-119895) after impact of

first time delay loop

10158401015840


second time delay loop119878(119894-119895) set of activities in a path consisting of seriesbranches119879119904 fuzzy duration of equivalent branch to series

branches1199011119879119904 occurrence probability of equivalent branch to

series branches1199012119879119904 success probability of equivalent branch to series

branches119879119901 fuzzy duration of equivalent branch to parallel


parallel branches1199012119879119901 success probability of equivalent branch to paral-

lel branches

33 Fuzzy Operations Used in the Proposed Solution Algo-rithm Regarding

1= (1198971 1198981 1199061 1199041) and

2= (1198972 1198982 1199062

1199042) as two positive trapezoidal fuzzy numbers the fuzzy

operators are calculated as follows [44]

Addition oplus 1oplus 2

= (1198971+ 1198972 1198981+ 1198982 1199061+ 1199062 1199041+ 1199042)

Substraction ⊖ 1⊖ 2

= (1198971minus 1199042 1198981minus 1199062 1199061minus 1198982 1199041minus 1198972)

Multiplication otimes 1otimes 2

asymp (1198971lowast 1198972 1198981lowast 1198982 1199061lowast 1199062 1199041lowast 1199042)

ℎ otimes 2= (ℎ lowast 119897

2 ℎ lowast 119898

2 ℎ lowast 119906

2 ℎ lowast 119904

2)

Division ⊘ 1⊘ 2asymp (

1198971

1199042

1198981

1199062

1199061

1198982

1199041

1198972

)

max 1 2

= ((1198971or 1198972) (1198981or 1198982) (1199061or 1199062) (1199041or 1199042))

(1)

4 Analytical Approach and Methodology

41 Conceptual Model In this section new parallel andreversible branches in the fuzzy GERT (PR-FGERT) are

Table 1 Added time in other branches separating from node (119894) SeeFigure 11

Qyt of branch (119894119894)occurrence

Occurrenceprobability Added time

0 1 minus 1199011119894119894

01 1199011

119894119894sdot (1 minus 1199011

119894119894)

119894119894

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 2 sdot

119894119894

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 119899 sdot

119894119894

presented to solve the problem defined in Section 2 Inthe presented analytical approach the proposed algorithmconsists of several steps so that a graphical presentation isshown in Figure 1 described in the following subsectionsHowever five main steps of the algorithm are as follows

42 Development of Relations and ExecutiveStages of Algorithm

421 Equivalent Making and Removing of Reversible-to-ItselfBranches Given Figure 2 for this type of branches threeparameters of fuzzy duration (119905

119894119894) occurrence probability

(1199011119894119894) and success probability (1199012

119894119894) are considered Since it is

not clear how many times these branches should be repeatedto free the node and node output branches (119894) thus theapproach of elimination of this type of branches is defined asfollows these branches are considered as a random geometricvariable calledGe(119909 1199011

119894119894) where119909 indicates number of branch

repeats (119894119894) Thus to eliminate this type of branches both ofmean duration that can be added and change of occurrenceprobability of other output branches from the node (119894) shouldbe calculated

Ge (119909 1199011119894119894) = (1minus1199011

119894119894) sdot 1199011119894119894

119909

119909 = 0 1 2 3 (2)

(i) Computing the average of added time in other sepa-rating branches from node (119894) by elimination of branch (119894119894) isas follows

According to Table 1 mean duration added can be calcu-lated as follows

119905119894119895= (0 sdot (1minus1199011

119894119894)) oplus ((1199011

119894119894) sdot (1minus1199011

119894119894) otimes 119894119894)

oplus (2 (1199011119894119894)2sdot (1minus1199011

119894119894) otimes 119894119894) oplus sdot sdot sdot

oplus (119899 (1199011119894119894)119899

sdot (1minus1199011119894119894) otimes 119894119894) oplus sdot sdot sdot

=infin

sum119909=0

oplus ([119909 (1199011119894119894)119909

sdot (1minus1199011119894119894)] otimes 119894119894)

(3)

According to the [(1 minus 1199011119894119894) otimes 119894119894] term independent of 119909

too we should exit it from summation and we have

= (infin

sum119909=0

119909 sdot (1199011119894119894)119909

) sdot (1minus1199011119894119894) otimes 119894119894 (4)


No

Yes

Yes

No

Yes

No

Priority and classification of loops

Qualitative description of project and extraction of certain and uncertain activities

Drawing of project network based on deterministic and probabilistic nodes

Estimation of occurrence

Determination of type of loops existingin the fuzzy network

Estimation of fuzzy time of

Application of loop delay parameters inthe effectible branches

Extraction of fuzzy network without loops

Equivalent making of loops andcomputation of parameters

Equivalent making of seriesbranches and computation of

Equivalent making of parallelbranches and computation of

Estimation of project completion fuzzy timebased on Figure 5

Estimation of success

Are thereparallelbranches

Are thereseries

branches

Are thereloops

performing the activities (ti-j) probability of activities (p1i-j) probability of activities (p2i-j)

parameters (tT119901 p1T119901 p2T119901 )


(tln(119900-119898) p1ln(119900-119898)

p2ln(119900-119898)

)

Figure 1 A graphical presentation of the proposed approach

By exiting one 1199011119894119894from summation we have

= (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1

) sdot 1199011119894119894sdot (1minus1199011

119894119894) otimes 119894119894 (5)

We assume that

Ψ = (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1

)by expanding997888997888997888997888997888997888997888997888997888rarr [Ψ=

1(1 minus 1199011

119894119894)2] (6)

By substituting of (6) in (5) and by simplifying we have

119905119894119895=

1199011119894119894

(1 minus 1199011119894119894)otimes 119894119894 (7)

Therefore after elimination of node (119894) the fuzzy durationresulting from (7) should be added to the duration of alloutput branches from node (119894) in accordance with (8)

1015840

(119896)= (119896)oplus[(

1199011119894119894

1 minus 1199011119894119894

)otimes 119894119894] forall119896 isin 119885

119884119894 (8)


ii

i

f

r

k

J

Figure 2 A display of delay branches

(ii) Computing the change of success probability in otherseparating branches from node (119894) by elimination of branch(119894119894) is as follows

To study the change of success probability in otherbranches from node (119894) by elimination of branch (119894119894) assumethat branch (119894-119895) with parameters (1199012

119894119895 1199011119894119895 119905119894119895) is a branch of

node (119894) Then we have Table 2According to Table 2 success probability can be calcu-

lated as follows

11990121015840119894119895= 1199012119894119895sdot (1minus1199011

119894119894) + 1199012119894119895sdot (1199012119894119894) sdot (1199011119894119894) sdot (1minus1199011

119894119894) + 1199012119894119895

sdot (1199012119894119894)2sdot (1199011119894119894)2sdot (1minus1199011

119894119894) + sdot sdot sdot + 1199012

119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899

sdot (1minus1199011119894119894) + sdot sdot sdot

=infin

sum119899=0

1199012119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899

sdot (1minus1199011119894119894)

= 1199012119894119895sdot (1minus1199011

119894119894)infin

sum119899=0

(1199012119894119894)119899

sdot (1199011119894119894)119899

(9)

Now by expanding of (9) and simplifying it we have

997904rArr 11990121015840119896= 1199012119896sdot (1minus1199011

119894119894) sdot (1+

(1199012119894119894) sdot (1199011119894119894)

1 minus [(1199012119894119894) sdot (1199011119894119894)])

forall119896 isin 119885119884119894

(10)

So we can conclude that success probability of eachbranch from node (119894) is multiplied in fixed value (1 minus 1199011

119894119894) sdot

(1 + ((1199012119894119894) sdot (1199011119894119894))(1 minus [(1199012

119894119894) sdot (1199011119894119894)]))

(iii) Computing the change of occurrence probability inother separating branches from node (119894) by elimination ofbranch (119894119894) is as follows

In order to calculate sum of probabilities of branchesseparating from node (119894) after elimination of branch (119894119894)becomes one the occurrence probability of each branchshould be multiplied by value of

(1+1199011119894119894

1 minus 1199011119894119894

) (11)

422 Equivalent Making of Parallel Branches between TwoNodes According to assumption in present paper there aremaximum three 119886 119887 and 119888 branches with the fuzzy times of119886= (1198861 1198862 1198863 1198864) 119887 = (1198871 1198872 1198873 1198874) and 119888 = (1198881 1198882 1198883 1198884)

between two nodes in parallel state Then for all activitiesbased on the type of input and output nodes the followingstates (Table 3) could be assumed

(1) Parallel Branches between Two Nodes with ldquoProbabilis-ticrdquo Output and ldquoExclusive-Orrdquo Input In this state amongdifferent ways to do an activity only one way should beselected and implemented To achieve the time of doingan equivalent activity we used the average times based onoccurrence probability since only one event occurs in eachmoment and on the other hand the branches are unknownMore occurrence probability of an activity would lead intoits higher share in equating In the meantime the time ofconducting each branch includes relevant time (in the caseof successful conclusion) and also added time (in the caseof nonconclusion of the branch) Since it was impossible toconduct other activities in that unit the relevant time wascomputed based on time average of other moods Thereforethe following can be extended by the above arguments

119886=

1199011119886

1199011119886+ 1199011119887+ 1199011119888

otimes[[[[

[

119886otimes1199012119886oplus (1minus1199012

119886)

otimes(

1199011119887

1199011119887+ 1199011119888

otimes [1199012119887otimes (119905119886oplus 119887) oplus (1 minus 1199012

119887) otimes (119905119886oplus 119887oplus 119888)]

oplus1199011119888

1199011119887+ 1199011119888


119888) otimes (119905119886oplus 119887oplus )]

)]]]]

]

(12)

Similarly other branches (119888 and 119887) can be computed Asthe result duration of equivalent branch is

119879119901=

[[[[[[[[[[[

[

1199011119886

1199011119886+ 1199011119887+ 1199011119888

otimes [119886oplus (1 minus 1199012

119886) otimes (

1199011119887

1199011119887+ 1199011119888

otimes (119887oplus (1 minus 1199012

119887) otimes 119888) oplus

1199011119888

1199011119887+ 1199011119888


119888) otimes 119887))]

oplus1199011119887

1199011119886+ 1199011119887+ 1199011119888


119887) otimes (

1199011119886

1199011119886+ 1199011119888


119886) otimes 119888) oplus

1199011119888

1199011119886+ 1199011119888


119888) otimes 119886))]

oplus1199011119888

1199011119886+ 1199011119887+ 1199011119888


119888) otimes (

1199011119886

1199011119886+ 1199011119887


119886) otimes 119887) oplus

1199011119887

1199011119886+ 1199011119887


119887) otimes 119886))]

]]]]]]]]]]]

]

(13)


1 2 3 4 n minus 1 n

Figure 3 A display of series branches

Table 2 Success probability of each branch from node (119894)


Occurrenceprobability

Success probabilityof branch (i-j)

0 1 minus 1199011119894119894

1199012119894119895

1 1199011119894119894sdot (1 minus 1199011

119894119894) 1199012

119894119895sdot (1199012119894119894)

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)2

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)119899

Concerning occurrence probability and success proba-bility of equivalent branch since branches are parallel andindependent occurrence probability is equal to probabilitiesaggregation (14) For success probability similar averagingtrend with similar times is used and we have

1199011119879119901= 1199011119886+1199011119887+1199011119888 (14)

1199012119879119901= [1199012119886+ (1minus1199012

119886) (1199012119887+ (1minus1199012

119887) 1199012119888)] (15)

(2) Parallel Branches between Two Nodes with ldquoDeterministicrdquoOutput and ldquoInclusive-Orrdquo Input In such case it is assumedthat there are three branches 119886 119887 and 119888 for an activityAll three branches start altogether and when one of them isconcluded the other shall be stopped

As we see to achieve equivalent activity time first thetime of current branches are getting defuzzy (based on (16))

Defuzzy operation is a reversing process which returns afuzzy distance to its crisp number

119889119886=(1198861 + 21198862 + 21198863 + 1198864)

6 (16)

Then these activities will be sorted from the lowest tothe highest based on defuzzy values Now we assign 1 for thelowest defuzzy value and mark other values

Sor min (119889119886 119889119887 119889119888) = (1198891 1198892 1198893) 997904rArr 1 2 3 (17)

Therefore we have

119879119901= 1199012

1 otimes 1 oplus (1minus11990121) otimes [119901

2

2otimes 2 oplus (1minus119901

22) otimes 3] (18)

Concerning occurrence probability and success probabil-ity when activities are parallel and independent occurrenceprobability is equal to the aggregation of standardized prob-abilities and success probability is a similar trend to activitiesduration

1199011119879119901= 1 (19)

1199012119879119901= 1199012

1 + (1minus11990121) [1199012

2+ (1minus1199012

2) 11990123] (20)

(3) Parallel Branches between Two Nodes with ldquoDeterministicrdquoOutput and ldquoAndrdquo Input Parallel branches show severalactivities All these several activities must be performed andno processors or successor can be considered When they arenot fully conducted it will not be possible to go to next stepSo to achieve duration of equivalent branch it is sufficient touse maximum fuzzy actor

119879119901= max (119905

119886 119887 119888) = [(1198861 or 1198871 or 1198881) (1198862 or 1198872 or 1198882)

(1198863 or 1198873 or 1198883) (1198864 or 1198874 or 1198884)] (21)

On the other hand based on the approach the suc-cess probability of equivalent branch resulted from multiplesuccess probability of all activities (22) since all activitiesshould be conducted to achieve the final event and to releasethe next node Also as for the occurrence probability sincethe activities are parallel and independent the occurrenceprobability of equivalent branch is as follows

1199012119879119901= 1199012119886sdot 1199012119887sdot 1199012119888 (22)

1199011119879119901= 1 (23)

423 EquivalentMaking of Series Branches A set of branchesis called series when the path between their source nodesincludes no diversion path (119878

(119894-119895)) (Figure 3) To estimate 3


Table 3 Six states of parallel branches between two different nodes

State Definition Figure

1 Parallel branches between two nodes with ldquoProbabilisticrdquo output andldquoExclusive-Orrdquo input

a

b

c

2 Parallel branches between two nodes with ldquoDeterministicrdquo output andldquoInclusive-Orrdquo input

a

b

c

3 Parallel branches between two nodes with ldquoDeterministicrdquo output andldquoAndrdquo input

a

b

c

4 Parallel branches between two nodes with ldquoProbabilisticrdquo output andldquoAndrdquo input

a

b

c

5 Parallel branches between two nodes with ldquoDeterministicrdquo output andldquoExclusive-Orrdquo input

a

b

c

6 Parallel branches between two nodes with ldquoProbabilisticrdquo output andldquoInclusive-Orrdquo input

a

b

c

main parameters of equivalent branch (119905119879119904 1199011119879119904 1199012119879119904) a set of

series branches is as follows119879119904= sum119896isin119878(119894-119895)

oplus 119896= [119905(1-2) oplus (2-3) oplus sdot sdot sdot oplus [(119899minus1)minus119899]]

1199011119904= prod119896isin119878(119894-119895)

1199011119896= [1199011(1-2) times119901

1(2-3) times sdot sdot sdot times 119901

1[(119899minus1)minus119899]]

1199012119879119904= prod119896isin119878(119894-119895)

1199012119896

= 1199012(1-2) times119901

2(2-3) times119901


2((119899minus1)minus119899)

(24)

424 Procedure of Removing Loops by Equivalent MakingExistence of reversible branches can be eventuated to forma loop and to execute one or several activities leading to delayand increase in time of progress and performance of activities(Figure 4) For removal of these loops and application ofdelay in a network a fuzzy time and occurrence probabilityequivalent to that loop will be computed and their impactsare applied to the activities affected by this loop

Parameters equivalent to a loop are equal to parametersof equivalent branch to the paths between the two nodes ofsource (119900) and end (119898) of the loop To get such values thefollowing steps can be assumed

(1) In order to reduce amount of computations first theparallel and series branches between two nodes of (119900)

o m

ln(o-m)

(p2ln(119900-119898)

p1ln(119900-119898)

tln(119900-119898) )

Figure 4 View of a loop formation

and (119898) in the main network made based on (13)ndash(24) shall be equalized and shall be applied in themain graph

(2) In the next stage the set of branches and nodesbetween two nodes of (119900) and (119898) which play a rolein connecting paths between these two nodes shallbe drawn based on the revised main network called119873ln(119900-119898) network

(3) If the equivalent branch connecting two nodes of (119900)and (119898) in119873ln(119900-119898) network is in a simple loop shapewe will go to step (5) However if the loop is in acomplex loop shape then we will go to step (4)

(4) The reviewed complicated loops are of subindepen-dent type in which all subsets of the main loop arein condition of independence relating to each otherand themain loop To start solution and application of


a complex loop impact the following shall be fol-lowed

(a) Classification of the loops is as follows

(i) To assume the main loop as a loop at zerolevel

(ii) To classify the internal loops into twogroups (a) Level 1 simple loops and (b)Level 1 complex loops

(iii) To classify the internal loops of Level 1complex loops into two groups (a) Level 2simple loops and (b) Level 2 complex loops

(iv) To continue the process of classificationuntil levels of all loops are specified

(b) Priority of the loops is as follows

The main assumption on priority of loopsshall be as followsThe priority shall belong to loops in whichthe longest distance of their source nodeto last level source node involves the mini-mum rate

(c) Selection and solution of reviewed loop basedon rendered priority in step (b) are as follows

First the lowest classified level in the loopshall be prioritized and based on suchpriority the loops of this level shall besolved in the selected complex loop andtheir impacts shall be applied based on step(6) Then process of prioritization shall beexecuted for the loop at one upper level inthis complex loopThe related impacts shallbe computed and applied on the basis ofpriority of the loops at this level of complexloop againThis process shall continue untilreaching the main loop of the selectedcomplex loop

(d) Review of main loop is as follows

After reviewing of all loops existing inLevel 1 and application of their impacts theimpact ofmain loop of complex loop (Levelzero) shall be computed and thenwewill goto step (5)Thus the delay impacts resultingfrom existence of the loops (simple andcomplex) in the network can be appliedand their equivalent can be made and theyshall be removed from the network

(5) Based on the parameters of the branch equal-ized for the paths between nodes (119900) and (119898)(119905(119900-119898) 119901

1

(119900-119898) 1199012

(119900-119898)) and parameters of reversible

branch (119905(119898-119900) 119901

1

(119898-119900) 1199012

(119900-119898)) the parameters equiva-lent to the loop (119905ln(119900-119898) 119901

1

ln(119900-119898) 1199012

ln(119900-119898)) can be com-puted as follows

ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)

(6) For the application of the impacts of the loop (ln(119900-119898))

in a network after computing parameters of the loopit would be assumed as a delay branch (119900119900) on node(119900) and its impacts will be applied as (26) in thenetwork It shall be noted that the impacts resultedfrom this loop shall be applied to parameters of timeand success probability on the output branches fromnode (119900)

1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)

Also to fix the probability of the occurrences in the outputbranches from node (119898) occurrence probability of thesebranches will be computed by the following so that the sumof occurrence probability of output branches from node (119898)remains at one

11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)

425 Proposed Algorithm to Resolve No-Loop FuzzyNetworksIntroduced problem solving method is based on equivalentmaking of parallel and series branches in each step ofsimplifying the fuzzy network and moving from the sourcenode to the sink (end) node by surveying different pathsbetween two nodes and to achieve a step in which only oneequal branch is built between the two nodes of source andsink Operation flowchart is shown in Figure 5

5 Application and Numerical ExampleA Case Study

This part of the project is conducted for the first time ina research institute without any ambiguity in defining theactions and estimating the time and probability of successwasconsidered as an applied example In addition to supportingmodel validity one can also determine the deficiencies andeven inabilities of current methods in treating with parallel


paths and combining them with loops as well as drawing andresolving the network According to Figure 6 and Table 4the project network contains needed inputs to resolve thenetwork

Summary of solution procedure based on the presentedalgorithm is as follows classification and prioritization of theloops existing in the network based on steps (a) and (b) areas follows

ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)

In the first step and based on Section 424 we equalize(eliminate) loops and compute new values of parametersConsequently a network without any loops is obtained asseen in Figure 7 as well as new input parameters based onTable 5

After three times the network without loop (Figure 7)will be simplified by equivalent making of series and parallelbranches (based on Sections 422 and 423) we have

1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)

Deterministic methods are used to schedule the projectso that limitation in drawing the network is proportionateto actual realities (such as regressive and parallel paths)Therefore there will be timescales with huge differences Toresolve output problem fuzzy CPM is used (109) To com-pare outputs from proposed algorithm and other methodsaverage of duration is computed and percentage of deviationcan be computed by real time of executing the project

According to Table 6 there is a huge difference (over twotimes) between fuzzy CPM output and real time of projecttermination which indicates the inability of this method toschedule RampD project Likewise the real time is almost inhigher border of project interval from the proposed model

It is noteworthy that the probability of project successwas computable in none of the current methods while it wascomputable in the proposedmodel and its obtained value (forcase study) was 59

51 Sensitivity Analysis In this paper for sensitivity analysisthe impact of loops occurrence probability on the projectcompletion time (PCT) and the success probability are exam-ined According to the logic ofGERTwith loops it is expectedthat with increasing the probability of loop occurrence thePCT increases and the success probability decreases and viceversa To do this the loop number 5-3 (ln

(3-5)) is consideredfor sensitivity analysis The results of the sensitivity analysisare shown in Table 7 and related charts (Figures 8 and 9)As can be seen increase (decreases) in the probability ofloop occurrence leeds to increase (decreases) in the PCT anddecrease (increase) in the success probability

6 Algorithm and Model Validation

To prove appropriate performance of the algorithmpresentedin this paper the results obtained by 3 articles includingscheduling of fuzzy GERT networks were selected and theproposed algorithmrsquos output was compared with the outputsof the articles These methods include Itakurarsquos method[15] which is one of the first and most valid methodsprovided in this area (Figure 10) Gavareshkirsquos method [1](Table 8) and Hasheminrsquos method [38] (Table 9) which arethe latest proposed methods Investigations revealed thatthe solutions found from the proposed algorithm in thispaper were very close to those found by Itakura and also byHashemin methods Because the proposed algorithm coversmore comprehensive forms of the network compared to thetwo methods mentioned more efficiency of this algorithmcan be confirmed

In the article presented by Hashemin [38] arrival timeat node 5 was regarded as the end of project time and wascalculated as 594 947 1559 and 1935 By comparing the twosolutions it can be seen that the proposed algorithm is closeto Hasheminrsquos solution and the two can be assumed equalConsequently the two algorithms in solving the problemhave an equal performance

HoweverHasheminrsquosmethod [38] is only suitable for spe-cial states of fuzzyGERT networks with specific nodes (nodeswith Exclusive-or receiver and Exclusive-or emitter) withoutloops and delay branches while the proposed algorithm inthis paper covers a wide range of networks

7 Conclusion and Future Research

In this paper it is attempted to make a brief study within anapplied comparison on some of introduced techniques bymaking a historical review on estimation of PCT in conditionof having uncertainty in the activities network durationand path As it is mentioned most of papers and scientificworks have been performed on two fields of fuzzy andprobability Researchers have solved problem of estimatingtime under uncertain conditions directly regarding to twoabove-mentioned approaches or through combining themwith linear and nonlinear planning models and simulationor through innovative solutions Generally in the conductedstudy it can be concluded that most of performed works arerelated to estimations under network and the best networks


Yes

NoYes

Finish

Are there parallelbranches between

two nodes

Equivalent making of parallel branchesbetween two nodes with the priority ofminimum distance from source node

No

Drawing a no-loop fuzzy network

(source node in left hand)

Start

Are thereseries branchesbetween nodes

Equivalent making of series branchesbetween nodes with the priority of

minimum distance from source node

Computation of parameters

Updating


Extraction of network witha single equivalent branch

Estimation of projectcompletion fuzzy time based

on end node(tT119901 p

1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )

Figure 5 The proposed algorithm to resolve no-loop fuzzy networks

0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321

Figure 6 Activity network of the RampD project


Table 4 Input parameters of activity network

Code of activity Duration (week) Success probability of activityloop Occurrence probability of activityloop0-1 (4 5 6 7) 1 11-2 (7 9 10 11) 09 12-3 (2 3 4 5) 09 13-4 (8 9 10 11) 085 063-13 (12 14 15 18) 08 044-5 (a) (2 3 4 5) 08 074-5 (b) (2 4 5 7) 08 024-5 (c) (6 7 8 10) 085 015-3 (0 0 0 0) 1 045-2 (0 0 0 0) 1 0255-13 (1 2 2 3) 09 03513-15 (1 1 2 2) 1 10-6 (4 5 6 8) 1 16-7 (2 3 3 4) 1 17-8 (4 5 6 7) 095 18-9 (3 4 4 5) 09 19-8 (0 0 0 0) 1 049-10 (d) (3 4 5 6) 09 0259-10 (e) (2 4 5 7) 08 03510-14 (1 1 1 1) 1 16-11 (2 3 3 4) 1 111-12 (6 7 8 9) 08 112-11 (0 0 0 0) 1 0312-12 (1 2 3 3) 1 03512-14 (1 2 2 3) 09 03514-15 (2 3 4 4) 1 115-16 (4 5 6 7) 1 116-1 (0 0 0 0) 1 0216-17 (1 2 2 3) 1 08

0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321

Figure 7 Activity network of the RampD project without any loops

have been developed for the uncertain conditions are relatedto the PERT and GERT networks

Since use of probabilistic distribution techniques in theprojects including large activity network and a dynamicenvironment full of uncertainty has been rather complicateddifficult and limited during the recent decades attentionto simulation and fuzzy approaches has been considerablyintensified and development of techniques in these fields isexpanded

Therefore in this paper it is tried to merge fuzzyand probability techniques in order to introduce a newapproach to solve scheduling of projects having networkswith parallel series and reversible cycle branches (RP-FGERT) for estimating of PCT and success probability Tomeasure the validity of algorithm a case study was solvedand related results were presented in Section 5 Then differ-ent methods were compared Provided results in Section 6show more accuracy and closeness to real time estimation


Table 5 Input parameters of activity network without any loops

Code ofactivity

Initial fuzzy duration ofactivity performance

Final fuzzy duration ofactivity performance

Success probabilityof activity

Occurrenceprobability of activity

0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1

Table 6 Comparative results and percentage of deviation

Fuzzy duration (week) Average of duration (month) Percentage of deviationFuzzy CPM (34 44 50 59) 109 542Proposed model (541 628 80 944) 173 273Real completion time (102 102 102 102) 238 mdash

Table 7 Results of sensitivity analysis

Variation percentageof occurrenceprobability

Variationpercentage of

the PCT


success probabilityminus70 minus144 63minus50 minus113 49minus30 minus74 32minus10 minus28 1210 32 minus1430 112 minus5150 229 minus10270 430 minus186

Thus the problems about estimating duration of research orRampDprojects can be solved in a betterway It is also resolvableand proper The complexities existing in other techniquesare highly prevented The proposed algorithm will be able toextend for different types of probabilistic and deterministicnodes with probable and certain inputs and outputs in future

Table 8 Proposed algorithmrsquos results Gavareshki methodrsquos resultsand CPM solution

Proposed algorithmrsquosresult

Gavareshki methodrsquosresult

CPMsolution

(36 45 45 63) (45 65 65 106) 35

Table 9 Results of Hasheminrsquos solution

Activity Fuzzy duration of activity Occurrenceprobability of activity

0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879

Meanwhile using different parallel paths between nodes ofthe above-mentioned types could be solved

In future proposed algorithm (method) would needresearch and development in the below areas

(i) On complex loops yet is opportunity for moreresearch and development on subdependent complexloops


07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y

Occurrence probability

Figure 8 Impact of variation probability of loop occurrence on suc-cess probability

110

100

90

80

70

60

50

0 02 04 06 08

PCT


Figure 9 Impact of variation probability of loop occurrence on thePCT

12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time

Figure 10 Fuzzy time of Itakura and the proposed algorithm

(ii) One can work on changing provided algorithm toflowcharts and create software

(iii) It is possible to use fuzzy numbers in estimatingoccurrence probability of activities and loops insteadof probabilistic functions

20

15

10

5

0

0 05 1

Expected iteration

Figure 11

(iv) One can use analysis of sensitivity and update sched-uling of activities network with loops

(v) To estimate cost of each activity and project comple-tion based on the proposed algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M H K Gavareshki ldquoNew fuzzy GERT method for researchprojects schedulingrdquo in Proceedings of the IEEE InternationalEngineering Management Conference (IEMC rsquo04) pp 820ndash824Singapore October 2004

[2] R Lachmayer M Afsari and R Hassani ldquoC method for alltypes of nodes in fuzzy GERTrdquo International Journal of ArtificialIntelligence and Neural Networks vol 5 no 1 pp 57ndash62 2015

[3] D Ivanov and B Sokolov Adaptive Supply Chain ManagementSpringer London UK 2010

[4] C You ldquoOn the convergence of uncertain sequencesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 482ndash4872009

[5] B Liu ldquoSome research problems in uncertainty theoryrdquo Journalof Uncertain Systems vol 3 no 1 pp 3ndash10 2009

[6] N Kuznetsov ldquoManaging the company in the setting ofimplementing large-scale development programsrdquo Asian SocialScience vol 11 no 1 pp 193ndash2003 2015

[7] K Oyama G Learmonth and R Chao ldquoApplying complexityscience to new product development modeling considerationsextensions and implicationsrdquo Journal of Engineering and Tech-nology Management vol 35 pp 1ndash24 2015

[8] P Patanakul A J Shenhar and D Z Milosevic ldquoHow projectstrategy is used in project management cases of new productdevelopment and software development projectsrdquo Journal of


Engineering and TechnologyManagement vol 29 no 3 pp 391ndash414 2012

[9] A A B Pritsker and G E Whitehouse ldquoGERT graphical eval-uation and review technique part II probabilistic and industrialengineering applicationsrdquo Journal of Industrial Engineering vol17 no 6 pp 293ndash301 1966

[10] A A B Pritsker and W W Happ ldquoGERT graphical evaluationand review technique part I Fundamentalsrdquo Journal of Indus-trial Engineering vol 17 no 5 pp 267ndash274 1966

[11] A A B Pritsker Modeling and Analysis Using Q-GERT Net-works Halsted Press New York NY USA 1977

[12] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash356 1965

[13] S Chanas and J Kamburowski ldquoThe use of fuzzy variables inPERTrdquo Journal of Fuzzy Sets and Systems vol 5 no 1 pp 11ndash191981

[14] I Gazdik ldquoFuzzy-network planningmdashFNETrdquo IEEE Transac-tions on Reliability vol 32 no 3 pp 304ndash313 1983

[15] H Itakura and Y Nishikawa ldquoFuzzy network technique fortechnological forecastingrdquo Fuzzy Sets and Systems vol 14 no2 pp 99ndash113 1984

[16] C S McCahon ldquoUsing Pert as an approximation of fuzzyproject-network analysisrdquo IEEE Transactions on EngineeringManagement vol 40 no 2 pp 146ndash153 1993

[17] C-H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[18] C-H Cheng ldquoFuzzy repairable reliability based on fuzzyGERTrdquoMicroelectronics Reliability vol 36 no 10 pp 1557ndash15631996

[19] S H Nasution ldquoFuzzy critical path methodrdquo IEEE Transactionson Systems Man and Cybernetics vol 24 no 1 pp 48ndash57 1994

[20] S Chang Y Tsujimura and T Tazawa ldquoAn efficient approachfor large scale project planning based on fuzzy Delphi methodrdquoFuzzy Sets and Systems vol 76 no 3 pp 277ndash288 1995

[21] M F Shipley A De Korvin and K Omer ldquoA fuzzy logicapproach for determining expected values a project manage-ment applicationrdquo Journal of the Operational Research Societyvol 47 no 4 pp 562ndash569 1996

[22] J R Wang ldquoFuzzy set approach to activity scheduling forproduct developmentrdquo Journal of the Operational ResearchSociety vol 50 no 12 pp 1217ndash1228 1999

[23] J Wang ldquoA fuzzy project scheduling approach to minimizeschedule risk for product developmentrdquo Fuzzy Sets and Systemsvol 127 no 2 pp 99ndash116 2002

[24] J R Wang ldquoA fuzzy robust scheduling approach for prod-uct development projectsrdquo European Journal of OperationalResearch vol 152 no 1 pp 180ndash194 2004

[25] J J Buckley ldquoFurther results for the linear fuzzy controllerrdquoKybernetes vol 18 no 5 pp 48ndash55 1989

[26] S Chanas and P Zielinski ldquoCritical path analysis in the networkwith fuzzy activity timesrdquo Journal of Fuzzy Sets and Systems vol122 no 2 pp 195ndash204 2001

[27] C-T Chen and S-F Huang ldquoApplying fuzzy method formeasuring criticality in project networkrdquo Information Sciencesvol 177 no 12 pp 2448ndash2458 2007

[28] S-P Chen and Y-J Hsueh ldquoA simple approach to fuzzycritical path analysis in project networksrdquoAppliedMathematicalModelling vol 32 no 7 pp 1289ndash1297 2008

[29] J Fortin P Zielinski D Dubois and H Fargier ldquoCriticalityanalysis of activity networks under interval uncertaintyrdquo Journalof Scheduling vol 13 no 6 pp 609ndash627 2010

[30] H Ke and B Liu ldquoFuzzy project scheduling problem and itshybrid intelligent algorithmrdquo Applied Mathematical Modellingvol 34 no 2 pp 301ndash308 2010

[31] W Huang S-K Oh and W Pedrycz ldquoA fuzzy time-dependentproject scheduling problemrdquo Information Sciences vol 246 pp100ndash114 2013

[32] A P Shibanov ldquoFinding the distribution density of the timetaken to fulfill the GERT network on the basis of equivalentsimplifying transformationsrdquo Automation and Remote Controlvol 64 no 2 pp 279ndash287 2003

[33] S S Hashemin and S M T Fatemi Ghomi ldquoA hybrid methodto find cumulative distribution function of completion timeof GERT networksrdquo Journal of Industrial Engineering Interna-tional vol 1 pp 1ndash9 2005

[34] K Kurihara and N Nishiuchi ldquoEfficient Monte Carlo simula-tion method of GERT-type network for project managementrdquoComputers and Industrial Engineering vol 42 no 2ndash4 pp 521ndash531 2002

[35] K Kurihara N Nishiuchi M Nagai and K Masuda ldquoBranch-ing probabilities planning of stochastic network for projectduration planningrdquo in Proceedings of the IEEE Conference onEmerging Technologies and Factory Automation (ETFA rsquo06) pp1333ndash1339 Prague Czech Republic September 2006

[36] R LachmayerM Afsari and B Sauthoff ldquoFuzzyGERTmethodfor Scheduling research projectsrdquo in Proceedings of the 9thInternational Industrial Engineering Conference pp 1ndash7 TehranIran 2013

[37] R Lachmayer and M Afsari ldquoMatlab method for exclusive-or nodes in fuzzy GERT networksrdquo World Academy of ScienceEngineering and Technology Computer and Information Engi-neering vol 2 no 1 2015

[38] S S Hashemin ldquoFuzzy completion time for alternative stochas-tic networksrdquo Journal of Industrial Engineering Internationalvol 6 no 11 pp 17ndash22 2010

[39] Q Shi andT Blomquist ldquoA new approach for project schedulingusing fuzzy dependency structurematrixrdquo International Journalof Project Management vol 30 no 4 pp 503ndash510 2012

[40] H C Martınez Leon J A Farris G Letens and A Hernan-dez ldquoAn analytical management framework for new productdevelopment processes featuring uncertain iterationsrdquo Journalof Engineering and Technology Management vol 30 no 1 pp45ndash71 2013

[41] H C M Leon J A Farris and G Letens ldquoImproving productdevelopment through front-loading and enhanced iterationmanagementrdquo in Proceedings of the Industrial EngineeringResearch Conference Reno Nev USA May 2011

[42] M T Hajiali M R Mosavi and K Shahanaghi ldquoEstimationof project completion time-based on a mixture of expert in aninteractive spacerdquoModernApplied Science vol 8 no 6 pp 229ndash237 2014

[43] K Shahanaghi andMTHajiali ldquoEstimation of project time andcost at completion using fuzzyKalmanfilter andARMAmodelrdquoManagement Business and Economics vol 2 no 1 2014

[44] M Verma and K K Shukla ldquoA new algorithm for solvingfuzzy constrained shortest path problem using intuitionisticfuzzy numbersrdquo International Journal of Artificial Intelligenceand Neural Networks vol 5 no 1 pp 38ndash42 2015

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


methods that is CPM (Critical Path Method) and PERT(Program Evaluation and Review Technique) were intro-duced and developed [6]

Project management in particular project planning is acritical factor in the success or failure of new product devel-opment (NPD) [7] and research and development (RampD)projects [8] Limited facilities of CPM and PERT methodsfor modeling of projects having complex and uncertainnetworks [6] caused the great scientists such as Pritsker andWhitehouse [9] and Pritsker and Happ [10] to introducethe probability GERT (Graphical Evaluation and ReviewTechnique) which is amethod consisted of flow graph theorymoment generation functions and PERT to solve problemswith probability activities Pritsker [11] developed GERTmethod and used it for different nodes and introduced useof simulation in that method Of course it is pointed outthat Pritsker and Whitehouse [9] used Maisonrsquos role whichhad been introduced in graph theory to solve probabilitynetworks The pointed out probability methods are all basedon probable distributions such as Beta (120573) or normal dis-tribution which are used for estimation of project activitiesperiod Thus to use probable distributions random samplesrepeatability and statistical deduction will be required

When fuzzy theory was introduced and developed byZadeh [12] scientistsrsquo and researchersrsquo attention in the fieldof network and science engineering was gradually caughtby the issue that they will be able to solve the problemof uncertainty in problems and also expressed problemsby a fuzzy approach Thus for the first time Chanas andKamburowski [13] introduced a new technique called FPERT(Fuzzy ProgramEvaluation and ReviewTechnique) in whichby using fuzzy theory the estimation of activities time andproject completion time had been shown by triangular fuzzynumbers Gazdik [14] presented another technique based onfuzzy sets and graph theory called FNET (fuzzy network)This is for estimation of activities time by use of algebraicoperators and by deductive method in the projects withoutdata Itakura and Nishikawa [15] applied fuzzy concepts inGERT networks for the first time and by replacing fuzzyparameters for probability parameters presented FGERT(FuzzyGraphical Evaluation andReviewTechnique)methodof solution is similar to probability GERT but it is differentbecause fuzzy theory was used to review GERT networksMcCahon [16] presented FPNA (Fuzzy Project NetworkAnalysis) technique to identify critical path and obtain totalfloat value and probability time for project completion Tenyears after presentation of his first measures on fuzzy GERTCheng [17 18] presented another method of solution Ofcourse this method was presented to solve problems of seriessystems reliability in which merely Exclusive-or nodes weredealt with Basis of his method is the same probable GERTbut it is only different in using fuzzy parameters instead ofprobable functions Nasution [19] dealt with developmentof FNET technique He solved fuzzy networks with themulticritical paths approach by using interactive differentialfuzzymethod in the network reversible computations Changet al [20] considered projects of large size planning networksWhen such projects are under uncertain conditions thenthe techniques presented to solve timed networks become

complicated and sometimes unsolvableThus they presenteda technique by merging comparison and composite methodsand also by using fuzzy Delphi which was efficient forsolving problems with the foregoing features Shipley et al[21] presented a technique called Belief in Fuzzy Probabilitiesof Estimated Time (BFPET) which is based on the fuzzylogic belief functions and fuzzy probability distributions anddevelopment principle Wang [22ndash24] dealt with develop-ment of fuzzy sets approach for planning product develop-ment projects of limited and imprecise data The approachhe adopted was in line with the works of Buckley [25] andTatish that is use of trapezoidal fuzzy numbers and theircomputational procedure Chanas and Zielinski [26] Chenand Huang [27] Chen and Hsueh [28] Fortin et al [29] andKe and Liu [30] dealt with the issue of critical path in thenetwork of projects scheduling problem that contains fuzzyduration times of activities All these studies represent activityduration times described by means of fuzzy sets howeverwithout any component of time dependency Huang et al[31] have studied the project scheduling problem in CPMnetworks with activity duration times that are both fuzzy andtime dependent

Some new analytical methods for determining the com-pletion time of GERT-type networks have been proposed byShibanov [32] Hashemin and Fatemi Ghomi [33] The mainproblem of these methods is high complexity of relationsand computations in a network without loops Kurihara andNishiuchi [34] solved probability GERT networks by MonteCarlo simulation and by this method they could analyzespecific kind of nodes present in GERT networks withoutloops (with loops [35]) Gavareshki [1] and Lachmayer et al[2 36 37] presented a new method to solve fuzzy GERTnetworks whose basis goes back to the method for solvingCPM networks that is moving forward and implementationof fuzzy literature and inference of the nodes Hashemin [38]provides an analytical procedure for this kind of networksby simplifying GERT networks Shi and Blomquist [39] dealtwith using matrix structures for solving time schedulingnetworks By combination of fuzzy logic and DSM (Depen-dency Structure Matrix) he could introduce a new approachfor solving problems on time scheduling of projects withuncertainty in periods of activities and uncertainty in over-lapping the activities Martınez Leon et al [40 41] presentan analytical framework for effective management of projectswith uncertain iterations in probable GERT networks Theframework is based on the Design Structure Matrix As theintroduced approachwas new it is capable of being developedin the fuzzy networks and uncertainty in the definition ofactivities

Usual techniques are not able to estimate project com-pletion time (PCT) of the projects that are executed for thefirst time or projects having computational problems [8 4243] Therefore only networking techniques with particulardefinitions on parallel nods and branches may be regressedand can create loops in fuzzy environment which provideproper timescale for these projects by using the capabilitiesof fuzzy sets and networks adapted to the attributes andfeatures of these projects Concerning the bottlenecks ofthese techniques a new method is developed to resolve














119894119895is






















(119894-119895)





to node (119900)






1015840



10158401015840








lel branches


1= (1198971 1198981 1199061 1199041) and

2= (1198972 1198982 1199062




= (1198971+ 1198972 1198981+ 1198982 1199061+ 1199062 1199041+ 1199042)






2 ℎ lowast 119898

2 ℎ lowast 119906

2 ℎ lowast 119904

2)


1198971

1199042

1198981

1199062

1199061

1198982

1199041

1198972

)

max 1 2

= ((1198971or 1198972) (1198981or 1198982) (1199061or 1199062) (1199041or 1199042))

(1)






0 1 minus 1199011119894119894

01 1199011

119894119894sdot (1 minus 1199011

119894119894)

119894119894

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 2 sdot

119894119894

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 119899 sdot

119894119894










Ge (119909 1199011119894119894) = (1minus1199011

119894119894) sdot 1199011119894119894

119909

119909 = 0 1 2 3 (2)



119905119894119895= (0 sdot (1minus1199011

119894119894)) oplus ((1199011

119894119894) sdot (1minus1199011

119894119894) otimes 119894119894)

oplus (2 (1199011119894119894)2sdot (1minus1199011


oplus (119899 (1199011119894119894)119899


=infin

sum119909=0

oplus ([119909 (1199011119894119894)119909


(3)



= (infin

sum119909=0

119909 sdot (1199011119894119894)119909



No

Yes

Yes

No

Yes

No















Are thereseries

branches

Are thereloops




(tln(119900-119898) p1ln(119900-119898)

p2ln(119900-119898)

)



= (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1

) sdot 1199011119894119894sdot (1minus1199011

119894119894) otimes 119894119894 (5)

We assume that

Ψ = (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1


1(1 minus 1199011

119894119894)2] (6)


119905119894119895=

1199011119894119894

(1 minus 1199011119894119894)otimes 119894119894 (7)


1015840

(119896)= (119896)oplus[(

1199011119894119894

1 minus 1199011119894119894


119884119894 (8)


ii

i

f

r

k

J




119894119895 1199011119894119895 119905119894119895) is a branch of


lated as follows

11990121015840119894119895= 1199012119894119895sdot (1minus1199011


119894119894) + 1199012119894119895


119894119894) + sdot sdot sdot + 1199012

119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899


=infin

sum119899=0

1199012119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899

sdot (1minus1199011119894119894)

= 1199012119894119895sdot (1minus1199011

119894119894)infin

sum119899=0

(1199012119894119894)119899

sdot (1199011119894119894)119899

(9)


997904rArr 11990121015840119896= 1199012119896sdot (1minus1199011

119894119894) sdot (1+

(1199012119894119894) sdot (1199011119894119894)

1 minus [(1199012119894119894) sdot (1199011119894119894)])

forall119896 isin 119885119884119894

(10)


119894119894) sdot

(1 + ((1199012119894119894) sdot (1199011119894119894))(1 minus [(1199012

119894119894) sdot (1199011119894119894)]))



(1+1199011119894119894

1 minus 1199011119894119894

) (11)




119886=

1199011119886

1199011119886+ 1199011119887+ 1199011119888

otimes[[[[

[


119886)

otimes(

1199011119887

1199011119887+ 1199011119888



oplus1199011119888

1199011119887+ 1199011119888



)]]]]

]

(12)


119879119901=

[[[[[[[[[[[

[

1199011119886

1199011119886+ 1199011119887+ 1199011119888


119886) otimes (

1199011119887

1199011119887+ 1199011119888


119887) otimes 119888) oplus

1199011119888

1199011119887+ 1199011119888


119888) otimes 119887))]

oplus1199011119887

1199011119886+ 1199011119887+ 1199011119888


119887) otimes (

1199011119886

1199011119886+ 1199011119888


119886) otimes 119888) oplus

1199011119888

1199011119886+ 1199011119888


119888) otimes 119886))]

oplus1199011119888

1199011119886+ 1199011119887+ 1199011119888


119888) otimes (

1199011119886

1199011119886+ 1199011119887


119886) otimes 119887) oplus

1199011119887

1199011119886+ 1199011119887


119887) otimes 119886))]

]]]]]]]]]]]

]

(13)


1 2 3 4 n minus 1 n






0 1 minus 1199011119894119894

1199012119894119895

1 1199011119894119894sdot (1 minus 1199011

119894119894) 1199012

119894119895sdot (1199012119894119894)

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)2

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)119899


1199011119879119901= 1199011119886+1199011119887+1199011119888 (14)

1199012119879119901= [1199012119886+ (1minus1199012

119886) (1199012119887+ (1minus1199012

119887) 1199012119888)] (15)




119889119886=(1198861 + 21198862 + 21198863 + 1198864)

6 (16)


Sor min (119889119886 119889119887 119889119888) = (1198891 1198892 1198893) 997904rArr 1 2 3 (17)

Therefore we have

119879119901= 1199012


2


22) otimes 3] (18)


1199011119879119901= 1 (19)

1199012119879119901= 1199012

1 + (1minus11990121) [1199012

2+ (1minus1199012

2) 11990123] (20)


119879119901= max (119905

119886 119887 119888) = [(1198861 or 1198871 or 1198881) (1198862 or 1198872 or 1198882)

(1198863 or 1198873 or 1198883) (1198864 or 1198874 or 1198884)] (21)


1199012119879119901= 1199012119886sdot 1199012119887sdot 1199012119888 (22)

1199011119879119901= 1 (23)







a

b

c


a

b

c


a

b

c


a

b

c


a

b

c


a

b

c




1199011119904= prod119896isin119878(119894-119895)

1199011119896= [1199011(1-2) times119901


1[(119899minus1)minus119899]]

1199012119879119904= prod119896isin119878(119894-119895)

1199012119896

= 1199012(1-2) times119901

2(2-3) times119901


2((119899minus1)minus119899)

(24)




o m

ln(o-m)

(p2ln(119900-119898)

p1ln(119900-119898)

tln(119900-119898) )




















1

(119900-119898) 1199012


branch (119905(119898-119900) 119901

1

(119898-119900) 1199012


1

ln(119900-119898) 1199012


ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)



1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)


11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)







ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















119894119895is






















(119894-119895)





to node (119900)






1015840



10158401015840








lel branches


1= (1198971 1198981 1199061 1199041) and

2= (1198972 1198982 1199062




= (1198971+ 1198972 1198981+ 1198982 1199061+ 1199062 1199041+ 1199042)






2 ℎ lowast 119898

2 ℎ lowast 119906

2 ℎ lowast 119904

2)


1198971

1199042

1198981

1199062

1199061

1198982

1199041

1198972

)

max 1 2

= ((1198971or 1198972) (1198981or 1198982) (1199061or 1199062) (1199041or 1199042))

(1)






0 1 minus 1199011119894119894

01 1199011

119894119894sdot (1 minus 1199011

119894119894)

119894119894

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 2 sdot

119894119894

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 119899 sdot

119894119894










Ge (119909 1199011119894119894) = (1minus1199011

119894119894) sdot 1199011119894119894

119909

119909 = 0 1 2 3 (2)



119905119894119895= (0 sdot (1minus1199011

119894119894)) oplus ((1199011

119894119894) sdot (1minus1199011

119894119894) otimes 119894119894)

oplus (2 (1199011119894119894)2sdot (1minus1199011


oplus (119899 (1199011119894119894)119899


=infin

sum119909=0

oplus ([119909 (1199011119894119894)119909


(3)



= (infin

sum119909=0

119909 sdot (1199011119894119894)119909



No

Yes

Yes

No

Yes

No















Are thereseries

branches

Are thereloops




(tln(119900-119898) p1ln(119900-119898)

p2ln(119900-119898)

)



= (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1

) sdot 1199011119894119894sdot (1minus1199011

119894119894) otimes 119894119894 (5)

We assume that

Ψ = (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1


1(1 minus 1199011

119894119894)2] (6)


119905119894119895=

1199011119894119894

(1 minus 1199011119894119894)otimes 119894119894 (7)


1015840

(119896)= (119896)oplus[(

1199011119894119894

1 minus 1199011119894119894


119884119894 (8)


ii

i

f

r

k

J




119894119895 1199011119894119895 119905119894119895) is a branch of


lated as follows

11990121015840119894119895= 1199012119894119895sdot (1minus1199011


119894119894) + 1199012119894119895


119894119894) + sdot sdot sdot + 1199012

119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899


=infin

sum119899=0

1199012119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899

sdot (1minus1199011119894119894)

= 1199012119894119895sdot (1minus1199011

119894119894)infin

sum119899=0

(1199012119894119894)119899

sdot (1199011119894119894)119899

(9)


997904rArr 11990121015840119896= 1199012119896sdot (1minus1199011

119894119894) sdot (1+

(1199012119894119894) sdot (1199011119894119894)

1 minus [(1199012119894119894) sdot (1199011119894119894)])

forall119896 isin 119885119884119894

(10)


119894119894) sdot

(1 + ((1199012119894119894) sdot (1199011119894119894))(1 minus [(1199012

119894119894) sdot (1199011119894119894)]))



(1+1199011119894119894

1 minus 1199011119894119894

) (11)




119886=

1199011119886

1199011119886+ 1199011119887+ 1199011119888

otimes[[[[

[


119886)

otimes(

1199011119887

1199011119887+ 1199011119888



oplus1199011119888

1199011119887+ 1199011119888



)]]]]

]

(12)


119879119901=

[[[[[[[[[[[

[

1199011119886

1199011119886+ 1199011119887+ 1199011119888


119886) otimes (

1199011119887

1199011119887+ 1199011119888


119887) otimes 119888) oplus

1199011119888

1199011119887+ 1199011119888


119888) otimes 119887))]

oplus1199011119887

1199011119886+ 1199011119887+ 1199011119888


119887) otimes (

1199011119886

1199011119886+ 1199011119888


119886) otimes 119888) oplus

1199011119888

1199011119886+ 1199011119888


119888) otimes 119886))]

oplus1199011119888

1199011119886+ 1199011119887+ 1199011119888


119888) otimes (

1199011119886

1199011119886+ 1199011119887


119886) otimes 119887) oplus

1199011119887

1199011119886+ 1199011119887


119887) otimes 119886))]

]]]]]]]]]]]

]

(13)


1 2 3 4 n minus 1 n






0 1 minus 1199011119894119894

1199012119894119895

1 1199011119894119894sdot (1 minus 1199011

119894119894) 1199012

119894119895sdot (1199012119894119894)

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)2

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)119899


1199011119879119901= 1199011119886+1199011119887+1199011119888 (14)

1199012119879119901= [1199012119886+ (1minus1199012

119886) (1199012119887+ (1minus1199012

119887) 1199012119888)] (15)




119889119886=(1198861 + 21198862 + 21198863 + 1198864)

6 (16)


Sor min (119889119886 119889119887 119889119888) = (1198891 1198892 1198893) 997904rArr 1 2 3 (17)

Therefore we have

119879119901= 1199012


2


22) otimes 3] (18)


1199011119879119901= 1 (19)

1199012119879119901= 1199012

1 + (1minus11990121) [1199012

2+ (1minus1199012

2) 11990123] (20)


119879119901= max (119905

119886 119887 119888) = [(1198861 or 1198871 or 1198881) (1198862 or 1198872 or 1198882)

(1198863 or 1198873 or 1198883) (1198864 or 1198874 or 1198884)] (21)


1199012119879119901= 1199012119886sdot 1199012119887sdot 1199012119888 (22)

1199011119879119901= 1 (23)







a

b

c


a

b

c


a

b

c


a

b

c


a

b

c


a

b

c




1199011119904= prod119896isin119878(119894-119895)

1199011119896= [1199011(1-2) times119901


1[(119899minus1)minus119899]]

1199012119879119904= prod119896isin119878(119894-119895)

1199012119896

= 1199012(1-2) times119901

2(2-3) times119901


2((119899minus1)minus119899)

(24)




o m

ln(o-m)

(p2ln(119900-119898)

p1ln(119900-119898)

tln(119900-119898) )




















1

(119900-119898) 1199012


branch (119905(119898-119900) 119901

1

(119898-119900) 1199012


1

ln(119900-119898) 1199012


ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)



1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)


11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)







ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1015840



10158401015840








lel branches


1= (1198971 1198981 1199061 1199041) and

2= (1198972 1198982 1199062




= (1198971+ 1198972 1198981+ 1198982 1199061+ 1199062 1199041+ 1199042)






2 ℎ lowast 119898

2 ℎ lowast 119906

2 ℎ lowast 119904

2)


1198971

1199042

1198981

1199062

1199061

1198982

1199041

1198972

)

max 1 2

= ((1198971or 1198972) (1198981or 1198982) (1199061or 1199062) (1199041or 1199042))

(1)






0 1 minus 1199011119894119894

01 1199011

119894119894sdot (1 minus 1199011

119894119894)

119894119894

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 2 sdot

119894119894

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 119899 sdot

119894119894










Ge (119909 1199011119894119894) = (1minus1199011

119894119894) sdot 1199011119894119894

119909

119909 = 0 1 2 3 (2)



119905119894119895= (0 sdot (1minus1199011

119894119894)) oplus ((1199011

119894119894) sdot (1minus1199011

119894119894) otimes 119894119894)

oplus (2 (1199011119894119894)2sdot (1minus1199011


oplus (119899 (1199011119894119894)119899


=infin

sum119909=0

oplus ([119909 (1199011119894119894)119909


(3)



= (infin

sum119909=0

119909 sdot (1199011119894119894)119909



No

Yes

Yes

No

Yes

No















Are thereseries

branches

Are thereloops




(tln(119900-119898) p1ln(119900-119898)

p2ln(119900-119898)

)



= (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1

) sdot 1199011119894119894sdot (1minus1199011

119894119894) otimes 119894119894 (5)

We assume that

Ψ = (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1


1(1 minus 1199011

119894119894)2] (6)


119905119894119895=

1199011119894119894

(1 minus 1199011119894119894)otimes 119894119894 (7)


1015840

(119896)= (119896)oplus[(

1199011119894119894

1 minus 1199011119894119894


119884119894 (8)


ii

i

f

r

k

J




119894119895 1199011119894119895 119905119894119895) is a branch of


lated as follows

11990121015840119894119895= 1199012119894119895sdot (1minus1199011


119894119894) + 1199012119894119895


119894119894) + sdot sdot sdot + 1199012

119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899


=infin

sum119899=0

1199012119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899

sdot (1minus1199011119894119894)

= 1199012119894119895sdot (1minus1199011

119894119894)infin

sum119899=0

(1199012119894119894)119899

sdot (1199011119894119894)119899

(9)


