Fuzzy critical path analysis based on centroid of centroids of fuzzy numbers and new subtraction method

Transcription

1 It. J. Mathematics i Operatioal Research, Vol. 5, No. 2, Fuzzy critical path aalysis based o cetroid of cetroids of fuzzy umbers ad ew subtractio method P. Phai Busha Rao* Departmet of Mathematics, GITAM Istitute of Techology, GITAM Uiversity, Visakhapatam , Idia peddipb@yahoo.com *Correspodig author N. Ravi Shakar Departmet of Applied Mathematics, GITAM Istitute of Sciece, GITAM Uiversity, Visakhapatam , Idia drravi68@gmail.com Abstract: This paper proposes a ew method to solve fuzzy critical path problems for a project etwork with activity times beig ormalised trapezoidal fuzzy umbers. is solved successfully to demostrate the validity of the proposed approach. Keywords: critical path method (CPM); fuzzy sets; trapezoidal fuzzy umbers; Referece to this paper should be made as follows: Busha Rao, P.P. ad Ravi Shakar, N. (2013) Fuzzy critical path aalysis based o cetroid of cetroids of fuzzy umbers ad ew subtractio method, It. J. Mathematics i Operatioal Research, Vol. 5, No. 2, pp Biographical otes: P. Phai Busha Rao is a Assistat Professor i the Departmet of Mathematics, GITAM Istitute of Techology, GITAM Uiversity, Visakhapatam, Idia ad also pursuig his PhD degree uder the guidace of N. Ravi Shakar i Departmet of Applied Mathematics, GITAM Istitute of Sciece, GITAM Uiversity, Visakhapatam, Idia. His research iterests are fuzzy optimisatio, decisio aalysis ad risk aalysis ad published about 11 papers i peer reviewed iteratioal jourals. N. Ravi Shakar is a Associate Professor i the Departmet of Applied Mathematics, GITAM Istitute of Sciece, GITAM Uiversity, Visakhapatam (Adhra Pradesh), Idia. His research iterests are applied group theory, rough set theory, fuzzy optimisatio, bio iformatics ad risk aalysis ad published about 26 papers i peer reviewed iteratioal jourals. 1 Itroductio Copyright 2013 Idersciece Eterprises Ltd.

2 206 P.P. Busha Rao ad N. Ravi Shakar cost, project maagemet has always bee a importat issue for several agecies ad idustrial orgaisatios. The etwork techiques used to hadle project aalysis are the Project Evaluatio ad Review Techique (PERT) ad the Critical Path Method (CPM). A project, ad more be performed accordig to some precedece costraits requirig that some activities caot start before some others are completed. Whe resource costraits are ot take ito accout, a project ca be represeted by a directed acyclic graph where the odes stad for activities ad the the completio time of the last task. Whe the activity times i a project are determiistic activities. There are also may cases where the activity times are ot determiistic, but radom assessmets, ad i this case, PERT, which is based o the probability theory, ca be employed. However, i real world applicatios, some activity times must be forecasted subjectively, like istead of stochastic assumptios which rely o historical data ad are out of reach i may cases to determie activity times. Hece, providig a precise estimate for the activity times at the begiig of the project may ot be possible. A alterative way to deal with situatios, ivolvig imprecisio i the data, is to employ the cocept of fuzziess proposed by Zadeh (1965), whereby the vague activity times ca be represeted by fuzzy sets. Several studies have ivestigated the case where the activity times i a project are ordial iformatio is available about the more or less plausible values of the duratios, which may or may ot be radom, the possibilistic approaches (Zadeh, 1978; Dubois ad Prade, 1988) ca be preferred to the probabilistic oes. Various approaches like the fuzzy shortest path ad the fuzzy PERT followed by the fuzzy CPM have bee proposed sice the late 1970s. The use of fuzzy umbers for project schedulig was started by Prade (1979) ad Dubois ad Prade (1980), followed by Chaas ad Kamburowski (1981) for the PERT etwork. Gazdik (1983) developed the FNET, which uses the theory of graphs besides fuzzy umbers. The above methods could determie possible values of earliest startig times by usig forward recursio, but failed to compute possible values of latest startig times, as the backward recursio takes the imprecisio of some duratios ito accout twice. If duratios are described by meas of fuzzy umbers ad as soo as the latest startig date of the last task is set to its earliest date, the backward recursio described i classical CPM gives egative times, which have o physical meaig. To overcome this problem, Kaufma ad Gupta (1988), Rommelfager (1994), Hapke et al. (1994) ad Hapke ad Slowiski (1996) proposed a backward recursio that relies o optimistic fuzzy subtractio, ad they provided good results. Their methods worked for geeral etworks. Nasutio (1994) was the oe who proposed a iteractive fuzzy subtractio i the backward calculatios ad resorted to symbolic computatios o variable duratio times, but this is highly combiatorial. McCaho ad Lee (1988) ad Yao ad Li (2000) the fuzzy arithmetic operatioal model to compute the latest startig time of each activity i a project etwork. Li ad Yao (2003) proposed a fuzzy critical path method based o preseted a algorithm to perform fuzzy critical path aalysis for project etwork problems.

3 Fuzzy critical path aalysis based o cetroid of cetroids 207 Zieliski (2005) has proposed polyomial algorithms for determiig possible values of latest startig times of activities. Che ad Huag (2007) preseted a aalytical method for measurig the criticality i a project etwork. Che (2007), Che ad Hsueh (2008) to critical path aalysis for a project etwork with activity times. Recetly, Liberatore (2008) has discussed some issues with critical path usig fuzzy activity times. times ad criticality of activities i etworks with miimal time lags ad imprecise a project etwork. Kumar ad Kaur (2011) proposed a ew subtractio method i fuzzy the latest startig times by backward recursio. Zareei at al. (2011) proposed a ew method for solvig critical path problem usig aalysis of evets. Although the method of obtaiig o-egative times for latest startig times have bee addressed earlier by several researchers, (1992) or Kaufma ad Gupta (1988). The proposed method is illustrated by a umerical shortcomig, a ew subtractio operatio is proposed i this paper, ad the proposed method is method to rak geeralised trapezoidal fuzzy umbers is preseted. I Sectio 4, a ew method of trapezoidal fuzzy umbers is proposed i Sectio 6. I Sectio 7, the advatages of the 2 Prelimiaries umbers ad arithmetic operatios of trapezoidal fuzzy umbers, which are the basis for the developmet of fuzzy set theory, are reviewed. 1980) ad Kaufma ad Gupta, 1985) are reviewed.

4 208 P.P. Busha Rao ad N. Ravi Shakar (Dubois ad Prade, 1980): Let X be a classical set of objects, called the uiverse. The characteristic fuctio f A of a crisp set A X assigs a value either 0 or 1 to each member i X. This fuctio ca be geeralised to a fuctio fa such that the value assiged to the elemet of the uiversal set X f : X 01,. The A assiged value idicates the membership grade of the elemet i set A. The fuctio is fa called the membership fuctio ad the set A each x X is called a fuzzy set., { ( ), } A = ( ) [ ] x f x x X f ( x)for A (Dubois ad Prade, 1980): A fuzzy set A real umbers R, is said to be a fuzzy umber if its membership fuctio has the followig properties: (i) A f ( λx+ (1 λ) y) mi ( f ( x), f ( y) ) x, y X, λ [ 0,1] A A A (ii) A x 0 R with f (iii) f A ( x)is piecewise cotiuous. ( x ) =1. (Dubois ad Prade, 1980): A fuzzy umber A is called positive fuzzy umber if its membership fuctio is such that f ( x)= 0, x< 0. A (Kaufma ad Gupta, 1985): A fuzzy umber A = ( abcdw,,, ; ) is said to be a trapezoidal fuzzy umber if its membership fuctio is give by A 0 wx ( a), a x b, b a w, b x c, f ( x) = Α wx ( d), c x d, c d 0, otherwise. where abcd,,, R. If w = 1, the A = ( abcd,,, ) is a ormalised trapezoidal fuzzy umber ad if 0 < w < 1, Α is said to be a geeralised or o-ormal trapezoidal fuzzy umber. (Kaufma ad Gupta, 1985): A trapezoidal fuzzy umber A = ( abcd,,, ) is said to be a zero trapezoidal fuzzy umber if ad oly if a=0, b=0, c=0, d=0. (Kaufma ad Gupta, 1985): A trapezoidal fuzzy umber A = ( abcd,,, ) is said to be a o-egative trapezoidal fuzzy umber if ad oly if a 0.

5 Fuzzy critical path aalysis based o cetroid of cetroids 209 (Kaufma ad Gupta, 1985): Two trapezoidal fuzzy umbers Α a, b, c, d ad B a2, b2, c2, d are said to be equal i.e. 2 A B a = a, b = b, c = c, d = d = ( ) 2.2 Arithmetic operatios betwee trapezoidal fuzzy umbers = ( ) I this subsectio, additio ad subtractio operatios betwee two trapezoidal fuzzy umbers take from Kaufma ad Gupta (1985) are reviewed. Let Α a1, b1, c1, d ad 1 B a2, b2, c2, d be two trapezoidal fuzzy umbers, the the 2 (i) Fuzzy umbers additio of Α ad B is deoted by A B ad is give by A B=( a, b, c, d ) ( a, b, c, d ) = ( ) = ( ) ( ) = a + a, b + b, c + c, d + d (ii) Fuzzy umbers subtractio of Α ad B is deoted by Α B ad is give by A B a1, b1, c1, d1 a, b, c, d = ( a d, b c, c b, d a ) = ( ) ( ) Sice the iceptio of fuzzy sets by (Zadeh, 1965), may authors have proposed differet methods which gives a satisfactory result to all situatios. It is true that fuzzy umbers are frequetly partial order, ad caot be compared like real umbers, which ca be liearly ordered. to rak fuzzy quatities are proposed by various researchers. Amog various rakig methods that are available i the literature, the cetroid-based rakig methods have their ow place i ito a crisp umber by usig a rakig fuctio which is based o the cetroid of the fuzzy umber. This rakig fuctio assigs a real umber to each fuzzy umber, so that a atural origi. Later o, rakig fuzzy umbers by area betwee the cetroid poit ad origial poit was proposed by Chu ad Tsao (2002). These methods produced biased results due to icorrect cetroid formula of the trapezoidal fuzzy umber, ad moreover, these methods failed to rak crisp umbers. Later o, several researchers addressed this problem, ad Wag et al. (2006) corrected the cetroid formula proposed by Cheg (1998). Most of the methods proposed so far are o- discrimiatig ad couter-ituitive, ad some produce differet rakigs for the same situatio, whereas some methods caot rak crisp umbers, which are a special case of

6 210 P.P. Busha Rao ad N. Ravi Shakar ew method of rakig geeralised trapezoidal fuzzy umbers which is based o the cetroid of cetroids of trapezoidal fuzzy umber ad spreads. 3.1 I this subsectio, a ew method to rak geeralised trapezoidal fuzzy umbers is preseted. To rak fuzzy quatities, each fuzzy quatity is coverted ito a real umber ad compared the whole of ay aalysis to a sigle umber, much of the iformatio is lost, ad most of the rakig methods cosider oly oe poit of view o comparig fuzzy quatities. Hece, a attempt is to be made to miimise this loss. The cetroid of a trapezoid is cosidered to be the balacig poit of the trapezoid triagle (APB), a rectagle (BPQC) ad agai a triagle (CQD). Let the cetroids of the be G1, G2 & G respectively. The cetroid of these cetroids 3 G1, G2 & G3 is take as the umbers. The reaso for selectig this poit as a poit of referece is that each cetroid poit: G 1 of triagle APB, G 2 of rectagle BPQC ad G 3 of triagle CQD is a balacig poit for each idividual better balacig poit for a geeralised trapezoidal fuzzy umber. Figure 1 Cetroid of cetroids Cosider a geeralised trapezoidal fuzzy umber Α = ( abcdw,,, ; ) (CQD) are: a+ 2b w G1 =, 3 3 ; b+ c w G2 =, 2 2 ad 2c+ d w G3 =, 3 3 respectively. Equatio of the lie GG 1 3 is w y = ad G 2 does ot lie o the lie GG

7 Fuzzy critical path aalysis based o cetroid of cetroids 211 Therefore, G 1, G 2 ad G 3 are o-colliear ad they form a triagle. G x y ( ) of the triagle with vertices G 1, G 2 ad G 3 of the, Α 0 0 geeralised trapezoidal fuzzy umber A = ( abcdw,,, ; ) as 2a+ 7b+ 7c+ 2d 7w G ( x, y, Α 0 0)= (1) As a special case, for triagular fuzzy umber A = ( abdw,, ; ) i.e., c = b the cetroid of cetroids is give by a+ 7b+ d 7w G ( x, y, Α 0 0)= 9 18 (2) The rakig fuctio of the geeralised trapezoidal fuzzy umber A = ( abcdw,,, ; ) which, R = = Α 2a 7b 7c 2d 7w x0 y (3) This is the area betwee the cetroid of the cetroids G ( x, y Α 0 0) origial poit. The mode of the geeralised trapezoidal fuzzy umber A = ( abcdw,,, ; ) mode = 1 w w b c dx b c 2 ( + ) = ( + ) (4) 2 0 The spread of the geeralised trapezoidal fuzzy umber A = ( abcdw,,, ; ) w 0 ( ) = ( ) spread = d a dx w d a The left spread of the geeralised trapezoidal fuzzy umber A = ( abcdw,,, ; ) (5) w left spread = b a dx w b a 0 ( ) = ( ) (6) The right spread of the geeralised trapezoidal fuzzy umber A = ( abcdw,,, ; ) is defied as: w right spread = d c dx w d c 0 ( ) = ( ) (7)

8 212 P.P. Busha Rao ad N. Ravi Shakar trapezoidal fuzzy umbers. Let A = ( a1, b1, c1, d1; w1) ad B = ( a2, b2, c2, d2; w2) be two geeralised trapezoidal fuzzy umbers. The workig procedure to compare Aad B is as follows: Step 1: Fid R A ad R B Case i: If R >R A B AB, Case ii: If R >R A B AB, { } = A ad miimum of{ AB, } = B { } = B ad miimum of { AB, } = A Case iii: If R >R A B compariso is ot possible, the go to step 2. Step 2: Fid mode A ad mode B Case i: If mode A > mode B { AB, } = B { AB, } = A ad miimum of Case ii: If mode A B AB, { AB, } = A { } = B ad miimum of Case iii: If mode A = mode B compariso is ot possible, the go to step 3. Step 3: Fid spread A ad spread B Case i: If spread A > spread B AB { AB, } = A {, } = B ad miimum of

9 Fuzzy critical path aalysis based o cetroid of cetroids 213 Case ii: If spread A < spread B { AB, } = B { AB, } = A ad miimum of Case iii: If spread A = spread B compariso is ot possible, the go to step 4. Step 4: Fid left spread A ad left spread B Case i: If left spread A > left spread B the AB, { } = A ad miimum of { AB, } Case ii: If left spread A < left spread B the AB, { } = B ad miimum of { AB, } Case iii: If left spread A = left spread B the for ormal trapezoidal fuzzy umbers { AB, } fuzzy umbers. = miimum of AB, = B = A { } A B ad go to step 5 for o-ormal Step 5 w 1 ad w 2 Case i: If w Case ii: If w Case iii: If w > w AB, 1 2 { } < w AB, 1 2 = w AB 1 2 = A ad miimum of { AB, } = B { } = B ad miimum of { AB, } = A {, = miimum of AB, A B } { }

10 214 P.P. Busha Rao ad N. Ravi Shakar 3.2 Some importat results procedure i Sectio 3.1, are preseted. Propositio liear fuctio for the ormalised trapezoidal fuzzy umber Α = ( abcd1,,, ; ) i.e. R = Α 2a 7b 7c 2d (8) If A = ( a1, b1, c1, d1) ad B = ( a2, b2, c2, d2) are two ormalised trapezoidal fuzzy umbers, the (i) R (ii) R (iii) R = R R k1a k2 B k1 A k2 B ; k1, k2 R = R A A ( A) ( A) = 0 Remark 1: Throughout the critical path aalysis, we cosider oly ormalised trapezoidal fuzzy umbers to represet activity duratios i a project etwork. Hece, i the rakig fuctio, we ca drop the quatity 7 w as it is commo to all ormalised trapezoidal fuzzy umbers. 18 Moreover, i the procedure of discrimiatig fuzzy umbers, step 5 is ot ecessary. Therefore, R Α 2a 7b 7c 2d = 18 (9) The role of a project maager is to miimise the time take for completio of the project. The critical path aalysis is based o the computatio of latest startig times of activities from the kowledge of the earliest edig time of the project. Whe duratios of the activities critical paths ad activities. However, precise iformatio about the duratio of the activities is seldom available i the real world. Hece there is always a ucertaity about the time duratios of activities i etwork plaig. This has led to the developmet of the Fuzzy Critical Path Method (FCPM). I this sectio, the otatios that are used throughout the critical path aalysis, some eviromet are preseted.

11 Fuzzy critical path aalysis based o cetroid of cetroids Notatios N = { 123,,,..., } : the set of all the odes i a project etwork A ij : the activity betwee odes i ad j FAT ij : the fuzzy ormal time of activity A ij FES j : the fuzzy earliest startig time of ode j FLF j j FTF ij S (j) F (j) NS (j) NP (j) P i P FCT ( P C ) A ij : the set of all successor activities of ode j : the set of all predecessor activities of ode j : the set of all odes coected to all successor activities of ode j i.e. NS( j) = { k / A S( j), k N} jk : The set of all odes coected to all predecessor activities of ode j i.e NP( j) = { i / A F( j), i N} : the i-th path ij : the set of all paths i a project etwork : the fuzzy completio time of the fuzzy critical path P C i a project etwork. 4.2 Properties Outlies of the importat properties that will be used i the proposed method ad to solve Set the iitial ode as zero for startig i.e. FES 1 = ( 0000,,, ), the the followig will hold. j i ij 1. FES = max{ FES FAT / i NP( j), j 1, j N} 2. FLF j = mi { FLF k FAT / k NS( j), j, j N} jk 3. FTF ij = ( FLF j FES i ) FAT ij,1 i < j ;, i j N. c ij i, j P c 4. FCT ( P ) = FAT, P P C.

12 216 P.P. Busha Rao ad N. Ravi Shakar A ij i a project etwork with the property mi { FTF A }, the the activity A ij is said to be a critical activity. ij 4.3 Proposed method ij path of a project etwork i a fuzzy eviromet. After idetifyig the activities i a project ad establishig the precedece relatioships of all activities alog with the fuzzy ormal time with respect to each activity, a project etwork is costructed. The steps of the proposed method are as follows: Step 1: Set FES 1 = ( 0000,,, ) ad calculate FES j, j = 23,,..., by usig the property of 1 i Sectio 4.2. Step 2: Set FLF = FES ad calculate FLF j j, = 1, 2,..., 1 by usig property 2 of Sectio 4.2. Step 3: Calculate FTF ij with respect to each activity i a project etwork by usig property 3 of Sectio 4.2. Step 4: Step 5: Fid fuzzy critical path P c by combiig all the fuzzy critical activities obtaied i step 4. Step 6: Calculate FCT ( P c ) of the fuzzy critical path P c of a project etwork obtaied i step 5 by I this sectio, a hypothetical project problem is preseted to demostrate the computatioal process of the fuzzy critical path aalysis that has bee proposed. The proposed method is Sectio 3 is used so that the fuzzy umbers ca easily be compared to the rakig fuctio used by Liag ad Ha (2004). Suppose there is a project etwork, as show i Figure 2, with the set of odes N = {1, 2, 3, 4}, ad the fuzzy activity time for each activity is take as a trapezoidal fuzzy umber. critical path ad fuzzy completio time of the project etwork, show i Figure 2, i which the fuzzy ormal time of each activity is represeted by the followig trapezoidal umbers. The reaso for usig the trapezoidal fuzzy umbers is that it is easy to use ad iterpret.

13 Fuzzy critical path aalysis based o cetroid of cetroids 217 of all the activities are i hours. Figure 2 Project etwork with fuzzy time duratio of each activity as trapezoidal fuzzy umber FAT 12 =(3,5,5,7), FAT 13 =(5,10,10,15), FAT =(1,3,4,5), 23 FAT 24 Solutio: =(2,4,5,6), FAT =(6,8,10,11) 34 completio time of the project etwork show i Figure 2 ca be obtaied by usig the followig steps: Step 1: By assumig FES 1 = ( 0000,,, ), the values of FES j, j=2, 3, 4 ca be obtaied as follows: FES 2 = FES 1 FAT 12 ( ) ( ) = ( 3557,,, ) = 0000,,, 3557,,, ( 0, 0, 0, 0) ( 5, 10, 101, 15),( 3, 5, 5, 7) ( 1, 3, 4, 5)} ( 0, 0, 0, 0) ( 5, 10, 101, 15),( 3, 5, 5, 7) ( 1, 3, 4, 5)} R ( ,,, ) = =, R (,,, ) = = Sice R ( ,,, ) > R( 48912,,, )

14 218 P.P. Busha Rao ad N. Ravi Shakar i.e., FES 3 = (5, 10, 10, 15) FES = max{ FES FAT, FES FAT } = max{( ,,, ) ( ,,, ),( 3557,,, ) ( 2456,,, )} = max{( 11, 18, 20, 26),( 5, 9, 10, 13)} R ( 11, 18, 20, 26) = , R ( 5, 9, 10, 13) = Sice R ( 11, 18, 20, 26) > R( 5, 9, 10, 13) {( 11, 18, 20, 26),( 5, 9, 10, 13)} = ( 11, 18, 20, 26) i.e. FES 4 = ( 11, 18, 20, 26) Step 2: By assumig FLF 4 = ( 11, 18, 20, 26), the values of FLF3, j = 321,, ca be obtaied as follows: FLF = FLF FAT = (11,18,20,26) (6,8,10,11) = (0,8,12,20) FLF 2 = mi { FLF 4 FAT, 24 FLF 3 FAT } 23 = mi {(11, 18, 20, 26) (2, 4, 5, 6), (0, 8, 12, 20) (1, 3, 4, 5)} = mi {(5, 13, 16, 24), ( 5, 4, 9, 19)} R (, ,, ) = 145., R( 54919,,, ) = 661. Sice R (, 51316,, 24) > R( 54919,,, ) So, miimum {(5, 13, 16, 24), ( 5, 4, 9, 19)} = ( 5, 4, 9, 19) i.e., FLF 2 = ( 5, 4, 9, 19) FLF 1 = mi { FLF 3 FT, 13 FLF 2 FAT } 12 = mi {(0, 8, 12, 20) (5, 10, 10, 15), ( 5, 4, 9, 19) (3, 5, 5, 7)} = mi {( 15, 2, 2, 15), ( 12, 1, 4, 16)} R ( 15, 2, 2, 15) = 0, R ( 12, 1, 4, 16) = Sice R ( 15, 2, 2, 15) <R ( 12, 1, 4, 16) So, miimum {( 15, 2, 2, 15), ( 12, 1, 4, 16)} = ( 15, 2, 2, 15) i.e., FLF 1 = ( 15, 2, 2, 15) Step 3: Calculate FTF ij with respect to each activity by usig property 3 FTF 12 = ( FLF 2 FES 1 ) FAT 12 = (( 5, 4, 9, 19) (0, 0, 0, 0)) (3, 5, 5, 7)

15 Fuzzy critical path aalysis based o cetroid of cetroids 219 = ( 12, 1, 4, 16) FTF 13 = ( FLF 3 FES 1 ) FAT 13 = ((0, 8, 12, 20) (0, 0, 0, 0)) (5, 10, 10, 15) = ( 15, 2, 2, 15) FTF 23 = ( FLF 3 FES 2 ) FAT 23 = ((0, 8, 12, 20) (3, 5, 5, 7)) (1, 3, 4, 5) = ( 12, 1, 4, 16) FTF 24 = ( FLF 4 FES 2 ) FAT 24 = ((11, 18, 20, 26) (3, 5, 5, 7)) (2, 4, 5, 6) = ( 2, 8, 11, 21) FTF 34 = ( FLF 4 FES 3 ) FAT 34 = ((11, 18, 20, 26) (5, 10, 10, 15)) (6, 8, 10, 11) = ( 15, 2, 2, 15) Step 4: R ( FTF 12 ) = R( 12, 1, 4, 16) = R ( FTF 13 ) = R( 15, 2, 2, 15) = 0 R ( FTF 23 ) = R( 12, 1, 4, 16) = R ( FTF 24 ) = R( 2, 8, 11, 21) = 9. 5 R ( FTF 34 ) = R( 15, 2, 2, 15) = 0 Hece, activities (1, 3) ad (3, 4) are fuzzy critical activities. Step 5: Combiig all the fuzzy critical activities obtaied i step 4, the fuzzy critical path is1 3 4( sayp c ). Step 6: Calculate FCT ( P c ) step 5 by usig property 4: of the fuzzy critical path P c of a project etwork obtaied i FCT ( Pc ) = FAT13 FAT34 = ( ,,, ) ( ,,, ) = ( ,,, ). (11, 18, 20, 26).

16 220 P.P. Busha Rao ad N. Ravi Shakar I a crisp theory, the additio ad subtractio operatios are always iverse operatios. It meas that A ad B always satisfy the relatio A + B B = A. I a fuzzy theory, the additio ad subtractio operatios are ot always iverse operatios i.e., A ad B B do ot satisfy the relatio ( A ) B = A. Therefore, the fuzzy backward pass has a FLF 2 from a fuzzy subtractio, amely {(0, 8, 12, 20) (1, 3, 4, 5)} = (-5, 4, 9, 19). It is clear that this trapezoidal fuzzy umber, which represets time, has a egative part i it, ad we kow that egative time has o physical meaig ad is ot feasible. To avoid this problem, we propose a ew subtractio betwee two trapezoidal fuzzy umbers. If A = ( a1, b1, c1, d1) ad B = ( a2, b2, c2, d2) are two trapezoidal fuzzy umbers, the the ew subtractio operatio R betwee A ad B as follows: A R B = ( max( 0, a1 d2), max( 0, b1 c2), max( 0, c1 b2), max( 0, d1 a2) ) (i) The result of this subtractio is a trapezoidal fuzzy umber with o egative part. We demostrate this ew subtractio operatio R Let A = (10, 15, 30, 40) ad B = (12, 13, 15, 20), the ( ) A R B = max( 0, a d ), max( 0, b c ), max( 0, c b ), max( 0, d a ) = (0, 0, 17, 28) I this sectio, we demostrate the advatages of usig the proposed subtractio tha the show that by usig the proposed subtractio, the shortcomig poited out i Sectio 5 for

17 Fuzzy critical path aalysis based o cetroid of cetroids 221 Solutio completio time of the project etwork show i Figure 2 ca be obtaied by usig the followig steps: Step 1: By assumig FES 1 = ( 0000,,, ), the values of FES j, j = 2, 3, 4 will be the same as i earliest startig times of the project etwork. FES 2 FES 3 = ( 3557,,, ) = (5, 10, 10, 15) FES 4 = ( 11, 18, 20, 26) Step 2: By assumig FLF 4 = ( 11, 18, 20, 26), the values of FLF3, j = 321,, ca be obtaied as follows: FLF = FLF R FAT = (11,18,20,26) R (6,8,10,11) = (0,8,12,20) FLF 2 = mi { FLF 4 R FAT, 24 FLF 3 R FAT } 23 = mi {(11, 18, 20, 26) R (2, 4, 5, 6), (0, 8, 12, 20) R (1, 3, 4, 5)} = mi {(5, 13, 16, 24), (0, 4, 9, 19)} R (, ,, ) = 145., R (, 04919,, ) = 694. Sice R (, 51316,, 24) > R(, ,, ) So, miimum {(5, 13, 16, 24), (0, 4, 9, 19)} = (0, 4, 9, 19) i.e., FLF 2 = (0, 4, 9, 19) FLF 1 = mi { FLF 3 R FAT, 13 FLF 2 R FAT } 12 = mi {(0, 8, 12, 20) R (5, 10, 10, 15), (0, 4, 9, 19) R (3, 5, 5, 7)} = mi {(0, 0, 2, 15), (0, 0, 4, 16)} R (,,, 00215) = 244., R (,,, 00416) = 333. Sice R (,,, 00215) < R(,,, 00416) So, miimum {(0, 0, 2, 15), (0, 0, 4, 16)} = (0, 0, 2, 15) i.e., FLF 1 = (0, 0, 2, 15) Step 3: Calculate FTF ij with respect to each activity by usig property 3 FTF 12 = ( FLF 2 R FES 1 ) R FAT 12

18 222 P.P. Busha Rao ad N. Ravi Shakar = ((0, 4, 9, 19) R (0, 0, 0, 0)) R (3, 5, 5, 7) = (0, 0, 4, 16) FTF 13 = ( FLF 3 R FES 1 ) R FAT 13 = ((0, 8, 12, 20) R (0, 0, 0, 0)) R (5, 10, 10, 15) = (0, 0, 2, 15) FTF 23 = ( FLF 3 R FES 2 ) R FAT 23 = ((0, 8, 12, 20) R (3, 5, 5, 7)) R (1, 3, 4, 5) = (0, 0, 4, 16) FTF 24 = ( FLF 4 R FES 2 ) R FAT 24 = ((11, 18, 20, 26) R (3, 5, 5, 7)) R (2, 4, 5, 6) = (0, 8, 11, 21) FTF 34 = ( FLF 4 R FES 3 ) R FAT 34 = ((11, 18, 20, 26) R (5, 10, 10, 15)) R (6, 8, 10, 11) = (0, 0, 2, 15) Step 4: Fid the fuzzy critical activity of a project etwork R ( FTF 12 ) = R ( 00416,,, ) = 333. R ( FTF 13 ) = R ( 00215,,, ) = 244. R ( FTF 23 ) = R ( 00416,,, ) = 333. R ( FTF 24 ) = R ( 0, 8, 11, 21) = R ( FTF 34 ) = R ( 00215,,, ) = 244. Hece activities (1, 3) ad (3, 4) are fuzzy critical activities. Step 5: Combiig all the fuzzy critical activities obtaied i step 4, the fuzzy critical path is1 3 4( sayp c ). Step 6: Calculate FCT ( P c ) step 5 by usig property 4: of the fuzzy critical path P c of a project etwork obtaied i FCT ( Pc ) = FAT13 FAT34 = ( ,,, ) ( ,,, ) = ( ,,, ).

19 Fuzzy critical path aalysis based o cetroid of cetroids 223 I this paper, we proposed ad developed a ew method based o the fuzzy set theory for method, duratios of the activities are cosidered to be positive trapezoidal fuzzy umbers o physical meaig, as they produce egative times. By applyig the ew subtractio operatio which may be called Rao-Ravi subtractio, proposed for o-egative trapezoidal fuzzy umbers, it is foud that it works well o fuzzy backward pass calculatios ad removes Aother advatage of the proposed method is that it removes multiple critical paths, as the discrimiate betwee all types of fuzzy umbers ad restricts multiple critical paths. duratios, but does ot cosider fuzziess i other parameters like resource requiremet. The The authors would like to thak the editor-i-chief ad the aoymous referee for providig very helpful commets ad various suggestios. Their isight ad commets have led to a improvemet i the quality as well as i presetig the paper i a better way. Refereces Fuzzy Sets ad Systems, Vol. 5, No.1, pp Che, C.T. ad Huag, S.F. (2007) Applyig fuzzy method for measurig criticality i project etwork, Iformatio Scieces, Vol. 177, No. 2, pp Che, S.P. (2007) Aalysis of critical paths i a project etwork with fuzzy activity times, Europea Joural of Operatioal Research, Vol. 183, No. 1, pp etworks, Applied Mathematical Modellig, Vol. 32, No. 7, pp Cheg, C.H. (1998) A ew approach for rakig fuzzy umbers by distace method, Fuzzy Sets ad Systems, Vol.95, No. 3, pp Chu, T.C. ad Tsao, C.T. (2002) Rakig fuzzy umbers with a area betwee the Cetroid poit ad origial poit, Computers ad Mathematics with Applicatios, Vol.43, No. 1 2, pp Dubois, D. (1979) Fuzzy real algebra: some results, Fuzzy Sets ad Systems, Vol. 2, No. 4, pp Dubois, D. ad Prade, H. (1988) Possibility Theory A Approach to Computerized Processig of Ucertaity, Pleum Press, New York. vs. copig with icomplete kowledge, Europea Joural of Operatioal Research, Vol. 147, No. 2, pp

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