Fuzzy Reliability and Fuzzy Availability of the Serial Process in Butter-Oil Processing Plant

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1 Journal of Mathematcs and Statstcs 5 (): 65-7, 2009 ISSN Scence Publcatons uzzy Relablty and uzzy Avalablty of the Seral Process n Butter-Ol Processng Plant Kuldeep Kumar, 2 Ja Sngh and Pawan Kumar Department of Mathematcs, Natonal Insttute of echnology, Kurukshetra, Haryana, Inda 2 Modern Insttute of Engneerng and echnology, Mohr, Kurukshetra, Inda Abstract: Problem statement: he purpose of ths study was to compute fuzzy relablty and fuzzy avalablty of the seral process n butter-ol processng plant for varous choces of falure and repar rates of sub-system. hs plant conssts of eght sub-systems out of whch two are supported by standby unts wth perfect swtch over devces and consdered that these two sub-systems never fal. he effect of coverage factor on the fuzzy avalablty also studed. Approach: In ths study the chapman- Kolmogorov dfferental equatons were formed usng mnemonc rule from the transton dagram of the butter-ol processng plant. hese equatons were solved for steady state recursvely and results were obtaned by computer program. Results: Result n the study analyzed fuzzy avalablty for varous values of system coverage factor, falure and repar rates. Industral mplcatons of the results also brefly dscussed. Concluson: he fndngs n the study suggested that the management of butterol processng plant s senstve sub-system s mportant to mprove ts performance. Key words: Manufacturng system, modelng, markov processes, fuzzy relablty INRODUCION Conventonal relablty theory consders the assumptons of the probablty theory and the bnary states of a component/system as workng or faled. Of late, there has been tremendous growth n the area of fuzzy set theory [7]. hs has changed the basc scenaro n relablty and concerned theores. hough conventonal relablty theory cannot be gnored, fuzzy relablty theory also needs to be consdered along wth t. In ths study, fuzzy relablty refers to profust relablty. Profust relablty approach [,8,9,0,] s based on the probablstc assumpton and the fuzzy state assumpton. he falure behavor of the system s fully assumed to follow probablty measures. he system s operatng and faled states are descrbed by fuzzy states. he system can be n any fuzzy state at any gven pont of tme. he system s falure or success s vewed n a fuzzy way. Many researchers appled ths concept on varous systems. Chowdhury and Msra [3] presented a method to fnd an expresson of fuzzy system relablty of a non-seres parallel network takng nto consderaton the specal requrements of fuzzy sets. Zuang [4] presented a method of relablty analyss n the presence of fuzzness attached to operatng tme. Ca [5] descrbed a method of fuzzy relablty for streetlghtng lamps replacement. Ca et al. [2] appled ths concept for evaluatng fuzzy relablty modelng of gracefully degradable computng systems. Pandey et al. [6] also found the profust relablty of a gracefully degradable system. In ths study, we propose to solve the governng dfferental equatons of the system to fnd the profust relablty and fuzzy avalablty of the seral processes n butter-ol processng plant usng both falure and repar rates. If any subsystem fals, then the system wll mmedately take reconfguraton operaton wthn no tme. he reconfguraton operaton wll detect and remove the faled subsystem from the system, however; all the other operatng subsystem wll contnue to operate as t s. he probablty of successful reconfguraton operaton s defned as coverage factor. We denote ths reconfguraton parameter or system coverage factor by c. he major motvaton of ths study s to brng fuzzy relablty theory nto a real-world applcable mantenance problem. MAERIALS AND MEHODS he relablty can be nterpreted as the probablty that no transton occurs from the system success state to the system falure state. Evdently, n the presence of Correspondng Author: Pawan Kumar, Department of Mathematcs, Natonal Insttute of echnology, Kurukshetra, Haryana, Inda el:

2 fuzzness attached to the system states, both the defnton of system falure and that of relablty should be modfed. In response to ths requrement, we ntroduce the concept of fuzzy relablty: Suppose a system wth n topologcal (non-fuzzy) states S,S 2,...,S n. Let U = {S,S 2,...,S n}denote the unverse of dscourse. On ths unverse we defne a fuzzy success state: S = {(S, µ (S )), =,2,...n} () S and a fuzzy falure state: Let: µ (S ) γ S(S ) = (5) µ (S ) + µ S(S ) hen γ S(S ) can be nterpreted as the grade of membershp of S, wth respect to S to. t s reasonable to say that the transton from S to S j makes the transton from S to occur to some extent f and only f the relaton γ S(S j) > γ S (S ) holds. We therefore defne: γ (S ) γ (S ) when γ (S ) > γ (S ) 0 otherwse S j S S j S µ S j) = (6) = {(S, µ (S )), =,2,...n} (2) as: hen the fuzzy nterval relablty can be expressed where, µ (S ) and µ (S ) are the correspondng S membershp functons, respectvely. In the conventonal relablty theory, one s nterested n the event of transton from the system success state to system falure state. Accordngly, here we are nterested n the event, denoted by S of the transton from the fuzzy success state to fuzzy falure state. Assume that the unverse U or the behavor of n system states s completely stochastcally characterzed n the tme doman, we defne: S does not occur n 0,t0 + t) = Pr the tmenterval [ t 0, t0 + t] (3) 0, t 0 +t) s referred to as the fuzzy nterval relablty of the system n [t 0, t 0 +t]. o compute the fuzzy nterval relablty, we must express S. Snce both S and are fuzzy states, the transtons between them are consequently fuzzy and thus S can be vewed as a fuzzy event [6]. Apparently S may occur only when some state transton occurs among the n system states {S,S 2,...,S n}, so S can be defned on the unverse: U = {m,, j =,2,...,n} (4) j where, m j represents the transton from state S to S j wth membershp functon: { µ ),, j =,2,...,n} S j.e., = {m, µ ),, j =,2,...,n}. S j S j 66,t + t) = µ )Pr 0 0 S j = j= n n [ + ] {m occurs durng t, t t } j 0 0 (7) where, m j s confned to be the transton from S to S j wthout passng va any ntermedate state. Let t 0 = 0; we have: 0, t0 + t) = ) (8) ) s referred to as the fuzzy relablty of the system at tme t. Here t s necessary to pont out that the applcablty of (4-6) s not confned to the area of relablty research. [hey can be used to defne any transton between two fuzzy states, provded that we recall that a transton between two fuzzy states s just a fuzzy event.]. Equaton 4-6 are a foundaton for characterzng the behavor of fuzzy stochastc process wth fuzzy states. [or the relablty purpose, we generally, but not necessarly, defne: µ (S ) = - µ (S ) =, 2, n]. Suppose that wthout passng va any ntermedate state, S j can not go to other states except S j- and µ (S j) < µ (S j ) for j =, 2 n. hen we have: n m ( j+ ) j occurs 0,t0 + t) = µ S ( j+ ) j)pr j= durng [ t 0, t 0 + t ] Suppose at tme t 0 the system s n S n. hen at tme t 0 +t the system s n S j mples that m n(n-) m (j+)j have occurred durng [t 0,t 0 + t]. So: S

3 j m( j+ ) joccurs at tme t0 + t the Pr = Pr durng[t 0,t 0 t] + = system s n S Also we note that: µ ) = µ ) + µ ) S j S k S kj granules. he buttermlk produced n ths process s pumped back to raw mlk slos and the butter granules are further processed n the machne so as to get homogeneous mass of butter. he homogeneous butter s taken out from machne nto butter trolleys and shfted to meltng vats. he CBM conssts of gearbox, motor and bearngs n seres. f µ < µ k < µ j (S ) (S ) (S ). hen we have: n at tme t0 + t the 0,t0 + t) = µ S nj).pr. j= system s n S at tme t + t Where: n C 0 = µ S nj).pr j= the system s n S C S j S j (9) µ ) = µ ), j =, 2 n (0) he system, notatons and assumptons: In ths study, we dscuss Butter-ol manufacturng plant whch conssts of eght sub-systems out of whch two pump and chller are supported by stand-by unts wth perfect swtch over devces and consdered that these two systems never fal. he mathematcal modelng s carred out for the remanng sx sub-systems that are prone to falure: Sub-system A (Separator): Chlled mlk from the chller s taken to the cream separator, where fats are separated from the mlk n the form of cream contanng 40-50% and the remanng skmmed mlk s stored n mlk slos for preparng mlk powder. It conssts of three components n seres, namely, motor, bearngs and hgh-speed gearbox. Sub-system B (Pasteurzer): Cream from the separator s pasteurzed n ths sub-system. Pasteurzaton s the process of heatng every partcle of cream to not less than 70 C. Its purpose s to destroy pathogenc organsms, to nactvate the enzymes present and to make possble removal of volatle flavors. here are two pasteurzers workng n parallel. If one fals the system works n reduced capacty. he pasteurzed cream s stored n double-jacketed cream storage tank for further processng. Sub-system C (Contnuous butter makng): Cream from the cream storage tank s pumped nto the Contnuous Butter Makng machne (CBM). he cream s churned n ths machne n order to get butter 67 Sub-system D (Meltng vats): hs sub-system conssts of a double-jacketed storage tank. Butter s melted n ths research at about 07 C very gently so that the water evaporates from the meltng butter. he meltng butter s then allowed to reman undsturbed for about half an hour. hs sub-system conssts of monoblock pumps, motors and bearng n seres. Sub-system E (Butter-ol clarfer): Butter-ol from meltng vats s taken out nto butter ol settlng tanks where t s allowed to settle for a few hours. After ths the fne partcles of butter-ol resdue are removed from the butter-ol and then butter ol s stored n storage tanks. Now, t s cooled to a temperature of C sutable for storage of butter-ol. hs sub-system conssts of motors and gearbox n seres. Sub-system (Packagng): In ths sub-system the packets of processed butter-ol are created usng a pouch-fllng machne. It s fll, flow and seal automatc machne. hs sub-system conssts of prnted crcut board and pneumatc cylnder n seres. Mathematcal formulaton of the system: o determne fuzzy relablty and long run fuzzy avalablty of a butter-ol manufacturng plant, the mathematcal formaton of the model s carred out usng mnemonc rule for sx sub-systems. ransent state: In order to fnd fuzzy relablty of ths system, we have formed a system of lnear dfferental equatons usng mnemonc rule from the transton dagram. Accordng to ths rule, the dervatve of the probablty of every state s equal to the sum of all probablty flows whch comes from other states to the gven state mnus the sum of all probablty flows whch goes out from the gven state to the other states. he dfferental equatons formed n ths way are known as the Chapman Kolmogorov dfferental equatons. Now the frst order dfferental equatons assocated wth the transton dagram (g. ): (t) + Y P (t) = β 2P 2(t) + β P 3(t) + β3p 4(t) + β P (t) + β P (t) + β P (t) ()