997904rArr 11990121015840119896= 1199012119896sdot (1minus1199011

119894119894) sdot (1+

(1199012119894119894) sdot (1199011119894119894)

1 minus [(1199012119894119894) sdot (1199011119894119894)])

forall119896 isin 119885119884119894

(10)


119894119894) sdot

(1 + ((1199012119894119894) sdot (1199011119894119894))(1 minus [(1199012

119894119894) sdot (1199011119894119894)]))



(1+1199011119894119894

1 minus 1199011119894119894

) (11)




119886=

1199011119886

1199011119886+ 1199011119887+ 1199011119888

otimes[[[[

[


119886)

otimes(

1199011119887

1199011119887+ 1199011119888



oplus1199011119888

1199011119887+ 1199011119888



)]]]]

]

(12)


119879119901=

[[[[[[[[[[[

[

1199011119886

1199011119886+ 1199011119887+ 1199011119888


119886) otimes (

1199011119887

1199011119887+ 1199011119888


119887) otimes 119888) oplus

1199011119888

1199011119887+ 1199011119888


119888) otimes 119887))]

oplus1199011119887

1199011119886+ 1199011119887+ 1199011119888


119887) otimes (

1199011119886

1199011119886+ 1199011119888


119886) otimes 119888) oplus

1199011119888

1199011119886+ 1199011119888


119888) otimes 119886))]

oplus1199011119888

1199011119886+ 1199011119887+ 1199011119888


119888) otimes (

1199011119886

1199011119886+ 1199011119887


119886) otimes 119887) oplus

1199011119887

1199011119886+ 1199011119887


119887) otimes 119886))]

]]]]]]]]]]]

]

(13)


1 2 3 4 n minus 1 n






0 1 minus 1199011119894119894

1199012119894119895

1 1199011119894119894sdot (1 minus 1199011

119894119894) 1199012

119894119895sdot (1199012119894119894)

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)2

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)119899


1199011119879119901= 1199011119886+1199011119887+1199011119888 (14)

1199012119879119901= [1199012119886+ (1minus1199012

119886) (1199012119887+ (1minus1199012

119887) 1199012119888)] (15)




119889119886=(1198861 + 21198862 + 21198863 + 1198864)

6 (16)


Sor min (119889119886 119889119887 119889119888) = (1198891 1198892 1198893) 997904rArr 1 2 3 (17)

Therefore we have

119879119901= 1199012


2


22) otimes 3] (18)


1199011119879119901= 1 (19)

1199012119879119901= 1199012

1 + (1minus11990121) [1199012

2+ (1minus1199012

2) 11990123] (20)


119879119901= max (119905

119886 119887 119888) = [(1198861 or 1198871 or 1198881) (1198862 or 1198872 or 1198882)

(1198863 or 1198873 or 1198883) (1198864 or 1198874 or 1198884)] (21)


1199012119879119901= 1199012119886sdot 1199012119887sdot 1199012119888 (22)

1199011119879119901= 1 (23)







a

b

c


a

b

c


a

b

c


a

b

c


a

b

c


a

b

c




1199011119904= prod119896isin119878(119894-119895)

1199011119896= [1199011(1-2) times119901


1[(119899minus1)minus119899]]

1199012119879119904= prod119896isin119878(119894-119895)

1199012119896

= 1199012(1-2) times119901

2(2-3) times119901


2((119899minus1)minus119899)

(24)




o m

ln(o-m)

(p2ln(119900-119898)

p1ln(119900-119898)

tln(119900-119898) )




















1

(119900-119898) 1199012


branch (119905(119898-119900) 119901

1

(119898-119900) 1199012


1

ln(119900-119898) 1199012


ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)



1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)


11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)







ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










No

Yes

Yes

No

Yes

No















Are thereseries

branches

Are thereloops




(tln(119900-119898) p1ln(119900-119898)

p2ln(119900-119898)

)



= (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1

) sdot 1199011119894119894sdot (1minus1199011

119894119894) otimes 119894119894 (5)

We assume that

Ψ = (infin

sum119909=1

119909 sdot (1199011119894119894)119909minus1


1(1 minus 1199011

119894119894)2] (6)


119905119894119895=

1199011119894119894

(1 minus 1199011119894119894)otimes 119894119894 (7)


1015840

(119896)= (119896)oplus[(

1199011119894119894

1 minus 1199011119894119894


119884119894 (8)


ii

i

f

r

k

J




119894119895 1199011119894119895 119905119894119895) is a branch of


lated as follows

11990121015840119894119895= 1199012119894119895sdot (1minus1199011


119894119894) + 1199012119894119895


119894119894) + sdot sdot sdot + 1199012

119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899


=infin

sum119899=0

1199012119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899

sdot (1minus1199011119894119894)

= 1199012119894119895sdot (1minus1199011

119894119894)infin

sum119899=0

(1199012119894119894)119899

sdot (1199011119894119894)119899

(9)


997904rArr 11990121015840119896= 1199012119896sdot (1minus1199011

119894119894) sdot (1+

(1199012119894119894) sdot (1199011119894119894)

1 minus [(1199012119894119894) sdot (1199011119894119894)])

forall119896 isin 119885119884119894

(10)


119894119894) sdot

(1 + ((1199012119894119894) sdot (1199011119894119894))(1 minus [(1199012

119894119894) sdot (1199011119894119894)]))



(1+1199011119894119894

1 minus 1199011119894119894

) (11)




119886=

1199011119886

1199011119886+ 1199011119887+ 1199011119888

otimes[[[[

[


119886)

otimes(

1199011119887

1199011119887+ 1199011119888



oplus1199011119888

1199011119887+ 1199011119888



)]]]]

]

(12)


119879119901=

[[[[[[[[[[[

[

1199011119886

1199011119886+ 1199011119887+ 1199011119888


119886) otimes (

1199011119887

1199011119887+ 1199011119888


119887) otimes 119888) oplus

1199011119888

1199011119887+ 1199011119888


119888) otimes 119887))]

oplus1199011119887

1199011119886+ 1199011119887+ 1199011119888


119887) otimes (

1199011119886

1199011119886+ 1199011119888


119886) otimes 119888) oplus

1199011119888

1199011119886+ 1199011119888


119888) otimes 119886))]

oplus1199011119888

1199011119886+ 1199011119887+ 1199011119888


119888) otimes (

1199011119886

1199011119886+ 1199011119887


119886) otimes 119887) oplus

1199011119887

1199011119886+ 1199011119887


119887) otimes 119886))]

]]]]]]]]]]]

]

(13)


1 2 3 4 n minus 1 n






0 1 minus 1199011119894119894

1199012119894119895

1 1199011119894119894sdot (1 minus 1199011

119894119894) 1199012

119894119895sdot (1199012119894119894)

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)2

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)119899


1199011119879119901= 1199011119886+1199011119887+1199011119888 (14)

1199012119879119901= [1199012119886+ (1minus1199012

119886) (1199012119887+ (1minus1199012

119887) 1199012119888)] (15)




119889119886=(1198861 + 21198862 + 21198863 + 1198864)

6 (16)


Sor min (119889119886 119889119887 119889119888) = (1198891 1198892 1198893) 997904rArr 1 2 3 (17)

Therefore we have

119879119901= 1199012


2


22) otimes 3] (18)


1199011119879119901= 1 (19)

1199012119879119901= 1199012

1 + (1minus11990121) [1199012

2+ (1minus1199012

2) 11990123] (20)


119879119901= max (119905

119886 119887 119888) = [(1198861 or 1198871 or 1198881) (1198862 or 1198872 or 1198882)

(1198863 or 1198873 or 1198883) (1198864 or 1198874 or 1198884)] (21)


1199012119879119901= 1199012119886sdot 1199012119887sdot 1199012119888 (22)

1199011119879119901= 1 (23)







a

b

c


a

b

c


a

b

c


a

b

c


a

b

c


a

b

c




1199011119904= prod119896isin119878(119894-119895)

1199011119896= [1199011(1-2) times119901


1[(119899minus1)minus119899]]

1199012119879119904= prod119896isin119878(119894-119895)

1199012119896

= 1199012(1-2) times119901

2(2-3) times119901


2((119899minus1)minus119899)

(24)




o m

ln(o-m)

(p2ln(119900-119898)

p1ln(119900-119898)

tln(119900-119898) )




















1

(119900-119898) 1199012


branch (119905(119898-119900) 119901

1

(119898-119900) 1199012


1

ln(119900-119898) 1199012


ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)



1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)


11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)







ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ii

i

f

r

k

J




119894119895 1199011119894119895 119905119894119895) is a branch of


lated as follows

11990121015840119894119895= 1199012119894119895sdot (1minus1199011


119894119894) + 1199012119894119895


119894119894) + sdot sdot sdot + 1199012

119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899


=infin

sum119899=0

1199012119894119895sdot (1199012119894119894)119899

sdot (1199011119894119894)119899

sdot (1minus1199011119894119894)

= 1199012119894119895sdot (1minus1199011

119894119894)infin

sum119899=0

(1199012119894119894)119899

sdot (1199011119894119894)119899

(9)


997904rArr 11990121015840119896= 1199012119896sdot (1minus1199011

119894119894) sdot (1+

(1199012119894119894) sdot (1199011119894119894)

1 minus [(1199012119894119894) sdot (1199011119894119894)])

forall119896 isin 119885119884119894

(10)


119894119894) sdot

(1 + ((1199012119894119894) sdot (1199011119894119894))(1 minus [(1199012

119894119894) sdot (1199011119894119894)]))



(1+1199011119894119894

1 minus 1199011119894119894

) (11)




