Avalable Onlne at http//jnrm.srbau.a.r Vol.1, No.3, Autumn 2015 Journal of New Researhes n Mathemats Sene and Researh Branh (IAU) Solvng Redundany Alloaton Problem wth Reparable Components Usng Genet Algorthm and Smulaton Method M. Shahrar * Faulty of Management, South Tehran Branh, Islam Azad Unversty, Tehran, Iran Reeved Sprng 2015, Aepted Autumn 2015 Abstrat Abstrat Relablty optmzaton problem has a wde applaton n engneerng area. One of the most mportant problems n relablty s redundany alloaton problem (RAP). In ths researh, we worked on a RAP wth reparable omponents and k-out-of-n sub-systems struture. The objetve funton was to maxmze system relablty under ost and weght onstrants. The am was determnng optmal omponents number of eah subsystem, nludng the optmal number of reparmen alloated to eah subsystem. Beause ths model belongs to Np. Hard problem, we used genet algorthm (GA) for solvng the presented model and response surfae methodology (RSM) was used for tunng of algorthm parameters. Also for alulatng the relablty of eah subsystem (and system relablty) we used a smulaton method. Fnally, a numeral example was solved to test the algorthm performane. Keywords Redundany Alloaton Problem, k-out-of-n sub-systems, Common Cause Falures, Genet Algorthm. *. Shahrar.mr@gmal.om
M. Shahrar /JNRM Vol.1, No.3 Autumn 2015 92 1- Introduton The ams of relablty systems are to fnd a better way for nreasng the lfe yle of omponents and systems. One of the most famous models n relablty s redundany alloaton problem (RAP) that tres to nrease the system relablty by parallelng some omponents to eah subsystem. Fyffe et al. [8] were the frst researhers who presented RAP. The objet of ther model was maxmzng the system relablty under ost and weght onstrants. They solved ther model by dynam programmng. Nakagawa and Myazak [13] presented a nonlnear problem for relablty optmzaton model. Ha. C, and Kuo [10] formulated RAP wth smlar subsystems as a non-onvex nteger programmng. Tllman et al. [16] revewed 1 studes that made on relablty problems wth dfferent redundany models and presented ommon relablty-redundany method. In 1992, Chern [3] proved that RAP belongs to Np. Hard problem beause of ts alulaton tme. Many heurst and meta-heurst methods were used for solvng ths problem at that tme. The heurst methods that were presented by Sharma and Venkateswarn [15], Aggarwal [2], Aggarwal et al. [1], Gopal et al. [9] and Nakagawa and Nakashma [1] were very smlar to eah other. Nakagawa and Nakashma [1] ompared the performane of NN, GAGI, and MSV heurst algorthms used for solvng general RAP. Ida et al. [12] and Yokota et al. [17] were the frst researhers who solved RAP wth seres-parallel onfguraton and dfferent falure modes wth genet algorthm (GA). Cot and Smth [7] presented the mathematal model of RAP for maxmzng system relablty under unertanty of omponents relablty and Cot, D.W. and Smth [] solved ths problem usng GA. Cot, D.W., and Smth [5] made some hanges on objetve funton of ths model and solved the new model wth GA. In ths study, we present a RAP wth seres-parallel onfguraton and k-out-of-n sub-systems. The omponents are reparable and eah reparman s only able to repar the omponents of a spef subsystem. The objetve funton s to maxmze system relablty and the system varables are number and type of eah subsystem omponents and reparman. The paper s dvded nto fve parts. The seond part s problem defnton. Thrd part deals wth solvng methodologes. A numeral example s presented n part four and the fnal part s onluson and further studes.
