Optimal Allocation of Complex Equipment System Maintainability

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1 Optmal Allocato of Complex Equpmet System ataablty X Re Graduate School, Natoal Defese Uversty, Bejg, , Cha edcal Protecto Laboratory, Naval edcal Research Isttute, Shagha, , Cha Emal:rexs841013@163.com Ya Kog ache Egeerg Departmet, Naval Petty Offcer Academy, Begbu, , Cha Emal:hgky@163.com Jwe Cu Departmet of ltary Logstcs ad Scece ad Techology Equpmet, Natoal Defese Uversty, Bejg,100091,Cha Emal:cjw@163.com Ka Zhag edcal Protecto Laboratory, Naval edcal Research Isttute, Shagha, , Cha Emal:zk@163.com L a edcal Protecto Laboratory, Naval edcal Research Isttute, Shagha, , Cha Emal:ml@163.com The mataablty allocato of equpmet system s a mult-objectve programmg problem wth multple costrats. Based o the defects of tradtoal algorthm, t s dffcult to coquer all the costrats at the same tme, ad t s dffcult for the result to reflect dfferet satsfacto levels o the targets. Ths paper proposes a method combato wth geetc algorthm (GA) ad satsfacto fucto. Frstly, use GA to solve the optmal soluto of the sgle target; secodly, get the satsfacto level of the total target through that of the sgle target; fally, obta satsfactory soluto of the problem wth GA. Results show that the method ca effectvely solve the defects above, whch s sgfcat to the optmal allocato of equpmet system mataablty. ISCC , December, 2015 Guagzhou, Cha Copyrght owed by the author(s) uder the terms of the Creatve Commos Attrbuto-NoCommercal-NoDervatves 4.0 Iteratoal Lcese (CC BY-NC-ND 4.0).

2 Complex Equpmet System ataablty X Re 1. Itroducto The tradtoal mataablty allocato methods, maly smple quattatve methods[1], s hard to apply to the hghly complex equpmet system. The mataablty allocato problem was studed whe the mmum mateace cost ad the shortest mateace tme are satsfed at the same tme[2]; however the models are wth complex olear characterstcs, especally whe the umber of parts equpmet systems creases [3], the soluto space wll be expoetally creasg. It s dffcult for the covetoal method to obta deal optmzato results (for example: the result ca't meet the costrat codtos). I addto, the covetoal method, whch the lear weghted method s used, fds t hard to gve reasoable weghted coeffcet the mult-objectve optmzato; therefore, the paper troduces a method combato wth the geetc algorthm (GA) ad the satsfacto fucto, ad the smulato example shows ts effectveess. 2. ataablty Allocato odel of Equpmet System (1) odel 1: the shortest mateace tme The costrats are: 1 m ct ( k ( ) TC r ct Tf ct ct ) C 0 (2.1) (2.2) L (2.3) ct ct s the mea mateace tme, ct s the mateace tme of the th part, s the falure rate of the th part. Formula (2.2) dcates the upper lmt of the mateace cost. Formula (2.3) dcates the upper lmt ad the lower lmt of the mateace tme of each part. (2) odel 2: the mmum mateace cost m C The costrats are: 1 1 ( k ( ct ) T ( C r ct f )) C 0 (2.4) 2

3 Complex Equpmet System ataablty X Re 1 1 ct * ct (2.5) L (2.6) ct C s mateace cost. Formula (2.5) dcates the upper lmt of the mea mateace tme. Formula (2.6) dcates the upper lmt ad the lower lmt of the mateace tme of each part. 3. Satsfacto Fucto of the Target The satsfacto fucto of the target takes the objectve fucto as the depedet varable, ad utlzes the objectve fucto s satsfacto level to represet the decso maker s satsfacto the target value. 0 dcates that the result s the most satsfed to the decso maker, ad 1 dcates the opposte meag. Suppose that there be a programmg model where mmum value s eeded for calculato. The value of objectve fucto R s betwee ad R m. Whe Rm satsfacto level equals 0. Rm R, the satsfacto level equals 1, ad Whe R Rmax, the I the mult-objectve programmg problems, the total satsfacto fucto s eeded based o each target satsfacto fucto. Suppose that satsfacto fuctos of target 1 ad target 2 be h 1( R1), h2( R2), ad the total satsfacto fucto s how as follows[4]. h( R) w1 w2 [ h1( R1)] [ h2( R2)] (3.1) w1, w 2 are the weghted coeffcets, whose value terval are [0, 1] to, reflect the mportace of dfferet goals. 1 dcates that the goal s the most mportat, 0 dcates that the goal ca be gored, ad s a operator to select a smaller umber. 4.Calculato ethod by Usg Geetc Algorthm Geetc algorthm (GA) s a kd of adaptve optmzato method, wth may advatages such as smart search ad gradual optmzato [5]. Here we troduce the detals of the method. 1) Ecodg Chromosome uses the bary codg. The mateace tme of each compoet s sgfed by a bary umber cotag 8 bts cosderg both the calculato precso ad the speed, so the whole legth of chromosomes s the equpmet system.) 8. ( s the total umber of compoets 3