119886=

1199011119886

1199011119886+ 1199011119887+ 1199011119888

otimes[[[[

[


119886)

otimes(

1199011119887

1199011119887+ 1199011119888



oplus1199011119888

1199011119887+ 1199011119888



)]]]]

]

(12)


119879119901=

[[[[[[[[[[[

[

1199011119886

1199011119886+ 1199011119887+ 1199011119888


119886) otimes (

1199011119887

1199011119887+ 1199011119888


119887) otimes 119888) oplus

1199011119888

1199011119887+ 1199011119888


119888) otimes 119887))]

oplus1199011119887

1199011119886+ 1199011119887+ 1199011119888


119887) otimes (

1199011119886

1199011119886+ 1199011119888


119886) otimes 119888) oplus

1199011119888

1199011119886+ 1199011119888


119888) otimes 119886))]

oplus1199011119888

1199011119886+ 1199011119887+ 1199011119888


119888) otimes (

1199011119886

1199011119886+ 1199011119887


119886) otimes 119887) oplus

1199011119887

1199011119886+ 1199011119887


119887) otimes 119886))]

]]]]]]]]]]]

]

(13)


1 2 3 4 n minus 1 n






0 1 minus 1199011119894119894

1199012119894119895

1 1199011119894119894sdot (1 minus 1199011

119894119894) 1199012

119894119895sdot (1199012119894119894)

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)2

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)119899


1199011119879119901= 1199011119886+1199011119887+1199011119888 (14)

1199012119879119901= [1199012119886+ (1minus1199012

119886) (1199012119887+ (1minus1199012

119887) 1199012119888)] (15)




119889119886=(1198861 + 21198862 + 21198863 + 1198864)

6 (16)


Sor min (119889119886 119889119887 119889119888) = (1198891 1198892 1198893) 997904rArr 1 2 3 (17)

Therefore we have

119879119901= 1199012


2


22) otimes 3] (18)


1199011119879119901= 1 (19)

1199012119879119901= 1199012

1 + (1minus11990121) [1199012

2+ (1minus1199012

2) 11990123] (20)


119879119901= max (119905

119886 119887 119888) = [(1198861 or 1198871 or 1198881) (1198862 or 1198872 or 1198882)

(1198863 or 1198873 or 1198883) (1198864 or 1198874 or 1198884)] (21)


1199012119879119901= 1199012119886sdot 1199012119887sdot 1199012119888 (22)

1199011119879119901= 1 (23)







a

b

c


a

b

c


a

b

c


a

b

c


a

b

c


a

b

c




1199011119904= prod119896isin119878(119894-119895)

1199011119896= [1199011(1-2) times119901


1[(119899minus1)minus119899]]

1199012119879119904= prod119896isin119878(119894-119895)

1199012119896

= 1199012(1-2) times119901

2(2-3) times119901


2((119899minus1)minus119899)

(24)




o m

ln(o-m)

(p2ln(119900-119898)

p1ln(119900-119898)

tln(119900-119898) )




















1

(119900-119898) 1199012


branch (119905(119898-119900) 119901

1

(119898-119900) 1199012


1

ln(119900-119898) 1199012


ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)



1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)


11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)







ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 n minus 1 n






0 1 minus 1199011119894119894

1199012119894119895

1 1199011119894119894sdot (1 minus 1199011

119894119894) 1199012

119894119895sdot (1199012119894119894)

2 (1199011119894119894)2

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)2

119873 (1199011119894119894)119899

sdot (1 minus 1199011119894119894) 1199012

119894119895sdot (1199012119894119894)119899


1199011119879119901= 1199011119886+1199011119887+1199011119888 (14)

1199012119879119901= [1199012119886+ (1minus1199012

119886) (1199012119887+ (1minus1199012

119887) 1199012119888)] (15)




119889119886=(1198861 + 21198862 + 21198863 + 1198864)

6 (16)


Sor min (119889119886 119889119887 119889119888) = (1198891 1198892 1198893) 997904rArr 1 2 3 (17)

Therefore we have

119879119901= 1199012


2


22) otimes 3] (18)


1199011119879119901= 1 (19)

1199012119879119901= 1199012

1 + (1minus11990121) [1199012

2+ (1minus1199012

2) 11990123] (20)


119879119901= max (119905

119886 119887 119888) = [(1198861 or 1198871 or 1198881) (1198862 or 1198872 or 1198882)

(1198863 or 1198873 or 1198883) (1198864 or 1198874 or 1198884)] (21)


1199012119879119901= 1199012119886sdot 1199012119887sdot 1199012119888 (22)

1199011119879119901= 1 (23)







a

b

c


a

b

c


a

b

c


a

b

c


a

b

c


a

b

c




1199011119904= prod119896isin119878(119894-119895)

1199011119896= [1199011(1-2) times119901


1[(119899minus1)minus119899]]

1199012119879119904= prod119896isin119878(119894-119895)

1199012119896

= 1199012(1-2) times119901

2(2-3) times119901


2((119899minus1)minus119899)

(24)




o m

ln(o-m)

(p2ln(119900-119898)

p1ln(119900-119898)

tln(119900-119898) )




















1

(119900-119898) 1199012


branch (119905(119898-119900) 119901

1

(119898-119900) 1199012


1

ln(119900-119898) 1199012


ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)



1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)


11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)







ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













a

b

c


a

b

c


a

b

c


a

b

c


a

b

c


a

b

c




1199011119904= prod119896isin119878(119894-119895)

1199011119896= [1199011(1-2) times119901


1[(119899minus1)minus119899]]

1199012119879119904= prod119896isin119878(119894-119895)

1199012119896

= 1199012(1-2) times119901

2(2-3) times119901


2((119899minus1)minus119899)

(24)




o m

ln(o-m)

(p2ln(119900-119898)

p1ln(119900-119898)

tln(119900-119898) )




















1

(119900-119898) 1199012


branch (119905(119898-119900) 119901

1

(119898-119900) 1199012


1

ln(119900-119898) 1199012


ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)



1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)


11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)







ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























1

(119900-119898) 1199012


branch (119905(119898-119900) 119901

1

(119898-119900) 1199012


1

ln(119900-119898) 1199012


ln(119900-119898) = (119900-119898) oplus (119898-119900)

1199011ln(119900-119898) = 1199011

(119900-119898) times1199011

(119898-119900)

1199012ln(119900-119898) = 1199012

(119900-119898) times1199012

(119898-119900)

(25)



1015840

(119896)= (119896)oplus[

[

(1199011ln(119900-119898)

1 minus 1199011ln(119900-119898)

)otimes ln(119900-119898)]

]

11990121015840(119896)

= 1199012(119896)sdot (1minus1199011

ln(119900-119898))

sdot [

[

1+(1199011ln(119900-119898) sdot 119901

2ln(119900-119898)

1 minus 1199011ln(119900-119898)

sdot 1199012ln(119900-119898)

)]

]

forall119896 isin 119885119884119900

(26)


11990111015840(1198961015840)= 1199011(1198961015840)(1+

1199011(119898-119900)

1 minus 1199011(119898-119900)

) forall1198961015840 isin 119885119884119898 (27)







ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












ln(8-9)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(1-16)

Zero Levele

Complex Loop

ln1(2-5)

Complex Loop

ln3(3-5)

Simple Loop

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ln0(11-12)

Zero Levele

Complex Loop

ln1(12-12)

Simple Loop(28)



1198790-17 = (5406 6862 8001 9437)

11990110-17 = 100

11990120-17 = 059

(29)













Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Yes

NoYes

Finish


two nodes


No



Start





Updating





1T119901 p2T119901 )

(tT119904 p1T119904 p2T119904 )


0 6 7 8 9

10

11 12

14

1716

a

b

c

d

e

15

13

54321





0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 6 7 8 9

10

11 12

14

1716d

e

15

13

54321







Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Code ofactivity





0-1 (4 5 6 7) (4 5 6 7) 1 11-2 (7 9 10 11) (168 2131 2438 2785) 084 12-3 (2 3 4 5) (757 1001 1223 1485) 082 13-4 (8 9 10 11) (1153 1333 1503 1689) 081 063-13 (12 14 15 18) (1553 1833 2003 2389) 076 044-5 (317 471 594 766) (317 471 594 766) 099 15-13 (1 2 2 3) (1 2 2 3) 09 113-15 (1 1 2 2) (1 1 2 2) 1 10-6 (4 5 6 8) (4 5 6 8) 1 16-7 (2 3 3 4) (2 3 3 4) 1 17-8 (4 5 6 7) (4 5 6 7) 095 18-9 (3 4 4 5) (5 667 667 833) 084 19-10 (d) (3 4 5 6) (3 4 5 6) 09 0429-10 (e) (2 4 5 7) (2 4 5 7) 08 05810-14 (1 1 1 1) (1 1 1 1) 1 16-11 (2 3 3 4) (2 3 3 4) 1 111-12 (6 7 8 9) (891 106 1229 1373) 068 112-14 (1 2 2 3) (154 308 362 462) 09 114-15 (2 3 4 4) (2 3 4 4) 1 115-16 (4 5 6 7) (4 5 6 7) 1 116-17 (1 2 2 3) (1 2 2 3) 1 1






the PCT







CPMsolution

(36 45 45 63) (45 65 65 106) 35



0-5 (606 990 1613 1974) 01210-7 (567 1004 1552 1931) 0879





07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










07

065

06

055

05

045

0 02 04 06 08

Succ

ess p

roba

bilit

y



110

100

90

80

70

60

50

0 02 04 06 08

PCT



12

1

08

06

04

02

010 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time




20

15

10

5

0

0 05 1

Expected iteration

Figure 11





References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article developing a mathematical model for...

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