Solvng Redundany Alloaton Problem wth Reparable Components Usng Genet Algorthm and Smulaton Method 93 2- Problem defnton In ths paper, we present a sngle-objetve RAP. The objetve funton of the problem s maxmzng system relablty under ost and weght onstrant. The subsystems are serally onneted and the omponents n eah subsystem are parallel. The onfguraton of subsystem s k-outof-n. The omponents are reparable and eah reparman an only work on the omponents of the alloated subsystem. The am of model s determnng the number and type of omponents n eah subsystem, nludng the number of reparmen alloated to eah subsystem. Beause alulatng the total repar tme of the omponents n eah subsystem n addton to the relablty of the system s very omplex, we used a smulaton method for alulatng them. 2-1- Model assumptons Some bas assumptons of model were as follows The system s seres-parallel, The omponents are dfferent based on ost and operaton rate, The omponents are reparable, It s possble to alloate dfferent omponent types for eah subsystem. 2-2- Nomenlatures R t System relablty at tme t, s Number of subsystems, Subsystems ndex, 1,2,, s, k v Mnmum number of omponents n subsystem, Maxmum number of omponents n subsystem, j Component type ndex, j 1,2,3, n j w j W j1 2 Number of omponent type j that alloated to subsystem, Weght of omponent type j that alloated to subsystem, Upper bound of system weght, Pre of omponent type j that alloated to subsystem, Pre of rerutment eah reparman for subsystem, The tme dependent repar ost of Components have onstant falure rate (CFR), The number of the subsystem s determnst, For eah subsystem, dfferent omponent types are avalable, 3 C m omponents that alloated to subsystem, Upper bound of system ost, Number of reparman rerutment for reparng omponents of subsystem,
M. Shahrar /JNRM Vol.1, No.3 Autumn 2015 9 t t Total reparng tme of omponent n subsystem, Msson tme of system. 2-3- Mathematal model The mathematal model s as follows Max R t s 1 R t S.t 2 k n j v 3 s 1 1 j 1 w j 1. n j W s s j1 j 2 3. 1 j 1 1. n. m t C 5 3- Solvng method We used smulaton method for determnng the total repar tme of the omponents n eah subsystem and alulatng relablty of eah subsystem as well as whole system, and a general GA for determnng the optmal system parameters. 3-1- Genet algorthm Genet algorthm (GA) s one of the most applable algorthms for solvng RAP. Ths algorthm was presented by Holland [11], n 1992. The pseudoode of ths algorthm presented n Fgure 1. Fg 1. Pseudoode of GA
Solvng Redundany Alloaton Problem wth Reparable Components Usng Genet Algorthm and Smulaton Method 95 3-1-1. Chromosome The hromosome used for llustratng the soluton of the presented model s presented n Fgure 2. 3-1-2. Algorthm operators We used rossover, mutaton operators for reatng new generaton and eltsm to have good solutons n new generaton. 3-1-3. Crossover operator For rossover operator, we selet two parents and then generate an nteger number between 1 and s. Next, we hange the frst part of the frst parent wth the seond part of seond parent and the seond part of the frst parent wth the frst part of seond parent as shown n Fgure 3. Fgure 2. Chromosome of the problem Fgure 3. Crossover operator
M. Shahrar /JNRM Vol.1, No.3 Autumn 2015 96 3-1-. Mutaton operator For mutaton operator, we selet one parent and then generate an nteger number between 1 and s. Then, we hange the frst part of the parent wth the seond part as shown n Fgure. Fgure. Mutaton operator 3-1-5. Penalty funton For evaluaton of eah hromosome, we onsder a ftness funton, whh onsders the relablty of eah hromosome. After generatng ntal hromosomes, rossover, and mutaton operator, the new hromosomes may be nfeasble. For avodng nfeasble seleton of hromosomes, we onsder a penalty funton for ftness funton. The penalty funton presented n Equatons 6 to 8 are as follows Ftness Funton R t 6 PF 1 PF 2 PF 1 MAX PF1 MAX s 1 j1 s 1 j1 j 1 W. n w s j 1 C n 2 3-2- Smulaton method j j,1. m. t 3,1 7 8 For alulatng ftness funton of eah reated hromosomes, we used a smulaton method. The flowhart of the event s presented n Fgures and 5.