4 g. : ranston dagram 2(t) + Y2P 2(t) = 2α 2cP (t) + β P 8(t) + β2p 9(t) (2) + β P (t) + β P (t) + β P (t) + β P (t) (t) + β P 3(t) = α ( c) P (t) (3) + (t) + β P + (t) = α ( c) P (t), = 3, 4, 5, 6 (4) 7+ (t) + β P 7+ (t) = α ( c) P 2(t), =, 2, 3, 4, 5, 6 (5) system are obtaned from the conon d when t, 0. In ths state, Eq. -5 reduce to the followng system of equatons. Here, we have used P for P (t ) =, 2 3: Y P = β 2 P 2 +β P 3 +β 3 P 4 +β 4 P 5 +β 5 P 6 +β 6 P 7 Y 2 P 2 = 2α 2 cp +β P 8 +β 2 P 9 +β 3 P 0 +β 4 P +β 5 P 2 +β 5 P 3 β P 3 = α 5 (-c) P β P + = α (-c) P, = 3, 4, 5, 6 β P 7+ = α (-c) P 2, =, 2, 3, 4, 5, 6 Let: Where: 2 ( ) 3 ( ) ( c) ( c) ( c) 2α c + α c + α c + Y = α4 + α5 + α6 Y 2 = [β 2 +α (-c) +α 2 (-c) +α 3 (-c) +α 4 (-c)+α 5 (- c)+α 6 (-c)] Wth ntal conons: P k (0) =, f k = 0, Otherwse he fuzzy relablty R P (t) of the system can be computed by: R (t) = P (t) + P (t) P 2 2 Steady state: In process ndustres, management s generally nterested n the long run fuzzy avalablty of the system. So the steady state probabltes of the system are also needed. Steady state probabltes of the 68 α ( c), 2 6 δ = = β Solvng these equatons recursvely, we get: P 2 = 2kc P where, k = α2 β2 P 3 = δ P P + = δ P, = 3, 4, 5, 6. P 7+ = 2kc δ P, =, 2, 3, 4, 5, 6 Now, usng the normalzng conon: We get: 3 = P = [ ] P = ( + 2kc)( + δ + δ + δ + δ + δ ) + 2kcδ Now we know long run fuzzy avalablty A ( ) can be calculated usng:

5 A( ) = P + P2 2 RESULS Numercal analyss: he effect of varous parameters on fuzzy avalablty s studed for steady state. If the falure, repar rates and coverage factors are altered, the fuzzy avalablty s affected. hs effect s shown n able -3. able : uzzy avalablty correspondng to falure rates of separator and CBM α 3 α c = c = c = c = c = c = c = c = c = c = c =.0 he value correspondng to all values of α and α 3 s J. Math. & Stat., 5 (): 65-7, Effect of falure rates of separator and Contnuous Butter Makng (CBM) on long run fuzzy avalablty: he effect on the system s studed by varyng the values of α and α 3. Let the values be α 2 = , α 4 = , α 5 = , α 6 = , β = 0.4, β 2 = 0.40, β 3 = 0.67, β 4 = 0.33, β 5 = 0.67, β 6 = able 2: uzzy avalablty correspondng to falure rates of separator and meltng vats α 4 α c = c = c = c = c = c = c = c = c = c = c =.0 he value of fuzzy avalablty correspondng to all values of α and α 4 s

6 able 3: uzzy Avalablty correspondng to falure and repar rates of Separator β α c = c = c = c = c = c = c = g. 2: Graph of coverage factor and fuzzy avalablty for separator and CBM g. 3: Graph of coverage factor and fuzzy avalablty for separator and meltng vats Let the values be α = 0.007, α 2 = , α 3 = , α 4 = , α 5 = , α 6 = , β = 0.4, β 2 = 0.40, β 3 = 0.67, β 4 = 0.33, β 5 = 0.67, β 6 = hs s shown n g. 2. c = c = c = c =.0 he value of fuzzy avalablty correspondng to all values of α and β s Now we see the effect of system coverage factor on the fuzzy avalablty wth the falure rates of the subsystem separator and contnuous butter makng. 70 Effect of falure rates of separator and meltng vats on long run fuzzy avalablty: he effect on the system s studed by varyng the values of α and α 4. Let the values be α 2 = , α 3 = , α 5 = , α 6 = , β = 0.4, β 2 = 0.40, β 3 = 0.67, β 4 = 0.33, β 5 = 0.67, β 6 = Now we wll see the effect of system coverage factor on the fuzzy avalablty. or ths let the values of falure and repar rates are: α = 0.008, α 2 = , α 3 = , α 4 = 0.002, α 5 = , α 6 = , β = 0.4, β 2 = 0.40, β 3 = 0.67, β 4 = 0.33, β 5 = 0.67, β 6 = 6.00 hs effect s shown n g. 3. Effect of falure and repar rate of separator on long run fuzzy avalablty: he effect on the system s studed by varyng the values of α and β. Let the values be α 2 = , α 3 = , α 4 = 0.004, α 5 = , α 6 = , β 2 = 0.40, β 3 = 0.67, β 4 = 0.33, β 5 = 0.67, β 6 = 6.00.