4 Complex Equpmet System ataablty X Re 2)Ftess fucto Use pealty factor to defe the ftess fucto cosderg costrats of the optmzato problem, see Type (4.1), whch s the maxmum objectve fucto, ad Type (4.2), whch s the mmum objectve fucto. N f N f (x) 1 f N 1 N f ( x) f s the ftess fucto, f (x) (4.1) (4.2) s the objectve fucto, s pushmet factor, N s a costat used to make sure that the ftess s postve. I the paper, N 300, 200. Whe the model 1 ad 2 are solved respectvely, the ftess fucto s establshed accordg to Type (4.2), ad the objectve fuctos dcate the mataablty tme, ad the mataablty cost. Whe the two models are solved sythetcally, the ftess fucto s establshed accordg to Type (4.1), ad the objectve fucto refers to the total satsfacto level, see Type (3.1) [6]. 5.Calculato ad Aalyss x meets the costrats x ca t meet the costrats x meets the costrats x ca t meet the costrats Supposg that a certa equpmet system have fve uts, the upper lmt for mateace cost s 1.45 mllo ad the upper lmt for average mateace tme s 0.75 hour. I ths algorthm, the populato sze s 5000, the bggest evoluto tmes s 100, the legth of the chromosome s 40, the tal crossover probablty s 0.7, the tal mutato probablty s Calculato by usg geetc algorthm odel 1 ad odel 2 are calculated separately by usg GA. To reduce the radom mpact o the results, ru the program fve tmes, ad the results are show Fg. 1 ad Fg. 2. From the two fgures, we ca see that the results are slghtly dfferet ad ted to be coverget every tme. 4

5 Complex Equpmet System ataablty X Re Fgure 1: Evolutoary Fgure of ea ateace Tme Accordg to the results above, obta the scopes of the two objectves that average mateace tme s 0.55~0.75, ad that mateace cost s 134.6~145; therefore, respectvely establsh the satsfacto of both fuctos (lear), as show (5.1 )~ (5.2), the total satsfacto fucto accordg to the type structure (3.1). 5.2 Comparso wth other methods h 1( ct ) 5 ct 3.75 (5.1) h 2( C) 0.096C (5.2) For comparso, Tables 1 ad 2 respectvely lst the result of Lgo software ad GA algorthm. It ca be see that, for a sgle goal programmg model, the soluto results of Lgo software ca ot meet the correspodg costrat codtos, ad GA ca solve the problem better. For mult-objectve programmg model, compare the soluto results whe the weghted factors are the same. The results of Lgo software ca ot reflect the cosstecy betwee mportace degrees of two goals, amely, the mportace degree of Goal 1 s obvously above that of Goal 2. (The satsfacto level of goal 1 s 0.97 ad the satsfacto level of goal 2 s 0.05.) We ca coclude that, t s hard for weghted factors to adjust the mportace degrees of dfferet goals the lear weghted method, whle the results obtaed from the satsfacto fucto s relatvely more reasoable. Fgure 2:Evolutoary Fgure of Cost ct1 ct 2 ct3 ct 4 ct 5 ct C h1 h2 odel odel w1=0.5,w2= Table 1 : Results by Usg GA Table 1 5

6 Complex Equpmet System ataablty X Re ct1 ct 2 ct3 ct4 ct5 ct C h1 h2 odel odel Table 2 :Results by Usg Lgo Software Soluto 6. Cocluso The paper studed the mataablty allocato model of equpmet system by usg GA ad satsfacto fucto. Compared wth the tradtoal methods, t solves the flaws that t s hard to fulfll the costrat codtos the sgle objectve programmg ad to reflect the sgfcace dfferece of dfferet goals. The method ad coclusos ca provde referece to the mataablty allocato of the equpmet system. Refereces [1] Ja-m Zhao, Hog Zhou. A optmal allocato method for system mataablty[j]. echacal scece ad techology. 19(2), (2000) (I Chese) [2] Yog Zhag, Xa-Hag Chao. A Study o mateace optmal dstrbuto of equpmet system based o heredtary algorthm[j]. East Cha electrc power. 12(1),85-87(2003) (I Chese) [3] Guag-Che Ba, Ya Che. Satsfactory soluto of mult-objectve optmzato of system relablty[j]. Systems egeerg ad electrocs. 22(8),85-87(2000) (I Chese) [4] L. N. Xg, Y. W. Che, Yag K W. A hybrd approach combg a mproved geetc algorthm ad optmzato strateges for asymmetrc travelg salesma problem[j]. Egeerg Applcatos of Artfcal Itellgece, 2008,21(8): [5]K. Kamall, L. J. Jag, J. Ye, et al. Usg Q-learg ad geetc algorthms to mprove the effcecy of weght adjustmets for optmal cotrol desg problems[j]. Joural of Computg ad Iformato Scece Egeerg, 2007,7(4): [6] L. N. Xg, Y. W. Che, K. W. Yag,A hybrd approach combg a mproved geetc algorthm ad optmzato strateges for asymmetrc travelg salesma problem[j]. Egeerg Applcatos of Artfcal Itellgece, 2008,21(8):