Redundancy Allocation for Series Parallel Systems with Multiple Constraints and Sensitivity Analysis

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1 IOSR Joural of Egieerig Redudacy Allocatio for Series Parallel Systems with Multiple Costraits ad Sesitivity Aalysis S. V. Suresh Babu, D.Maheswar 2, G. Ragaath 3 Y.Viaya Kumar d G.Sakaraiah e (Mechaical Egg Dept, Adhiyamaa College of Egg, Hosur , T. N. State) Idia 2 (Pricipal, M.N.R.College of Egg, Sagareddy Dist, Hyderabad-5 72, A.P.State, Idia 3 (Pricipal, Adhiayamaa College of Egg, Hosur 6359, T.N.State) Idia. d.(pricipal, Sri Bhagava Mahaveer Jai College of Egg, Bagalore, K.A. State,) Idia. e. (Professor, Mech Egg Dept G.Pulla Reddy College of Egg, Kurool, A.P.State,) Idia. Abstract- The mai obective of this paper is to make redudacy allocatio for Series Parallel Systems with multiple costraits i order to determie the compoet reliabilities (r ), the umber of compoets i each stage (x ), stage reliability (R ) ad the System Reliability (R s ) i each stage for the give Cost ad Weight costraits to maximize the System Reliability. The system used is a Series Parallel cofiguratio by usig optimizatio techiques, such as Lagragea Multiplier Method ad Dyamic Programmig. The Reliability Model has bee developed for Cost ad Weight costats. The authors i their work make a attempt to egotiate the impact of Cost ad Weight as costats for the Mathematical Fuctio r c d b π.ta The developed models are hady with high applicatio value particularly 2 i the case of Itegrated Reliability Model for redudat systems with Series Parallel cofiguratio. Geerally reliability is treated as the fuctio of Cost but i ay give practical situatio apart from cost other costrait like Weight will have hidde impact o the reliability of the system. I this model the Lagragea techique is implemeted to determie the umber of compoets as itegers ad the variatio i Cost ad weight is foud more, this leads to itroduce Dyamic Programmig techique by takig the umber of compoets as real umbers. The model has yielded very ecouragig results ad it ca be applied to ay type of system, simple or complex. The advatage of this model is very flexible ad requires little processig time. Keywords: System Reliability; Stage Reliability; Series Parallel System; Multiple costraits;. INTRODUCTION This Paper treats a System with may stages i Series Parallel cofiguratio. To build high reliability i to a system, a Desig Egieer usually resorts to redudat uits for each stage, but must stay with i the resources available, i.e. costraits improved o the desig, such as Cost ad Weight. The optimum redudacy depeds o Reliability, Cost ad Weight etc. of each stage. The reliability of a System ca be maximized subect to the resource costraits to determie the optimum umber of redudat compoets for each stage, whe the reliability of each compoet is kow i other situatios, the reliability of the system ca be maximized subect to the resource costrait to determie the reliability of the compoets i the system whe the umber of Redudat uits i each stage is kow. As o Today the literature o maximizatio of System Reliability problems are cosidered, there is o much work reported o Itegrated Reliability Model for Redudat Systems with multiple costats. I this sceario the authors wat to make a attempt to optimize the Reliability of a System with Multiple Costats.To study ad optimize the Itegrated Reliability Model for Redudat Systems with Multiple Costraits is cosidered with Cost ad Weight as costats, for the give kow mathematical fuctio r π.ta 2 c b d 2. STATEMENT OF THE PROBLEM To determie the ukows i.e. the umber of compoets (x ), the compoet reliabilities (r ) the stage reliability (R ) at each stage for a give multiple costats to maximize the system reliability. Though Cost has direct relatio i maximizig System Reliability, the idirect impact of weight as o additioal costrait i optimizig the Reliability of a Redudat System presets a ovel begiig i the metioed area of research. The Series Parallel Systems are cosidered with Cost ad Weight as costraits to maximize the Reliability of a redudat system as its obective fuctio. 3. ASSUMPTIONS OF THE MODEL. All the compoets i each stage are assumed to be idetical. 2. The compoets are assumed to be statistically idepedet i.e. the failure of oe compoet does ot affect the performace of the other compoets i the system. ISSN: P a g e

