Internatonal Journal of Mathematcal rchve-(), 0, 477-474 valable onlne through www.jma.nfo ISSN 9 5046 FUZZY CRITICL PTH METHOD (FCPM) BSED ON SNGUNST ND CHEN RNKING METHOD ND CENTROID METHOD Dr. S. Narayanamoorthy* & S. Maheswar ssstant Professor, Department of ppled Mathematcs, Bharathar Unversty, Combatore-64046 Research Scholar, Department of ppled Mathematcs, Bharathar Unversty, Combatore-64046 (Receved on: -0-; Revsed & ccepted on: 8--) BSTRCT Ths paper presents a method for fndng crtcal path n the fuzzy project network. Trangular fuzzy numbers are used to represent actvty tmes n the project network. We used rankng method and defuzzfcaton formula on fuzzy numbers. In Sanguansat and Chen Rankng Method we used set of fuzzy numbers and centrod method we used ndvdual fuzzy numbers. numercal example s gven and we compared the trangular fuzzy number for Sanguansat and Chen Rankng Method and centrod method va graph. Keywords: FCPM, fuzzy project network, trangular fuzzy numbers, rankng method, defuzzfcaton. I. INTRODUCTION The crtcal path s the one from the start of the project to fnsh of project where the slack tmes are all zeros. project network s defned as a set of actvtes that must be performed accordng to precedence constrants statng whch actvtes must start after the completon of specfed other actvtes. The operaton tme for each actvty n the fuzzy project network s characterzed as a postve trangular fuzzy number. In accordance wth CPM, the forward pass yelds the fuzzy earlest-start and earlest-fnsh tmes The backward pass s performed to calculate the fuzzy latest-start and latest fnsh tmes [6], [9] and [0]. Besdes, CPM has proved very valuable n evaluatng project performance and dentfyng bottlenecks. Thus, CPM s a vtal tool for the plannng and control of complex projects. path through a project network s one of the routes from the startng node to the endng node. ccordng to the crtcal path, the decson-maker can control the tme and the cost of the project. The length of a path s the sum of the duratons of the actvtes on the path. The project duraton equals the length of the longest path through the project network. The longest path s called the crtcal path n the network.cpm has been used n busness management, factory producton, etc. [], [] extended the fuzzy arthmetc operatons model to compute the latest startng tme of each actvty n a project network. [] and [8] used fuzzy arthmetc operatons to compute the earlest startng tme of each actvty n a project network. In ths paper, we used Sanguansat and Chen rankng method and defuzzfcaton formula for trangular fuzzy numbers and appled to the float tme for each actvty n the fuzzy project network to fnd the crtcal path. In ths paper we compared these two methods and whch one s more effectve to determnng the crtcal path n the fuzzy project network.. PRELIMINRIES Defnton.. Let X be a set. fuzzy set on X s defned to be a functon : X [ 0,] or µ : X [ 0,] Equvalently, a fuzzy set s defned to be the class of objects havng the followng representaton {(, µ ( ) : )} where X [ ] x x x X µ : 0,, s a functon called the membershp functon of. Defnton.. The fuzzy number s a fuzzy set whose membershp functon ( x ). µ ( x ) s pecewse contnuous; µ satsfes the followng condtons. fuzzy set of the unverse of dscourse X s convex. fuzzy set of the unverse of dscourse X s called a normal fuzzy set f x X, µ ( x) Correspondng author: Dr. S. Narayanamoorthy* ssstant Professor, Department of ppled Mathematcs, Bharathar Unversty, Combatore-64046. Internatonal Journal of Mathematcal rchve- (), Nov. 0 477