Solvng Redundany Alloaton Problem wth Reparable Components Usng Genet Algorthm and Smulaton Method 97 Component falure Is any workng onponent? System fnshng workng Is any dle reparman? Addng one unt to queue of omponents watng to alloate to reparman Alloatng a reparman to fled omponent Subtratng one unt from avalable reparman Fg. Component falure dagram of smulaton Repar of one omponent Addng one unt to workng omponents Is any faled omponent n que? Addng one unt to dles reparman Alloatng one omponent to reparman Subtratng one unt from dle reparmans Fgure 5. Component Repar dagram of smulaton 3-3- Parameter tunng We used response surfae methodology (RSM), for algorthm parameter tunng. The algorthm parameters are populaton sze ( npop ), rossover operator probablty ( P ), mutaton operator probablty ( P m ), and algorthm teraton ( MaxIt). The boundares and optmum values of the algorthm parameters are presented n Table 1.
M. Shahrar /JNRM Vol.1, No.3 Autumn 2015 98 Table 1. The boundares and optmum values of algorthm parameters Optmal value Upper value Lower value npop 50 100 100 P 0. 0.7 0.63 P m 0.1 0.3 0.1 MaxIt 100 200 200 - Numeral example For llustratng the performane of the presented algorthm, we solve a numeral example. Ths example has been already solved by Cot [7]. The parameters of examples are presented n Table 2. The other parameters are ( n max 6), W 220, C 500 and t 100. The values of 2 and are alulated n Equatons 09 and 3 10. Choe 1 (j=1) Choe 2 (j=2) Choe 3 (j=3) Choe (j=) 1 1 w1 2 2 w 2 3 3 w 3 w 1 0.005320 1 3 0.000726 1 0.0099 2 2 0.008180 2 5 2 0.008180 2 8 0.000619 1 10 0.0031 1 9 - - - 3 0.013300 2 7 0.011000 3 5 0.0120 1 6 0.00660 0.00710 3 5 0.01200 6 0.00683 5 - - - 5 0.006190 2 0.00310 2 3 0.00818 3 5 - - - 6 0.00360 3 5 0.005670 3 0.00268 2 5 0.00008 2 7 0.010500 7 0.00660 8 0.0039 5 9 - - - 8 0.015000 3 0.001050 5 7 0.01050 6 6 - - - 9 0.002680 2 8 0.000101 3 9 0.00008 7 0.00093 3 8 10 0.01100 6 0.006830 5 0.001050 5 6 - - - 11 0.00390 3 5 0.003550 6 0.00310 5 6 - - - 12 0.002360 2 0.007690 3 5 0.013300 6 0.011000 5 7 13 0.002150 2 5 0.00360 3 5 0.006650 2 6 - - - 1 0.011000 6 0.00830 7 0.003550 5 6 0.00360 6 9
Solvng Redundany Alloaton Problem wth Reparable Components Usng Genet Algorthm and Smulaton Method 99 2 2 3.. j1 3 j1 j1 j1 ; ; 1,3,6,9,12,1 2,,5,7,8,10,11,13 9 2 3 ; 1,2,,1 10 10 We solved the presented example wth GA usng smulaton methods and the optmal soluton for ths example wth 20 tmes runnng the algorthm s presented n Fgure 6. The total ost and weght of the problem are C 99. 9375 and W 219 and the optmal system relablty s R 100 0. 9963. 5- Conluson and further studes In ths paper we presented a RAP wth seres-parallel onfguraton and k-out-of-n subsystems. The omponents are reparable and eah reparman s only able to repar the omponents of spef subsystem. The objetve funton s maxmzng system relablty and the system varables are number and type of eah subsystem omponents and reparman. We used smulaton method for alulatng system relablty and GA for solvng the numeral example. As we expeted, the relablty of the system s greater than the relablty of the system wth non-reparable omponents. For further studes dfferent assumptons an be onsdered. For example, omplex onfguraton an be onsdered for subsystems. Also dfferent meta-heurst algorthms an be used for solvng the presented model to ompare ther results wth the results of ths algorthm. Fgure 6. The optmal hromosome of problem