2 IOSR Joural of Egieerig 3. A compoet is either i workig coditio or oworkig coditio. r π c d.ta (4) 2 b Where c is cost costrait ad b, d are costats. 7. PROBLEM FORMULATION System Reliability for the give cost fuctio R R s = (5) FIG : SERIES-PARALLEL CONFIGURATION 4. NOMENCLATURE: R s = System Reliability. R = Stage Reliability, <R < F= Lagragea fuctio r = Reliability of each compoet i stage, <r <. x = No. of compoets i stage. c =Cost coefficiet of each compoet i stage w = Weight coefficiet of each compoet i stage. C o = Maximum allowable System Cost. W o = Maximum allowable System Weight. b = Scalig factor for stage used i the fuctio d = Shapig factor for stage used i the fuctio p = Costat used i weight fuctio. q = Costat used i weight fuctio. 5. MATHEMATICAL MODEL Cosider that there are statistically idepedet stages i Series with x statistically idepedet i each stage. System Reliability for the give cost fuctio R R s = Subected to x = ( r ) () c. x C (2) w. x W (3) No egativity restrictio x is a iteger ad r, R > 6. MATHEMATICAL FUNCTION Cost co efficiet of each compoet i stage is derived from the followig relatioship betwee Cost ad Reliability. Cost coefficiet of each uit i stage is derived from the followig relatioship betwee cost ad reliability π c d r.ta (6) 2 b d r c b. ta (7) π/2 Sice cost costrait is liear i x c.x C (8) Similarly weight costrait is also liear i x w.x W (9) Substitutig equatios (6) ad (7) i (8) ad (9) we get the followig relatio d r b. ta.x C π/2 () q r p. ta.x W π/2 () The umber of compoets at each stage x is give through the relatio x l(r l(r ) ) (2) Maximize R [ ( r ) ] (3) s x ISSN: P a g e

3 IOSR Joural of Egieerig Subect to the costraits d r l( R ) b. ta. C π/2 l( r ) (4) q r l( R ) p. ta. W π/2 l( r ) (5) 8. LAGRANGEAN METHOD Solvig the proposed formulatio usig Lagragea method. F R 2 s p b. ta r /( / 2) l( ) q R. tar /( / 2). W d. l( r ) l( R ) C l( r ) (6) where λ ad λ 2 are Lagragea multipliers ad F beig Lagragea fuctio. The umber of compoets i each stage (x ), optimum compoet reliability (r ), stage reliability (R ) ad the system reliability (R s ) are derived from the Lagragea method. The method provides real valued solutio with referece to cost ad weight. The statioary poit ca be obtaied by differetiatig the Lagragea fuctio with respect to R, r, λ, ad λ 2 9. RESULTS AND DISCUSSIONS The followig reliability desig tables related to cost ad weight are calculated by usig the compoet reliabilities ad the umber of compoets derived from Lagragea method. 9. Case Study: Cosider the case of a Mechaical system with three stages for which the compoet Reliability is give by the equatio (4). To determie the optimum compoet reliability, stage reliability, umber of compoets i each stage ad the System Reliability ot to exceed the system cost Rs.25, Weight of the system 3kg. The compoet Reliabilities, Stage Reliabilities, Number of compoets i each stage ad the System Reliability are determied by solvig the above mathematical fuctio by usig MATLAB Versio 7. ad are preseted i the followig tables. 9.2 Cost ad Weight as costraits: 9.2. Reliability Desig Without x roudig off: Table I. Reliability desig relatig to Cost i (Rs): Stage r R x c c. x Total Cost 25. Table II. Reliability desig relatig to Weight (Kg): Stage r R x c c. x Total Cost 3. System Reliability Reliability Desig with x roudig off: The reliability desig is reestablished by cosiderig the values of to be itegers (by roudig off the value of to the earest iteger) ad the relevat results relatig to cost ad weight are preseted i the followig table, further givig the iformatio by calculatig the variatio due to cost ad weight ad the system reliability (before ad after roudig off).table: 3Reliability desig relatig to Cost i Rupees. Table III. Reliability desig relatig to Cost i (Rs) : Stage r R x c c. x Total Cost Table IV. Reliability desig relatig to Weight (Kg): Stage r R X W W. X Total Cost 39.8 System Reliability (R s ) =.84 Variatio i total Cost = 7.82% Variatio i total Weight = 6.6% Variatio i System Reliability = 4.88%. DYNAMIC PROGRAMMING: To optimize the desig by usig Dyamic Programmig the same case problem discussed i the precedig chapter has bee cosidered by takig the values of Compoet Reliabilities (r ), the umber of compoets i each stage (x ), Stage Reliabilities (R ) ad the System Reliability (R s ) as iputs. This Approach is particularly useful i optimizig the desig with the values of x s to be itegers, which are highly appreciated for practical implemetatio to real life problems. The umber of compoets, which was take as a real umber has bee chaged to a iteger. The output has come i two stages with correspodig Stage Reliability is show i Table V. ISSN: P a g e

4 IOSR Joural of Egieerig Table V DYNAMIC PROGRAMMING STAGE : No.of Stage Reliability Compoets No.of Compoets Table VI DYNAMIC PROGRAMMING STAGE 2: Stage Reliability No.of Compoets Table VII DYNAMIC PROGRAMMING STAGE 3: Stage Reliability RELIABILITY DESIGN- COST: From the Dyamic Programmig tables the maximum System Reliability is.672 with a total COST of Rs ad the correspodig optimal values are as show below. Table VIII: Reliability desig relatig to Cost i (Rs): STAGE r R x c c.x T T A L C O S T RELIABILITY DESIGN - WEIGHT: From the Dyamic Programmig tables the maximum System Reliability is.672 with a total WEIGHT of 39.8 ad the correspodig optimal values are as show below. Table IX: Reliability desig relatig to Weight i (Kg) STAGE r R x w x.w T T A L W E I G H T 39.8 System Reliability =.672 Variatio i Total Cost = 7.82% Variatio i Total Weight = 6.6% Variatio i System Reliability = 25.4 %.3 Sesitivity Aalysis: It is observed that whe the iput data of compoet reliability is icreased percet there percet icrease i system reliability. Similarly whe percet decrease i iput data there will be 5.5 percet decrease ISSN: P a g e

5 IOSR Joural of Egieerig i system reliability is observed. Whe oe factor is varied, keepig other factors costat, variatio i Cost ad Weight is as show i Table X. Table X Sesitivity Aalysis: Variatio i compoet reliability(r ) For % icrease COST 26.5% icreases WEIGHT 27.35% icreases System Reliability 28% icreases For % decrease COST 29.78% icreases WEIGHT 22.3% icreases System Reliability 5.6% icreases The aalysis cofirms that the cost ad weight are more sesitive to iput data. 2. CONCLUSION Primarily this paper is focused i allocatig redudacy uits with multiple costraits for a reliability system, where i the developmet of itegrated reliability model is discussed i detail. The paper ifers that the multiple costraits problem is first treated through Lagragea method where this method provided a real valued solutio ad as such may be ifeasible for practical implemetatio. For this reaso, the problem solved by Dyamic Programmig, which proved a ideal solutio to take the umber of compoets i iteger values ad to fid the exact system reliability. The variatio of Cost, Weight ad System Reliability is aalyzed with respect to compoet reliability i the form of Sesitivity aalysis. This model ca also be further ivestigated for differet mathematical fuctios of iterest ad also ca be applied for Parallel Series cofiguratio systems, where the applicatio of these models for such systems will be feasible oly whe the cost of the system is very low. [5] Dhigra.A.K., Optimal Apportiomet of Reliability ad Redudacy i Series Systems uder Multiple Obectives, IEEE Trasactios o Reliability, Vol.-4, No.4, December 992, PP [6] Flehiger.B.J., System reliability as a fuctio of system age, Effects of itermittet compoet usage ad periodic maiteace, preseted at 959 IRE Natioal Covetio, New York, March 959. [7] Gopal.K., Aggrawal.K.K, ad Gupta.J.S, A ew method for solvig Reliability Optimizatio problem, IEEE Trasactios o Reliability, Vol.R-29, No., April 98, PP [8] Gordo.R., Optimum Compoet Redudacy for maximum System Reliability, Operatios Research., Vol.5, March 957, PP [9] Deb.K., Optimizatio for Egieerig Desig: Algorithms ad Examples, Pretice-Hall of Idia private limited, New Delhi, 995. [] Hwag.C.L., Lai.K.C, Tillma.F.A. & Fa.L.T. Optimizatio of System Reliability by the sequetial ucostraied miimizatio techique, IEEE Trasactios o Reliability, Vol.R-24, No.2, Jue 975, PP [] S.V.Suresh Babu, Dr.D.Maheswar ad Dr.G.Ragaath, Optimizatio of System Reliability for Redudat Systems with Multiple Costraits, Iteratioal Joural o Advaced Scieces i Egieerig ad Techology (IJAEST), August, 2, Page No.99-22, Vol.No.2. Issue No.2. [2] Misra.K.B. A Method of Solvig Redudacy Optimizatio problems, IEEE Trasactios o Reliability. Vol.R-2, No.5, August 97, PP-7-2. [3] Rosario Romera, Jose E.Valdes ad Romulo I.Zequeira, Active-Redudacy Allocatio i Systems, IEEE Trasactios o Reliability, Vol.53, No.3, Sep 24. REFERENCES: [] Misra.K.B., Dyamic Programmig formulatio of Redudacy allocatio problem, Iteratioal Joural of Math Educatioal Sciece, Techology (U.K), Vol.2, July-September 97(a), PP [2] Balagurusamy.E. Reliability Egieerig, TMH, 984. [3] Tillma.F.A., Hwag.C.L., ad Way Kuo., Optimizatio of System Reliability, Marvel Dekker, New York,98. [4] Kuo.W, Li.H, Xu.Z, ad Zhag.W, Reliability Optimizatio with Lagrage Multiplier ad Brach ad Boud Techique, IEEE Trasactios o Reliability, Vol.R-36, No.5, December 987, PP ISSN: P a g e