Redundancy-Allocation in Pharmaceutical Plant Deepika Garg*, Kuldeep Kumar**,G.L.Pahuja***

Transcription

1 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, Redudacy-Allocato Pharmaceutcal Plat Deepka Garg*, Kuldeep Kumar**,G.L.Pahua*** *RESEARCH SCHOLAR,DEPT. OF MATHEMATICS,N.I.T.,KURUKSHETRA ** PROFESSOR, DEPT. OF MATHEMATICS,N.I.T., KURUKSHETRA ***PROFESSOR,DEPT. OF ELECTRICAL ENGINEERING,N.I.T.,KURUKSHETRA ABSTRACT I preset paper three heurstcs algorthms to optmze the problem of costraed redudacy allocato complex system are descrbed ad used to allocate redudacy a maufacturg system amely pharmaceutcal plat. Computatoal procedures of proposed algorthms are outled. These algorthms are appled to fd the best redudacy strategy, combato of compoets, ad levels of redudacy for each subsystem order to maxmze the system relablty uder cost costrats.results of these algorthms are compared to get best possble soluto for the proposed problem. Key Words Redudacy, Optmzato, Heurstc Method, Relablty.. INTRODUCTION The prmary goal of relablty egeerg s to mprove the system relablty. Tavakkol-moghaddam et.al. explaed that the tal desg actvty, the redudacy allocato s a drect way of ehacg system relablty. The redudacy allocato problem (RAP) s a complex combatoral optmzato problem, whch s very mportat may dustral applcatos. Blloet 2 exemplfed that RAP cossts of determg the umber of each subsystems whch compose the system so that relablty of complete system s maxmzed, cosderg certa costrats such as cost ad the weght of the subsystem.i the formulato of RAP of a seres parallel structure, the system cossts of compoets seres, ad for each compoet multple elemet choces are used parallel. Ths relablty desg problem has geerally bee formulated by cosderg actve redudacy. Km ad Yum 3 accetuated that redudacy optmzato problem s usually formulated as a o-lear teger problem whch s geeral dffcult to solve due to the cosderable amout of computatoal efforts requred to fd a exact optmal soluto. Therefore varous heurstcs methods have bee developed to solve RAP. Varous researchers Sharma ad Vekateswara 4, Aggarwal et.al. 5,,Nakagawa ad Nakashma 6,Gopal et.al. 7 appled heurstcs methods to solve RAP. Several researchers Kuo et.al. 9, Nakagawa ad Myazak 0,Tllma et.al., Kuo ad Prasad 2, Kuo et.al. 3, Ge et al 4 revewed varous applcatos of heurstcs methods solvg RAP.I ths paper,three heurstc algorthms are preseted for solvg RAP. These algorthms are used to solve the redudacy allocato problem pharmaceutcal plat. 2. LITERATURE SURVEY To deal wth RAP, Several heurstc methods are avalable the lterature relablty systems. Almost all of them are teratve algorthms yeldg a allocato each terato. Ushakov 5 developed a heurstc method for optmal allocato of redudaces subect to cost costrat. Aggarwal 6 proposed a heurstc method for optmal redudacy allocato for a geeral system. Most of the proposed methods were applcable to problems havg ay umber of costrats whch eed ot be lear. Kohda ad Ioue 7 preseted a heurstc method for redudacy optmzato for geeral coheret system. It was a teratve algorthm startg wth a feasble soluto.it dd ot requre the creasg ature of costrat fucto. Hwa ad Yum 3 developed a method for solvg costraed redudacy optmzato problems complex systems. The proposed method allowed excurso over a bouded feasble rego whch ca allevate the rsk of beg trapped at a local optmum. Dghua 8 proposed sh s method to solve the problem of costraed redudacy optmzato complex system usg mmal path set.pahua ad Subramayam 9 developed a heurstc algorthm (GLP) to solve the problem of costraed redudacy optmzato ad clamed that t yelded mproved results over Sh s 8 method terms of soluto qualty ad cosderably ehaced success rates for dverse problems. GLP algorthm cossts of two steps: ) selecto of ISSN:

2 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, optmal path 2) selecto of optmal compoet the chose path subect to ay umber of costrats. Salet feature of heurstc algorthms s that each terato, a stage s selected o the bass of a sestvty factor ad allocato at that stage s creased by uty. The defto of ths factor vared wth the method (Kuo et al. 3 ). Furthermore redudacy allocato problem s cosdered for varous system structures such as seres, parallel, mxed cofgurato ad geeral/complex systems. The seres structure s most commo structure that s used most system desgs, thus seres system redudacy allocato problem s solved ths paper, usg heurstc algorthm. 3. MATERIALS AND METHODS 3. SYSTEM DESCRIPTION The Pharmaceutcal plat cossts of varous uts vz. Weghg Mache, Sfter Mache, Mass Mxer, Graulator, Flud Bed Dryer, Octagoal Bleder, Rotary Compresso Mache, Coatg Mache, Ar compressor, Strp Packg Mache.These subsystems are arraged seres.. Itally dfferet raw materals are weghed accordg to the master formula wth the help of weghg mache.the ths mxture s placed to the Shfter. Shfter s used for sevg of raw materal.after sevg, raw materal s trasferred to Mass Mxer for proper mxg ad the graulato s doe wth the help of graulator, the these wet graules are to dred up wth the help of Flud Bed Dryer.After dryg of graules they are shfted to Octagoal bleder for lubrcato,the lubrcated graules are compressed wth the help of compresso mache.the coatg of compressed tablets are doe wth the help of coatg mache ad hereafter coated tablets are ready for fal packg. 3.2 NOMENCLATURE AND ASSUMPTIONS NOMENCLATURE Coherece: A property of a system or subsystem, as defed by Km,Yum 3. Path set: A set of compoets such that, f all compoets the set operate, the system s guarateed to operate. Mmal path set: A path set such that, f ay compoet s removed from the set, the remag compoets o loger form a path set. NOTATIONS x th Subsystem/Compoet of system R (x ), Q (x ) Relablty, urelablty of subsystem-x R s (x) System relablty umber of th subsystems x (x l,.x ) x* (, 2, 3. 0 ) s optmal soluto R Dfferece relablty of th subsystem by addg oe more redudat compoet g (x ) th resource- cosumed by th subsystem C maxmum of resource- P t mmal path set of the system umber of subsystems k umber of costrats m umber of mmal path sets h(.) a fucto that yelds the system relablty, based o uque subsystems, ad whch depeds o the cofgurato of the subsystems ASSUMPTIONS. The system ad all of ts subsystems are coheret. 2. There are subsystems the system. Subsystem structure (other tha coherece) s ot restrcted. 3. All compoet states are mutually-statstcally depedet. ISSN:

3 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, All costrats are separable ad addtve amog compoets. Each costrat s a creasg fucto of x for each subsystem. 5. Redudat compoets ca ot cross subsystem boudares. 3.3.REDUDANCY ALLOCATIOM PROBLEM IN PHARMACETUCIAL PLANT I case of pharmaceutcal plat, problem s to maxmze relablty wth cost costrat.cosderg cost costrat C =Rs (Here umber of costrat s oe.e k=,hece = ) Problem s to maxmze Rsx hrx,.,rxr x Subect to: g x * Where g (x ) s cost of th subsystem ad s total umber subsystems of th subsystem. Relablty ad cost of each compoet as gve by plat maagemet are: Table : Relablty ad cost of each subsystem of pharmaceutcal plat Subsystem x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 0 Relablty of Subsystem R (x ) Cost of subsystem g (x ) ALGORITHM I ths algorthm selecto crtero (S ) s defed as mmum stage relablty.takg to accout ths selecto crtera.e. sestve factor, redudacy allocato problem s solved the followg way. ALGORITHM STEP Italze KK=0, K=0, E(0 )=0 STEP 2 Italze umber of every subsystems equal to [ = for (<=<=) (Number of x ))] STEP 3 Calculate relablty of each subsystem [R x (Q x ) ] STEP 4 Fd the compoet wth mmum relablty a) Fd such that R x m{ R x, E(KK)forall 0KKK}] b) If ad 2 are such that R (x ) R (x 2 ) m{ R 2 x, E(KK)forall 0KKK}] The check f f g(x ) g(x ) 2 The take = otherwse = 2 STEP 5 Check f costraed s volated o addg oe more redudat subsystem th subsystem ISSN:

4 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, [ g x ( )g x C ] a. The f costraed s volated the remove ths compoet from selecto lst [ K=K+, E[K]=J] go to step 6 b.if o costraed s volated the check f by addg ths compoet relablty mproves up to desrable exted[ R >0.000 ] c.f by addg ths compoet relablty mproves up to desrable exted( R >0.000) add compoet ad crease o. of that compoet by uty [( = +)] go to step 3 d.if by addg ths compoet relablty ot mproves up to desrable exted( R <=0.000 ) remove ths compoet from selecto lst [ K=K+, E[K]=J] go to step 6. STEP 6 a.if all subsystem are ow excluded from further cosderato[check f K>0 ]), the x*=(, 2, 3. 0 ) s the optmal soluto; b. else go to step 3 STEP 7 7. Calculate the system relablty, R s (x*) by usg equato IMPLEMENTATION OF ALGORITHM Table 2: Usg algorthm Number of compoet each subsystem Cosumed Resources g(x)* Subsystem selecto factor S S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S ? ? ? ? ? ? ? ISSN:

5 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, ? ? ? # Algorthm stop here A redudat compoet s added to ths subsystem Ths chose compoet s removed from further cosderato as cost costrat s volated? The cost costrat s volated # Ths chose compoet s removed from further cosderato as R s ot as per desre redudat compoet s ot added to ths subsystem due to accessblty of lower cost compoet of same selecto factor. Optmal soluto s x*=(2,,,,2,2,2,,,) Rs(x*)= ALGORITHM 2 I ths algorthm selecto crtera (S )s defed as gve below R x S b x g x C..e rato of subsystem relablty to percetage of resources cosumed. Takg ths selecto crtera.e sestve factor, algorthm s descrbed as follows STEPS OF ALGORITHM 2 STEP Italze KK=0, K=0,E(0 )=0 STEP 2 Italze umber of every subsystem equal to [ = for (<=<=) (Number of x ))] STEP 3 a. calculate relablty of each subsystem [R x (Q x ) ] b.calculate Selecto factor for each subsystem R x S b x g x C STEP 4 Fd the compoet wth maxmum selecto factor [choose such that b x max{ bx, E(KK) for all 0 KK K}] If ad 2 are such that b (x ) b (x 2 ) m{ b 2 x, E(KK) for all 0 KK K}] The check f f g(x ) g(x ) 2 ISSN:

6 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, = Else = 2 STEP 5 Check f costraed s volated o addg oe more redudat subsystem th [ g x ( )g x C ] subsystem a. The f costraed s volated the remove ths compoet from selecto lst [ K=K+, E[K]=J] go to step 6 b.if o costraed s volated the check f by addg ths compoet relablty mproves up to desrable exted[ R >0.000 ] c.f by addg ths compoet relablty mproves up to desrable exted( R >0.000) add compoet ad crease o. of that compoet by [( = +)] go to step 3 d.f by addg ths compoet relablty ot mproves up to desrable exted( R <=0.000 ) remove ths compoet from selecto lst [ K=K+, E[K]=J] go to step 6. STEP 6 a.if all subsystem are ow excluded from further cosderato[check f K>0 ]), the x*=(, 2, 3. 0 ) s the optmal soluto; b. else go to step 3 STEP 7 Calculate the system relablty, R s (x*) by usg equato IMPLEMENTATION OF ALGORITHM 2 Number of compoet each subsystem Cosumed Resources g(x)* Table 3:Usg algorthm 2 Subsystem selecto factor S S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S # # # # # # ? # # ISSN:

7 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, ? # # # # ? # # ? # # ? # # ? # # ? # # ? # # Algorthm stop here Optmal soluto s x*=(2,4,2,,,,2,,,) Rs(x*)= ALGORITHM 3 I ths algorthm selecto crtera (S )s chose as maxmum of R S bx g x C where R Q x R x..e rato of chage subsystem relablty to percetage cosumed resources whch s proposed by Pahua ad Subramayam 8. ALGORITHM 3 STEP Italze KK=0, K=0,E(0 )=0 STEP 2 Italze umber of every subsystem equal to [ = for (<=<=) (Number of x ))] STEP 3 Calculate relablty ad selecto factor of each subsystem a. calculate relablty of each subsystem R x Q x b. calculate Selecto factor for each subsystem R S bx g x C where R Q x R x STEP 4 Fd the compoet wth maxmum selecto factor [choose such that ISSN:

8 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, b x max{ b x, E(KK) for all 0 KK K} If ad 2 are such that b(x ) b(x ) m{ b x, E(KK) for all 0 KK K}] 2 The check f f g(x ) g(x ) 2 = Else = 2 STEP 5 Check f costraed s volated o addg oe more redudat subsystem th [ g x ( )g x C ] subsystem a. The f costraed s volated the remove ths compoet from selecto lst [ K=K+, E[K]=J] go to step 6 b.if o costraed s volated the check f by addg ths compoet relablty mproves up to desrable exted[ R >0.000 ] c.f by addg ths compoet relablty mproves up to desrable exted( R >0.000) add compoet ad crease o. of that compoet by [( = +)] go to step 3 d.f by addg ths compoet relablty ot mproves up to desrable exted( R <=0.000 ) remove ths compoet from selecto lst [ K=K+, E[K]=J] go to step 6. STEP 6 a. If all subsystem are ow excluded from further cosderato[check f K>0 ]), the x*=(, 2, 3. 0 ) s the optmal soluto; b. else go to step 3 STEP 7 Calculate the system relablty, R s (x*) by usg equato IMPLEMENTATION OF ALGORITHM 3 Table 4:Usg algorthm 3 Number of compoet each subsystem Cosumed resources g(x)* Subsystem selecto factor S S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S ? ISSN:

9 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, ? ? ? ? ? ? ? # # ? # Algorthm stops here Optmal soluto s x*=(2,2,2,,2,,2,,,) Rs(x*)= RESULTS AND DISCUSSIONS By usg algorthm Optmal soluto of RAP (equato ad 2 ) s x*=(2,,,,2,2,2,,,) ad Rs(x*)= By applyg algorthm2 Optmal soluto for same s x*=(2,4,2,,,,2,,,) ad Rs(x*)= From algorthm 3 Optmal soluto for ths problem s x*=(2,2,2,,2,,2,,,) ad Rs(x*)= Out of threes three algorthm, algorthm2 gves mmum relablty(0.5828) ad algorthm gves the maxmum relablty (0.6996).Hece best soluto for ths problem s x*=(2,,,,2,2,2,,,) ad Rs(x*)= CONCLUSION Result of these algorthms appled o the gve system demostrate that algorthm2 gves mmum relablty(0.5828) ad algorthm gves the maxmum relablty (0.6996).However t has bee cocluded by pahua 9 that algorthm3 yeld the best result whereas algorthm gves worst results the terms of relablty mprovemet.however the performace of the algorthms based upo heurstcs methods s ot optmal always ad same method may gve dfferet results for dfferet problems. Also eve for the same problem, solved wth dfferet data, success rate of the method may dffer from problem to problem To coclude, t s assumed that redudacy allocato usg heurstc methods always mproves the system relablty that may be true optmal /ear optmal soluto. 6.ACKNOWLEDGEMENT Authors are very thakful to Plat maagemet of Altarsr Pharmaceutcal Pvt. Ltd., Roorkee, for provdg us suffcet formato ad also for formatve dscussos essetal for coductg relablty optmzato through redudacy allocato of cocer plat. Results of the same have also bee dscussed wth Plat maagemet.results are foud to be hghly beefcal for the system desgg. ISSN:

10 Deepka Garg et. al. / Iteratoal Joural of Egeerg Scece ad Techology Vol. 2(5), 200, REFERENCES: [] R.Tavakkol-Moghaddam,J.Safar,F.Sassa. Relablty optmzato of a seres-parallel systems wth a choce of redudacy strateges usg a geetc algorthm, Relablty egeerg ad system safety 93: (2008). [2] A. Blloet.Redudacy allocato for seres-parallel systems usg teger Lear programmg,ieee Trasacto o Relablty 57(3): (2008). [3] J Km Hwa,J.Yum Bog.A heurstc method for solvg redudacy-optmzato problems complex systems, IEEE trasacto o relablty 42(4): (993). [4] J. Sharma ad K.V.Vekteswara. A drect method for maxmzg the system relablty, IEEE trasacto o relablty R20(20): (97). [5] K.K.Aggarwal,J.S.Gupta,K.B.Msra. A ew heurstc algorthm for solvg a redudacy optmzato problem, IEEE Trasacto o relablty R-24: 86-87(975). [6] Y.Nakagawa,K.Nakashma. A heurstc method for determg Optmal relablty allocato, IEEE Trasacto o relablty R- 26: 56-6(977). [7] K.Gopal,K.K.Aggarwal,J.S Gupta. A mproved algorthm for relablty optmzato,ieeetrasacto for Relablty R- 27: (978) [8] Dghua Sh. A New heurstc Algorthm for costraed redudacy-optmzato complex systems, IEEE Trasacto o relablty R-36(36)(5):62-623(987). [9] W.Kuo,C.L. Hwag,F.A.Tllma. A ote o heurstc methods optmal system relablty, IEEE Trasacto o relablty R- 27: (978). [0] Y.Nakagawa,S.Myazak.A expermetal comparso of the heurstc methods for solvg relablty optmzato problems, IEEE Trasacto for relablty R- 30: 8-84 (98) [] F.A.Tllma, C.L.Hwag,W.kuo. Optmzato of system relablty,marker Dekker 985 [2] W.Kuo ad V.R. Prasad. A aotated overvew of system- relablty optmzato,ieee Trasacto o relablty 49(2):76-87(2000). [3] W. Kuo,V.R.Prasad,F.A.Tllma,C.Hawag. Optmal relablty desg fudametal ad applcato,cambrdge Uversty Press, Lodo, 200 [4] M.Ge,YS Yu.Soft computg approach for relablty optmzato :state of-art survey, Relab. Eg.Syst. Safety 9(9): (200). [5] I.A.Ushako. A heurstc method of optmzato of the redudacy of multfucto systems, Egeerg Cyberetcs 0(4): 62-63(972) [6] K.K. Aggarwal. Redudacy optmzato geeral systems, IEEE Trasacto o relablty R- 25: (976). [7] T.Kohda ad K.Ioue. A relablty optmzato method for complex systems wth the crtero of local optmalty, IEEE Trasacto o relablty R-3(3)(5):09-(982) [8] G.L. Pahua ad A.G.Subramayam.Redudacy allocato complex systems, Iteratoal coferece o tellgece system ad etworks (ISN-2008),(Klawad): 32-36(2008). [9] G.L.Pahua,(2005), Relablty aalyss ad optmzato-recet ad ew approaches,p.h.d.thess,k.u.k. ISSN: