Scholars Journal of Physics, Mathematics and Statistics

Transcription

1 Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp Scholars Joural of Phsics, Mathematics ad Statistics Sch. J. Phs. Math. Stat. 015 A: Scholars Academic ad Scietific Publishers SAS Publishers A Iteratioal Publisher for Academic ad Scietific Resources ISSN Prit ISSN Olie Solvig Full Fu Critical Path Aalsis i Project Networks Usig Liear Programmig Problems M. Jaalakshmi Assistat Professor, Departmet of Mathematics, School of Advaced Scieces, VIT Uiversit, Vellore-14, Idia. *Correspodig Author: M. Jaalakshmi m.jaalakshmi@vit.ac.i Abstract: A ew method for fidig fu optimal solutio, the maimum total completio fu time ad fu critical path for the give full fu critical path FFCP problems usig crisp liear programmig LP problem is proposed. I this proposed method, all the parameters are represeted b triagular fu umber. The fu optimal solutio of the FFCP problems obtaied b the proposed method, do ot cotai a egative part of the values of the fu decisio variables. This paper will preset with great clarit of the proposed method ad illustrate its applicatio to FFCP problems occurrig i real life situatios. Kewords: Full fu critical path problem, liear programmig problem, fu liear programmig problem, Boud ad Decompositio method 1. INTRODUCTION Critical path method CPM techiques have become widel recogied as valuable tools for the plaig ad schedulig of large projects.cpm is the oe from the start of the project to fiish of project where the slack times are all eros. The purpose of the CPM is to idetif critical activities o the critical path so that resources ma be cocetrated o these activities i order to reduce project legth time. The successful implemetatio of CPM requires the availabilit of clear determied time duratio for each of the activit. However, i practical situatios this requiremet is usuall hard to fulfill, sice ma of activities will be eecuted for the first time. To deal with such real life situatios, Zadeh [1] itroduced the cocept of fu set. Sice there is alwas ucertait about the time duratio of activities i the etwork plaig, due to which fu critical path method FCPM was proposed sice the late 1970s. For fidig the fu critical path, several approaches are proposed over the past ears. Gadik[]have developed a fu etwork of ukow project to estimate the activit duratios ad used fu algebraic operatios to fid the duratio of the project ad its critical path. A chapter of the book Kaufma ad Gupta [3] devoted to the CPM i which activit times are represeted b triagular fu umbers. Caho ad Lee [4] have developed a ew approach to calculate the fu completio project time. Nasutio [5] have proposed a method to compute total floats ad the critical paths i a fu project etwork. Yao ad Li [6] have itroduced a method for rakig fu umbers without the eed for a assumptios ad have used both positive ad egative fu values to defie orderig which is the applied to fu project etwork. Dubois et al [7] have eteded the fu arithmetic operatioal model to calculate the latest startig time of each activit i a fu project etwork. Che[8] have proposed a approach based o the etesio priciple ad liear programmig formulatio to critical path aalsis i fu project etworks. Che ad Hsueh [9]have preseted a simple approach to solve the CPM problems with fu activit times beig fu umbers o the basis of the liear programmig formulatio ad the fu umber rakig method that are more realistic tha crisp oes. Yakhchali ad Ghodspour [10] have itroduced the problems of determiig possible values of earliest ad latest startig times of a activit i etworks with miimal time lags ad imprecise duratios that are represeted b meas of iterval or fu umbers. Amit Kumar ad Parmpreet Kaur [11] have proposed a method to fid fu optimal solutio for FFCP with a ew represetatio of triagular fu umbers with the help of fu liear programmig model. Ravi Shakar et al.[] have itroduced a ew defuificatio formula for trapeoidal fu umber ad appl to the float time slack time for each activit i the fu project etwork to fid the critical path. Usha Madhuri et al.[13] have proposed a ew fu liear programmig model is proposed to fid fu critical path ad fu completio time of a fu project. Available Olie: 144