M. Shahrar /JNRM Vol.1, No.3 Autumn 2015 100 Referenes [1] Aggarwal, K. K, Gupta, J. S. and Msra, K. B., A New Heurst Crteron for Solvng a Redundany Optmzaton Problem, IEEE Transatons on Relablty 1975; Vol. 2, pp. 86-87. [2] Aggarwal, K. K., Redundany Optmzaton n General Systems, IEEE Transaton on Relablty, 1976; Vol. 25, pp. 330-332. [3] Chern, M. S., On the Computatonal Complexty of Relablty Redundany Alloaton n a Seres System, Operaton Researh Letters 1992; Vol. 11, pp. 309-315. [] Cot, D. W. and Smth, A., Consderng Rsk Profles n Desgn Optmzaton for Seres Parallel Systems, Proeedngs of the Relablty & Mantanablty Symposum, Phladelpha, 1997. [5] Cot, D. W. and Smth, A., Redundany Alloaton to Maxmze a Lower Perentle of the System Tme-to- Falure Dstrbuton, IEEE Transatons on Relablty 1998; Vol. 7, pp. 79-87. [6] Cot, D. W. and Smth, A., Stohast Formulatons of the Redundany Alloaton Problem, Proeedngs of the Ffth Industral Engneerng Researh Conferene, Mnneapols, 1996. [7] Cot, D. W., Maxmzaton of System Relablty wth a Choe of Redundany Strateges, IIE Transatons 2003; Vol. 35, No. 6, pp. 535-5. [8] Fyffe, D. E., Hnes, W.W. and Lee, N.K., System Relablty Alloaton and Computatonal Algorthm, IEEE Transatons on Relablty 1968; Vol. 17, pp.6-69, [9] Gopal, K., Aggarwal, K.K., and Gupta, J. S., An Improved Algorthm for Relablty Optmzaton, IEEE Transatons on Relablty 1978; Vol. 27, pp. 325 328. [10] Ha, C. and Kuo, W., Relablty Redundany Alloaton An Improved Realzaton for Non-onvex Nonlnear Programmng Problems, European Journal of Operatonal Researh 2006; Vol. 171, pp. 2-38. [11] Holland, J., 1992, Adaptaton Natural and Artfal ststems, Unversty of Mhgan [12] Ida, K., Gen, M. and Yokota, T., System Relablty Optmzaton wth Several Falure Modes by Genet Algorthm, Proeedng of the 16th Internatonal Conferene on Computers and Industral Engneerng, Ashkaga of Japan, 199.82. [13] Nakagawa, Y. and Myazak, S., Surrogate Constrants Algorthm for
Solvng Redundany Alloaton Problem wth Reparable Components Usng Genet Algorthm and Smulaton Method 101 Relablty Optmzaton Problems wth Tow Constrants, IEEE Transaton on Relablty 1981; Vol.30, pp. 175-180. [1] Nakagawa, Y. and Nakashma, K., A Heurst Method for Determnng Optmal Relablty Alloaton, IEEE Transaton on Relablty 1977; Vol. 26, pp. 156-161. [15] Sharma, J. and Venkateswarn, K.V., A Dret Method for Maxmzng the System Relablty, IEEE Transatons on Relablty 1971; Vol. 20, pp. 256-259. [16] Tllman, F. A., Hwang, C.L. and Kuo, W., Determnng Component Relablty and Redundany for Optmum System Relablty, IEEE Transatons on Relablty, 1977; Vol. 26, pp. 162-165. [17] Yokota, T., Gen, M. and Ida, K., System Relablty of Optmzaton Problems wth Several Falure Modes by Genet Algorthm, Japanese Journal of Fuzzy Theory and systems 1995; Vol. 7, pp. 117-1.
M. Shahrar /JNRM Vol.1, No.3 Autumn 2015 102