Behavior Analysis of a Biscui Making lan using Markov Regeneraive Modeling arvinder Singh & Aul oyal Deparmen of Mechanical Engineering, Lala Lajpa Rai Insiue of Engineering & Technology, Moga -, India E-mail : aulmech7@yahoo.com Absrac Modern producs ranging from simple componens o complex should be designed o be opimal and reliable. The challenge of modern engineering is o ensure ha manufacuring coss are reduced and design cycle imes are minimized while achieving requiremens for performance and reliabiliy. This paper explains a mehodology o sudy he ransien behavior of repairable mechanical biscui shaping Sysem peraining o a biscui manufacuring plan. The mehodology for deermining he availabiliy of he sysem is based on Markov Modeling. The effec of repair rae on mos vulnerable iems of he sysem is examined o realize he highes level of performance. The failure and repair raes of he differen subcomponens of he sysem are aken as consan. robabiliy consideraions a various sages of he sysem give differenial equaions, which are solved using Laplace Transform o obain he sae probabiliies. Increase in availabiliy of sysem confers many benefis such as more profi, improved delivery performance and reduced lead imes. Keywords Manufacuring sysem availabiliy, prioriy performance measures, produciviy, reliabiliy. I. INTRODUCTION Reliabiliy/availabiliy/mainainabiliy analyses of producion sysems peraining o process indusries have assumed ever-increasing imporance in he recen pas. This can benefi he indusry in erms of higher produciviy and lower mainenance coss. In he recen pas quie a good number of sudies have been carried ou by researchers []-[] in he field of reliabiliy/availabiliy/mainainabiliy, and several mehods have been proposed for reliabiliy analysis of indusrial sysems under prevenive/predicive mainenance. A number of performance measures are used in he process indusry as indicaors o describe he performance of a plan regarding is reliabiliy and mainainabiliy. Commonly used indicaors are: on sream facor, on sream facor wih slowdown, availabiliy (inheren, achievable or operaional), urnaround rae, annualized urnaround index, rouine mainenance cos index ec. Clearly mos of hese indicaors are used mainly in he operaional sage while a few of hem can be used o evaluae differen designs a he early sages of design. lan availabiliy is commonly considered in he design sage for screening differen design alernaives. Availabiliy, in general, is defined as he abiliy of an iem o perform is required funcion a a saed insan of a ime or over a saed period of ime. A biscui manufacuring plan siuaed in Ludhiana, India is chosen for sudy. In his paper a subsysem of he plan, which is a coninuous producion sysem, is considered and he availabiliy analysis of he complex mechanical sysem under preempive resume prioriy repair is carried ou. Laplace ransform is used for solving differenial equaions o obain sae probabiliies. Numerical resuls based upon he rue daa colleced from indusry are presened o illusrae he seady sae behavior of he sysem under differen plan condiions. II. SYSTEM DESCRITION In Biscui Shaping Sysem, biscuis are shaped from he dough shee, which is supposed o wound over he dies of a roary machine. Iniially iler machine serves he purpose of iling ou he dough from he rolleys o he dough hopper. In his hopper, dough is ransformed ino dough shees wih he help of blades. The conveyor ransfers his dough shee furher o cuer where cuing of dough shee ino predefined size have been done. Furher desired size dough shee is conveyed o roary machine hrough meal deecor. The meal 8 ISSN : 8, Volume-, Issue-,
Inernaional Journal on Theoreical and Applied Research in Mechanical Engineering (IJTARME) deecor deecs unwaned/foreign paricles from he dough shee before subsequen operaion. On roary machine differen sizes and shapes of biscuis can be produced wih he help of inerchangeable dies. The figure- represens he working of his sysem. Fig. : rocess Flow Diagram III. NOTATIONS AND ASSUMTIONS NOTATIONS Subsysem A: Uni namely iler machine subjeced o Subsysem B: Uni namely dough hopper subjeced o Subsysem C&C: Two unis namely conveyor and cuer working parallel in he plan subjeced o minor & major failure. Subsysem D: Uni namely roary machine subjeced o λ, λ : Failure raes of subsysems C & C resp. λ, λ, λ : Failure raes of subsysems A, B and D resp., : Repair raes of subsysems C & C resp.,, : Repair raes of subsysems A, B and D resp. g: The subsysem /uni is good bu no operaive. r: The subsysem /uni is under repair. qr: The subsysem/uni is queuing for repair. i (): Sae probabiliy ha he sysem is in i h sae a ime. S: Laplace ransform variable Dash (`) : Represen derivaives w.r... ASSUMTIONS. All he unis are iniially operaing and are in good sae.. Each uni is as good as new afer repair.. The failure and repair evens are saisically independen.. In case of he failure of iler machine, he dough will be fed manually o he dough hopper. So iler machine is he leas prioriy iem for repair acion. The sysem can work a full capaciy even when his subsysem is under repair acion.. There is single repair faciliy and hus repair is performed on prioriy basis if he failed iems are more. 6. The subsysem namely Meal Deecor (M) never fails. 7. Whenever a uni fails, is repair begins immediaely. 8. Maximum wo subsysems will come o failed sae a he same ime because repair of failed uni begins immediaely and is carried quie quickly. IV. AVAILABILITY ANALYSIS OF THE SYSTEM robabiliy consideraion gives he following differenial equaions associaed wih he sae ransiion diagram as shown in figure-. ( ) o 6 7 8 Fig. : Sae Transiion Diagram ( ) ISSN : 8, Volume-, Issue-,
Inernaional Journal on Theoreical and Applied Research in Mechanical Engineering (IJTARME) ( ) ( ) ( ) ( ) 6 6 ( ) 7 7 ( ) 8 8 ( ) Wih iniial condiions a ime = () = for i= i () = for i i Taking Laplace ransform of above equaions, he probabiliy ransform are: ( s ) A (s) A i ( s ) i For i = o A s s 7 A s s s 6 8 s Availabiliy funcion A () for he sysem is given as A ( s) ( s) ( s) ( s) ( ) s A (s) ( s) ( s ) A. where STEADY STATE BEHAVIOR OF THE SYSTEM The Seady sae behavior of he sysem can be analyzed by seing, d, he sae d probabiliies are: o 6 7 8 6 7 8 On solving equaions recursively, we ge i H H H H i B o For i = o H H H 6 HH H 7 H H H 8 HH H H H B H H H H Using normalizing condiion, we ge i H i i i The Overall seady sae availabiliies of he sysem running wih eiher full capaciy is given by: A o O H Inversion of A (s ) gives he availabiliy funcion A ( ) ISSN : 8, Volume-, Issue-,
Inernaional Journal on Theoreical and Applied Research in Mechanical Engineering (IJTARME) H H i i NUMERICAL RESULTS The Overall seady sae availabiliy of he sysem by aking =., =., =., =., =., =., =., =., =., =. is evaluaed as A O o =.7 H V. AVAILABILITY ANALYSIS The effec of various parameers on availabiliy is sudied. If he failure rae and repair raes are alered, he availabiliy is affeced. This effec is shown in following ables obained for availabiliy of biscui making sysem.. Effec of failure rae of Conveyor-I on availabiliyao: Taking =., =., =., =., =., =., =., =., =.. Table. Seady sae availabiliy versus failure rae of Conveyor-I....8 A.7.8.86.7677. Effec of failure rae of Cuer on availabiliy Ao: Taking =., =., =., =., =., =., =., =., =.. Table. Seady sae availabiliy versus failure rae of Cuer.....8 A.7..86.87. Effec of failure rae of Dough hopper on availabiliy Ao: Taking =., =., =., =., =., =., =., =., =.. Table.Seady sae availabiliy versus failure rae of dough hopper.... A.7..88.88. Effec of failure rae of Roary machine on availabiliy Ao: Taking =., =., =., =., =., =., =., =., =. Table Seady sae availabiliy versus failure rae of roary machine....8 A.7.86.88.7. Effec of repair rae of Conveyor-I on availabiliy Ao: Taking =., =., =., =., =., =., =., =., =. Table. Seady sae availabiliy versus repair rae of Conveyor-I..66..6 A.7..66. 6 Effec of repair rae of Cuer on availabiliy Ao: Taking =., =., =., =., =., =., =., =., =.. Table 6. Seady sae availabiliy versus repair rae of Cuer.... A.7..8.8 7 Effec of repair rae of Dough hopper on availabiliy Ao: Taking =., =., =., =., =., =., =., =., =.. Table 7 Seady sae availabiliy versus repair rae of Dough hopper..66..6 A.7..67.8 8. Effec of repair rae of Roary machine on availabiliy Ao: Taking =., =., =., =., =., =., =., =., =.. Table 8 Seady sae availabiliy versus repair rae of roary machine..8.6. A.7.77.. VI. RESULTS AND DISCUSSION From analysis par i is being found ha increase in he failure rae of conveyor-i, cuer, dough hopper and roary machine reduces he availabiliy of he sysem. Table o highlighs he effec of failure rae of ISSN : 8, Volume-, Issue-,
Inernaional Journal on Theoreical and Applied Research in Mechanical Engineering (IJTARME) conveyor-i, cuer, dough hopper and roary machine on he long run availabiliy of he biscui making sysem. On he oher hand he repair raes of he consiuen componen increase he availabiliy of he sysem. This effec is shown in ables o 8.The respecive improvemens in he availabiliy of he sysem are.%,.8%,.% and.7% on increasing he repair rae of conveyor-i, cuer, dough hopper and roary machine from. o.6,. o.,. o.6 and. o. repairs per hour respecively. VII. REFERENCES [] Singh I.., A Complex Sysem having Four Types of Componens wih revenive Repea rioriy Repairs, Micron reliabiliy, 8, Vol., No. 6, pp. -6. [] I.. Singh, reempive repea prioriy repairs and failures of non-failed componens during sysem failure of a complex sysem, Micron reliabiliy,, pp. 6-. [] R.T. Islamov, Using Markov Reliabiliy Modelling for Muliple Repairable Sysems, Reliabiliy engineering and sysem safey,, pp. 8 [] Q. Microelsen, Use of Reliabiliy echnology in he process indusry, Reliabiliy engineering and sysem safey, 8, pp. 7-8 [] upa., Singh J., and Singh I.., Behavioral Analysis of Duplex Casing Sysem under revenive Resume rioriy Repair Discipline a Case Sudy, Indusrial Engineering Journal Vol. XXXIII No., December. [6]. upa, J. Singh, and I.. Singh, Mainenance lanning based on erformance Analysis of 7-ou-of-: Chemical-sysem: a Case Sudy,, Inernaional Journal of Indusrial Engineering,, pp. 6-7. [7]. upa, A. upa, Availabiliy improvemen of a producion sysem under prevenive mainenance, Indusrial Engineering Journal, India, 6. Vol. XXXV No.8, pp. -6. [8] D. Dyer, Unificaion of Reliabiliy / Availabiliy / Repairabiliy Model for Markov Sysems, IEEE Transacions on Reliabiliy, 8, Vol. 8. ISSN : 8, Volume-, Issue-,