2 Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp PRELIMINARIES We eed the followig mathematical orietated defiitios of fu set, fu umber ad membership fuctio ad also, defiitios of basic arithmetic operatio o fu triagular umbers which ca be foud i Zadeh [1] ad Kaufma ad Gupta [14]. Defiitio.1 Let A be a classical set ad A be a membership fuctio from A to [0,1]. A fu set A with the membership fuctio A is defied b A, A : A ad A [0,1]. Defiitio. A real fu umber a1, a, fuctio a a satisfig the followig coditios: i a a is a cotiuous mappig from R to the closed iterval [0, 1], ii a a 0 for ever a, a 1 ], iii a a is strictl icreasig ad cotiuous o [ a 1, a ], iv a a = 1 for ever a [ a, ], v a a is strictl decreasig ad cotiuous o [ a, a 3 ] ad vi a a 0 for ever a [ a 3, ]. a is a fu subset from the real lie R with the membership Defiitio.3A fu umber a is a triagular fu umber deoted b 1, a, umbers ad its membership fuctio a is give below: a1 / a a1 for a1 a a / a for a 0 otherwise where a a. 1 Defiitio.4 Let a 1, a, ad b1, b be two triagular fu umbers. The i a1, a, b 1, b = a1 b1, a b, b3. ii a1, a, b 1, b = a1 b3, a b, b1. iii k a1, a, = ka 1, ka, k, for k 0. iv k a1, a, = ka 3, ka, ka1, for k 0. a1b1, ab, b3, a1 0, v a 1, a, b1, b a1b3, ab, b3, a1 0, 0, a1b3, ab, b1, 0. Let FR be the set of all real triagular fu umbers. a where a1, a ad are real Defiitio.5 Let A a1, a, ad B b1, b be i F R, the i A B iff ai b i, i 1,, 3 ii A B iff ai b i, i 1,, 3 iii A B iff ai b i, i 1,, 3 ad A 0 iff a i 0, i 1,, 3. Defiitio.6 Let A a1, a, be i F R. A is said to be positive if a i 0, i 1to3. Defiitio.7 Let A a, a, be i F R. A is said to be iteger if a 0, i 1to3 are itegers. 1 i Available Olie: 145

3 Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp Cosider the followig crisp liear programmig problems with m iequalit/equalit costraits ad variables ma be formulated as follows: Maimie c T c where j 1 X T Z c X A X {,, } b 0,, j 1 ad B bi m1 A is a oegative crisp vector, m are oegative real vectors for all 1 i m ad1 j. a is a oegative real crisp matri ad 3. LP FORMULATION OF CRISP CRITICAL PATH AND FULLY FUZZY CRITICAL PATH PROBLEMS The CPM is a etwork-based method desiged to support i the plaig, schedulig ad cotrol of the project. Its objective is to costruct the time schedulig for the project. Two basic results provided b CPM are the total duratio time eeded to complete the project ad the critical path. Oe of the efficiet approaches for fidig the critical paths ad total duratio time of project etworks is LP. The LP formulatio assumes that a uit flow eters the project etwork at the start ode ad leaves at the fiish ode. I this sectio the LP formulatio of CCP problems is reviewed ad also the LP formulatio of FFCP problems is proposed. The liear programmig model discussed i the book writte b Taha[15] is reviewed i this sectio which ca be foud i Amit Kumar ad Parmpreet Kaur [11].Cosider a project etwork G N, A cosistig of a fiite set N 1,,..., of odes evets ad A is the set of activities i,. Deote t as the time period of activiti,. The LP formulatio of crisp critical path problems is as follows: Maimie t i, A 1, j: A i : A i i : A 1, j : j, k A, i 1, k, is the decisio variable deotig the amout of flow i activit i, i, ad the costraits represets the coservatio of flow at each ode. A, t is the time duratio of activit There are several real life problems i which a decisio maker ma be ucertai about the precise values of activit time. Suppose time parameters t ad, i, A are imprecise ad are represeted b fu umbers t ad, i, A respectivel. The the FFCP problems ma be formulated ito the fu liear programmig FLP problem: Maimie t A 1, j: A i : A i i : A j : j, k A 1,, i 1, k, is a o-egative real umber i, A. Available Olie: 146

4 Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp Suppose all the parameters t ad are represeted b a, b, c tpe triagular fu umbers t, t, t ad,, are respectivel the the LP formulatio of FFCP problems, proposed i the Sectio, ma be writte as: Maimie t, t, t A,,,, 1,1,1, j: A i : A 1 j 1 j 1 j, i i : A, jk j : j, k A,, 1,1,1, i i, jk, jk, i 1, k,,, is a o-egative triagular fu umber i, A.,, Now, sice is a triagular fu umber, the t, j 1,,..., 3.1 The relatio 3.1 is called bouded costraits. Jaalakshmi ad Padia [16] have proposed a method amel, boud ad decompositio method to a full fu liear programmig FFLP problem to obtai a optimal fu solutio.now, b usig Boud ad Decompositio method, the above FFCP problems ca be decomposed ito three crisp LP problems amel, middle level problem MLP, upper level problem ULP ad lower level problem LLP as follows: MLP Maimie Z = j1 middle value of A t, t, t,, Costraits i the decompositio problem i which at least oe decisio variable of the MLP occurs ad all decisio variables are o-egative itegers. ULP Maimie Z 3 = upper valu e of t, t, t,, j1 A upper value of t, t, t,, Z j1 A Costraits i the decompositio problem i which at least oe decisio variable of the ULP occurs ad are ot used i MLP all variables i the costraits ad objective fuctio i ULP must satisf the bouded costraits replacig all values of the decisio variables which are obtaied i MLP ad all decisio variables are oegative itegers. ad LLP Miimie Z 1 = j1 lower valu e of A t, t, t,, lower value of t, t, t,, Z j1 A Costraits i the decompositio problem i which at least oe decisio variable of the LLP occurs which are ot used i MLP ad ULP all variables i the costraits ad objective fuctio i LLP must satisf the Available Olie: 147

5 Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp bouded costraits replacig all values of the decisio variables which are obtaied i MLP ad ULP ad all decisio variables are oegative itegers. where Z is the optimal objective value of MLP. Now, we propose a ew algorithm to fid the fu optimal solutio, the maimum total completio fu time ad fu critical path for FFCP problems. The steps of the proposed method are as follows: STEP 1:Formulate the give FFCP problems ito FFLP problems. STEP:Usig Boud ad Decompositio method, costruct MLP, ULP ad LLP problems from FFLP obtaied i Step 1. STEP 3: Usig eistig liear programmig techique, solve the MLP problem, the the ULP problem ad the, the LLP problem i the order ol ad obtai the values of all real decisio variables, ad t ad values of all objectives are Z 1, Z ad Z3. Let the decisio variables values be, ad t, i, A ad objective values be 1, Z ad Z3 Z. STEP 4: The fu optimal solutio to the FFLP problems is,, t, i, A ad the maimum fu 1 3 objective is Z, Z, Z. STEP 5: Fid the fu critical path b combiig all the activities i, such that 1,1,1 ad the correspodig maimum total completio fu time is the maimum fu objective value obtaied i Step 4. REMARKS 3.1 I the case of FFCP problems ivolvig trapeoidal fu umbers ad variables decompose it ito four crisp LP problems ad the, we solve the middle level problems secod ad third problems first. The, solve the upper level ad lower level problems ad the, the fu critical path ad maimum total completio fu time is obtaied ivolvig trapeoidal fu umbers ad variables. Now, the proposed method for fidig fu critical path ad maimum total completio fu time ivolvig triagular fu umber for FFCP problem is illustrated usig the followig umerical eamples. EXAMPLE 3.1 The problem is to fid the fu critical path ad maimum total completio fu time of the project etwork, show i Figure 3.1, i which the fu time duratio of each activit is represeted b a triagular fu umbers t 3, 4, 5 t 3.8, 3, 3. t 4, 5,6 4 t 1.8,,. Fig 3.1 Available Olie: 148

6 Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp Now we ca covert the give FFCP problem ito liear programmig problem as: Maimie Z to [ 3, 4, 5,,.8, 3, 3. 3, 3, 3 4, 5, 6 4, 4, 4 1.8,,.,, ],, 1,1,1 3, 3, 3 4, 4, 4,,,, 3, 3, 3 4, 4, 4,, 1,1,1,,,,,,,,,, , are o-egative triagular fu umbers. Now, usig Boud ad Decompositio method, the above LP ca be decomposed ito three crisp LP problems. Now, the problem P is give below: Maimie Z = , 3, 4, 0. subject Solvig the problem P b liear programmig techique, the optimal solutio is ad Maimie Z =9 ad the alterate solutio is ad Maimie Z =9. Now, the problem P 3 is give below: Maimie Z Z , 3, 4, 0 ad,, 0., 3 4 Solvig the problem P 3 b liear programmig techique with ad Z =9, the optimal solutio is 1, 3 1, 4 0, 1 ad Maimum Z 3 =10.40 ad its alterate solutio is 1, 3 0, 4 1, 0 ad Maimie Z 3 =11. Now, the problem P 1 is give below: Maimie Z Z Available Olie: 149

7 Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp , 3, 4, 0 ad,, 0., 3 4 Solvig the problem P 1 b liear programmig techique with ad Z =9, the optimal solutio is ad Maimum Z 1 =7.6 ad the alterate solutio is ad Maimie Z 1 =7. Therefore, the fu optimal solutio is 1,1,1, 3 1,1,1, 4 0, 0, 0, 1,1,1, the fu critical path is1 3 4 ad the maimum total completio fu time is 7.6, 9, Usig the alterative fu optimal solutio 1,1,1, 3 0, 0, 0, 4 1,1,1, 0, 0, 0, the fu critical path is 1 4 ad the maimum total completio fu time is 7, 9, 11. Hece, i this problem, the fu critical paths are time completio fu time are 7.6, 9, 10.4 ad 7, 9, 11 respectivel. ad 1 4, the correspodig maimum 4. CONCLUSION A ew method has bee proposed to fid the fu critical path ad fu completio time of a fu project usig crisp LP problems. The advatage of the proposed method is that the values of the fu decisio variables do ot cotai a egative part ad fu rakig fuctios were ot used.sice, the proposed method is based ol o crisp LP problem it is ver eas to solve FFCP problems havig more umber of fu variables with help of eistig computer software. REFERENCES 1. Zadeh LA Fu Sets. Iformatio ad Cotrol, : Gadik I Fu etwork plaig-fnet. IEEE Trasactios o Reliabilit, : Kaufma A, Gupta MM Fu Mathematical Models i Egieerig ad Maagemet Scieces. Elsevier, Amsterdam, Mc Caho CS, Lee ES Project etwork aalsis with fu activit times. Computer ad Mathematics with Applicatios, : Nasutio SH Fu Critical Path Method. IEEE Trasactios o sstems, Ma ad Cberetics, : Yao JS, Li FT Fu Critical Path method based o siged distace rakig of fu umbers. IEEE Trasactios o Sstems, Ma ad Cberetics- Part A: Sstems ad Humas, : Dubois D, Fargier H, Galvago, V O latest startig times ad float i activit etworks with ill-kow duratios. Europea Joural of Operatioal Research, : Che SP Aalsis of critical paths i a project etwork with fu activit times. Europea Joural of Operatioal Research, : Che SP, Hsueh YJ A simple approach to fu critical path aalsis i project etworks. Applied Mathematical Modellig, : Yakhchali SH, Ghodspour SH Computig latest startig times of activities i iterval-valued etworks with miimal time lags. Europea Joural of Operatioal Research, : Amit Kumar, Parmpreet Kaur A New method for fu critical path aalsis i project etworks with a ew represetatio of triagular fu umbers. Applicatios ad Applied Mathematics, : Ravi Shakar N, Sireesha V, Phai Busha Rao P A Aaltical Method for Fidig Critical Pathi a Fu Project Network. It. J. Cotemp. Math. Scieces, : Usha Madhuri K, Pardha Saradhi B, Ravi Shakar N Fu Liear Programmig Model for Critical Path Aalsis. It. J. Cotemp. Math. Scieces, 013 8: Kaufma A, Gupta MM Itroductio to Fu Arithmetics: Theor ad Applicatios. Va Nostrad Reihold, New York, Taha H.A Operatios Research: A Itroductio. Pretice-Hall, New Jerse, Jaalakshmi M, Padia P A New Method for Fidig a Optimal Fu Solutio Full Fu Liear Programmig Problems. Iteratioal Joural of Egieerig Research ad Applicatios, 0 4: Available Olie: 150