Dr. S. Narayanamoorthy* & S. Maheswar/ FUZZY CRITICL PTH METHOD (FCPM) BSED ON SNGUNST ND CHEN./ IJM- (), Nov.-0. Defnton.. fuzzy number wth membershp functon n the form µ x a, a x < b, b a c x x, b x c, c b 0, otherwse ( ) s called a trangular fuzzy number ( a, b, c). Defnton.4. Let and be two trangular fuzzy numbers parameterzed by ( a, a, a ) and ( b, b, b ) respectvely. The smplfed fuzzy number arthmetc operatons between the trangular fuzzy numbers and are as follows: Fuzzy numbers addton : ( a, a, a ) ( b, b, b ) ( a + b, a + b, a + b ) Fuzzy numbers subtracton : ( a, a, a ) ( b, b, b ) ( a b, a b, a b ) For example: Let and be two trangular fuzzy numbers, where (,, ) and (5, 6, 7). Then, (,, ) (5, 6, 7) (6, 8, 0) (,, ) (5, 6, 7) ( -7, -5, -). FUZZY CRITICL PTH LGORITHM Consder the fuzzy project network, where the duraton tme of each actvty n a fuzzy project network s represented by trangular fuzzy numbers. Step: Calculate E s s and E f s by usng ths equaton and f s E E t from ths we can obtan the earlest fuzzy event tme. { } Step : Calculate L f s and L s s by usng ths equaton f f L Mn L j t j and s f L L t j P ( ) we can obtan the latest fuzzy event tme. from ths Step : Calculate F T s for each actvty (,j) usng T F s s L E or F f f T L E From ths we can obtan the total fuzzy float tme of each actvty. Step 4: Rank the total slack fuzzy tme of each path usng Sanguansat and Chen rankng method and centrod method. Step 5: The path havng mnmum rank score n step 4 s the crtcal path for Sanguansat and Chen rankng and centrod method mnmum value s the crtcal path. 4. RNKING METHOD ND DEFUZZIFICTION FORMUL ON TRINGULR FUZZY NUMBERS Here we are usng Sanguansat and Chen Rankng Method and Centrod formula on Trangular Fuzzy Numbers. trangular fuzzy number s shown n fg (), and t can be parameterzed by a trplet (a, b, c). 0, IJM. ll Rghts Reserved 478
Dr. S. Narayanamoorthy* & S. Maheswar/ FUZZY CRITICL PTH METHOD (FCPM) BSED ON SNGUNST ND CHEN./ IJM- (), Nov.-0. Fgure:. Membershp functon curve for Trangular fuzzy number 4. Sanguansat and Chen Rankng Method ssume that there are n fuzzy numbers,,..., n. Step: Transform each trapezodal fuzzy number ( a, a, a ) a a a k k k,,, ( a, a, a) denotes the upper bound of a j, n and j. nto a standardzed fuzzy number, where k Max a, a denotes the absolute value of a j and j j j a j Step : Calculate standard devaton of each standardzed fuzzy number j ( ) aj a,, Where a denotes the mean of the values a, a, a and n. rea L and rea R of each standard fuzzy number, where ( a + ) + ( a + ) real ( a + ) + ( a+ ) rear Step : Calculate the areas Then, calculate the areas rea + L and ( a ) ( a) + + real a + a + rear ( ) ( ) Step 4: Calculate the values rea + rea L R + + L R XD rea + rea + and rea + R of each standardzed fuzzy number XD of each standardzed fuzzy number Step 5: Calculate the rankng score ( ) of each standardzed fuzzy number Score Where k f k ( ), where, XD, () + XD + k XD otherwse and n 0, IJM. ll Rghts Reserved 479
Dr. S. Narayanamoorthy* & S. Maheswar/ FUZZY CRITICL PTH METHOD (FCPM) BSED ON SNGUNST ND CHEN./ IJM- (), Nov.-0. 4. Centrod method Let ( a, b, c) be a trangular fuzzy number. Then from the defnton of the centrod method we may get the followng formula for centrod method as a+ b+ c Centrod ( ) () 5. NUMERICL EXMPLE Consder the followng Fg. () shows the network representaton of a fuzzy project network each actvty duraton s represented by Trangular Fuzzy Numbers. Fgure:. fuzzy project Network 5. Sanguansat and Chen Method In ths method we used trangular fuzzy number for each path n the fuzzy project network, and total fuzzy slack tme for each path s gven n table (). Table: Total fuzzy slack tme for each path n fuzzy project network Path Total fuzzy slack tme (a, b, c) Total fuzzy slack tme ( a+c, b, b+c ) From the above table fuzzy slack tmes are changed from ( abc,, ) to ( a c, b, b c) Let {,, } be set of fuzzy slack tmes to rank ths method. (8, 8, 4), (9, 8, 6), (58, 9, 74), 4 (9, 4, 7), 5 (0, 0, 7) k Max a j k 74 j (0.4, 0.4, 0.5675), a 0.5 (0.570, 0.4, 0.85), a 0.5405 (0.787, 0.98, ), a 0.75 4 (0.98, 0.89, 0, 648), a 4 0.5 5 (0, 0, 0.5), a 5 0.666 --5-7 (-6,8,4) (8,8,4) -4-5-7 (-6,8,5) (9,8,6) -4-7 (,9,45) (58,9,74) -4-6-7 (-,4,4) (9,4,7) --6-7 (-7,0,7) (0,0,7) + + to apply ths rankng method. 0, IJM. ll Rghts Reserved 4740
Dr. S. Narayanamoorthy* & S. Maheswar/ FUZZY CRITICL PTH METHOD (FCPM) BSED ON SNGUNST ND CHEN./ j 0.868, ( ) aj a 0.09, ( a ) ( a ) + + + real rea L.4, rea 5L 0.44, rea L.85, L ( a ) ( a ) + + + rear rea R.405, rea 5R.5 rea R.405, R ( a ) ( a) + + real rea + L 0.7568, rea + L 0.649, L ( a) ( a) + + rear IJM- (), Nov.-0. 0.095, 4 0.884 5 rea.5877, rea 4L.904, rea.6959, rea 4R.769, rea + 0.4, rea + 4L 0.7095, 5L rea + R 0.5946, rea + R 0.457, rea + R 0.04, rea + 4R 0.70, rea + 5R 0.75 rea + rea, XD rea + rea + + + Score L R ( ) XD + XD + k L R rea + By usng rea L, rea R, rea + L, rea + R and from ths we can get the values of Score( ) values. The values are calculated and tabulated n the below table (). XD, and Table: Chen rank of total fuzzy slack tme for each path Path XD Score( ) --5-7.6506.54 0.597-4-5-7.7904.0676 0.546-4-7.86 0.76 0.54-4-6-7.567.45 0.69 --6-7.5.75 0. Here the path havng mnmum rank score s --6-7. So the crtcal path for the fuzzy project network s --6-7. 5. Centrod Method For trangular fuzzy numbers we calculated the fuzzy actvty tme and fuzzy slack tme and defuzzfed values. The values are tabulated n table (). 0, IJM. ll Rghts Reserved 474
Dr. S. Narayanamoorthy* & S. Maheswar/ FUZZY CRITICL PTH METHOD (FCPM) BSED ON SNGUNST ND CHEN./ IJM- (), Nov.-0. Table: Defuzzfed values of slack tme for fuzzy project Path Fuzzy actvty tme for paths Slack tme (a, b, c) Defuzzfed value usng eqn () --5-7 (,8,) (-6,8,4) -4-5-7 (,8,) (-6,8,45) 9-4-7 (8,,4) (,9,45) 9-4-6-7 (,6,0) (,4,4) 4. --6-7 (9,4,8) (-7,0,7) 0 In table () we used defuzzfed formula and we are gettng the mnmum value s 0. The path --6-7 s the crtcal path for the network. So the requred path for the fuzzy project network s --6-7. Graph.. Comparson between Sanguansat & Chen Rankng Method and Centrod Method usng Trangular fuzzy number. 6. CONCLUSION In ths paper we have computed total fuzzy slack tme for each path n the fuzzy project network to fnd the crtcal path for a gven fuzzy project network. numercal example has partcularly provded to explan the rankng method and defuzzfcaton method for trangular fuzzy numbers. From the graph we can conclude that, comparng two methods Sanguansat and Chen Method s better to fnd the crtcal path n a fuzzy project network. REFERENCES [] huja, H.N., Dozz, S.P., and bourzk, S.M., Project Management, New York: wley, (994). [] Dubos, D., and Prade, H., Possblty Theory: n pproach to Computerzed Processng of Uncertanty. New York: Plenum, (988). [] Hapke, M., and Slownsk, R., Fuzzy prorty heurstcs for project schedulng, Fuzzy sets and systems, vol. 8, no., (996), pp.9-99. [4] Rav Shankar, N., Sreesha, V., and Phan Bushan Rao, P., n analytcal method for fndng crtcal path n a fuzzy project network. Int.Journal of Math. Scences, Vol.5, (00), no.0, 95-96. [5] Natarajan,.M., Balasubraman, P., and Tamlaras,., Operatons Research, Dorlng Kndersley (Inda) Pvt.Ltd.ISBN-8-7-0000-. (006). 0, IJM. ll Rghts Reserved 474
Dr. S. Narayanamoorthy* & S. Maheswar/ FUZZY CRITICL PTH METHOD (FCPM) BSED ON SNGUNST ND CHEN./ IJM- (), Nov.-0. [6] Rav Shankar, N., Sreesha, V., and Phan Bushan Rao, P., (00), Crtcal path analyss n the fuzzy project network. dvances n fuzzy mathematcs ISSN 097-5X, Vol.5, no., 85-94. [7] Shakeela Sathsh, and Ganesan, K., smple approach to fuzzy crtcal path analyss n project networks. Int. journal of Scentfc & Eng. Research. Vol., (0). [8] Svarenthnamohan, R., Operatons Research, Tata McGraw-Hll Publshng Company Lmted. ISBN- 0-07-0-5996-9. (005). [9] Twar, N.K., and Shshr, Shandlya, K. Operatons Research, Prentce-Hall of Inda.ISBN-8-0- 966-X. (006). Source of support: Nl, Conflct of nterest: None Declared 0, IJM. ll Rghts Reserved 474