7 g. 4: Graph of Coverage actor and fuzzy avalablty for separator wth dfferent falure and repar rates Now we wll see the effect of system coverage factor on the fuzzy avalablty. or ths let the values of falure and repar rates are: α = 0.009, α 2 = , α 3 = , α 4 = 0.004, α 5 = , α 6 = , β = 0.45, β 2 = 0.40, β 3 = 0.67, β 4 = 0.33, β 5 = 0.67, β 6 = hs effect s shown n g. 4. DISCUSSION rom able, we see that as the falure rate of contnuous butter makng machne ncreases the fuzzy avalablty of the system decreases slowly and f we ncrease the falure rate of the separator the fuzzy avalablty decreases rapdly correspondng as the system coverage factor ncreases. rom the plot n g. 2 t s clear that as the coverage factor ncreases, fuzzy avalablty ncreases. rom able 2, we see that as the falure rate of the sub-system meltng vats ncreases, the fuzzy avalablty of the system decreases slowly but as we ncreases the falure rate of the subsystem separator ncreases, the fuzzy avalablty of the system decreases rapdly. rom the plot n g. 3, t s clear that as the coverage factor ncreases, fuzzy avalablty ncreases. able 3 shows that ncrease n falure rate of separator decreases the fuzzy avalablty of the system rapdly and ncrease n the repar rate of separator ncreases the fuzzy avalablty of the system. rom the plot n g. 4, t s clear that as the coverage factor ncreases, fuzzy avalablty ncreases. CONCLUSION Analyss of fuzzy avalablty of butter-ol processng plant can help n ncreasng the producton of the butter-ol. A comparatve study of able -3 and g. -4 reveals that sub-system A,.e., separator has maxmum effect on the long run fuzzy avalablty of the complete system. he effect of system coverage factor correspondng to values of falure and repar rate of sub-system a on long run fuzzy avalablty of system has also been presented graphcally n g. 4. Other J. Math. & Stat., 5 (): 65-7, sub-systems are almost equally effectve. Numerc results show that all the fuzzness, system coverage factor and mantenance have sgnfcant effects on the fuzzy avalablty of butter-ol processng plant. Hence, t s recommended that management should pay more attenton to sub-system A so that the overall performance of the system may mprove. ACKNOWLEDGEMEN he researchers are grateful to the revewers for ther crtcal evaluaton and suggested revsons for further mprovement of the research. REERENCES. Ca, K.Y., 996. Introducton to uzzy Relablty. Kluwer Academc Publshers, Norwell, MA., USA., ISBN: , pp: Ca, K.Y., C.Y. Wen and M.L. Zhang, 99. uzzy relablty modelng of gracefully degradable computng systems. Relabl. Eng. Syst. Safe., 33: Chowdhury, S.G. and K.B. Msra, 992. Evaluaton of fuzzy relablty of a Non-seres parallel network. Mcroelect. Relabl., 32: Zuang, H.Z., 995. Relablty analyss method n the presence of fuzzness attached to operatng tme. Mcroelect. Relabl., 35: DOI: 0.06/ (94)0073-L 5. Ca, K.Y. and C.Y. Wen, 990. Street-Lghtng lamps replacement: A fuzzy vewpont. uzzy Sets Syst., 37: DOI: 0.06/065-04(90) Pandey, D. and S.K. yag, Profust relablty of a gracefully degradable system. uzzy Sets Syst., 58: DOI: 0.06/j.fss Zadeh, L.A., 968. Probablty measures of fuzzy events. J. Math. Anal. Appl., 23: measures%20of%20fuzzy%20events%20968.pdf 8. Verma, A.K. et al., uzzy-relablty Engneerng: Concepts and Applcatons. st Edn., Narosa Publshng House, UK., ISBN: pp: Ca, K.Y., C.Y. Wen and M.L. Zhang, 99. Survvablty ndex for CCNs: A measure of fuzzy relablty. Relabl. Eng. Syst. Safe., 33: Ca, K.Y., C.Y. Wen and M.L. Zhang, 99. uzzy varables as a bass for a theory of fuzzy relablty n the possblty context. uzzy Sets Syst., 42: DOI: 0.06/065-04(9)9043-E. Ca, K.Y., C.Y. Wen and M.L. Zhang, 993. uzzy states as a bass for a theory of fuzzy relablty. Mcroelect. Relabl., 33: