A Cost-Effective Strength-Stress Reliability Modeling and Optimization in Engineering Design

Transcription

1 A Cot-Effectve trength-tre Relablt Modelng and Optmzaton n Engneerng Degn Mohammad T. Khaawneh, hannon R. Bowlng, ttcha Kaewkuekool, and Bung Rae Cho Department of Indutral Engneerng Clemon Unvert Clemon, outh Carolna Abtract Becaue of manufacturng varablt, the trength of a component ma var gnfcantl and unpredctabl. When a component put nto ue, t ubjected to a tre, whch alo unpredctable. If the trength of the component uffcent to wthtand tre, then the component functonal. Otherwe, the component fal mmedatel. Achevng hgher relablt requre hgher cot. Th paper frt how two relablt optmzaton model ung normall dtrbuted trength and tre: one to maxmze relablt ubject to maxmall allowable cot aocated wth relablt, whle the other to mnmze the cot ubject to a requred mnmum relablt level. The paper then proceed b howng how thee two conflctng decon crtera are compred n a jont manner b preentng a propoed cot-relablt comprome model. A numercal example along wth entvt anal preented and compared wth prevou work. Keword trength, tre, Relablt, Optmzaton, Engneerng degn. Introducton Qualt, a derable charactertc that a product or ervce hould poe, a ke factor leadng to bune ucce, growth, and an enhanced compettve poton. Becaue of toda' compettve market, t often not onl derable but alo necear to maxmze the relablt of a product to enure cutomer atfacton and product ucce. The challenge that bune face not onl to develop product that are relable but alo take nto conderaton the cot factor. Achevng hgh level of relablt whle mnmzng cot often poe problem and lmtaton for engneer durng the degn tage. Therefore, a cot-relablt comprome wll alwa ext n the context of tem degn. A prmar goal of relablt and degn engneer to chooe the bet tructural and mechancal degn, conderng factor uch a cot, relablt, weght and volume (ee Kapur and Lamberon, 977). The queton then are of how to ncorporate thee factor nto a model that wll acheve optmum reult. Becaue the factor that nfluence the falure of a product are often probabltc n nature, t mportant to ncorporate the randomne of the degn varable nto a model that optmze the fnal product. However, th randomne motvate ome degner to beleve that component falure ma be entrel elmnated b ung a preconceved margn a a afet factor. The relablt of a component an mportant factor that need to be condered at earler tage of degn. It ha been proven that conventonal degn methodologe ma not be adequate from a relablt pont of vew. A new probabltc degn methodolog ha been ntroduced and t explctl dentfe all the mportant degn parameter and varable. The two mportant random varable that have been condered are tre and trength. Hence, determnng the probablt dtrbuton for thee varable a ke tep n calculatng component relablt. For a certan mode of falure, the relablt of a component wth repect to the partcular mode of falure the probablt that the trength of the component greater than the tre actng on the component. Therefore, the cotrelablt comprome nvolve ncorporatng nonlnear optmzaton model wth two random varable tre and trength a decon varable.. Reearch Motvaton The frt tep n optmzng relablt-cot n tem degn developng tattc about tre and trength dtrbuton. Conder the cae n whch trength and tre are ndependent and normall dtrbuted. Informaton about a trength dtrbuton can be obtaned from materal properte whle thoe about a tre dtrbuton can be obtaned from load tattc htor.

2 Two model have been ntroduced n the lterature to addre uch knd of problem (Kapur, 975). In eence, the objectve of the frt model to fnd optmal proce parameter (.e. mean and tandard devaton) for the tre and trength parameter b mnmzng total cot whle mantanng a dered mnmum relablt level. The objectve of the econd model to fnd optmal tattc of tre and trength b maxmzng relablt whle mantanng mnmum cot budget avalable. The mathematcal formulaton of thee model are tated below: Nonlnear Optmzaton Model I mn TC C( ) + C ( ) + C3 ( ) + C4( ).t. Z + Nonlnear Optmzaton Model II () max Z ( ) / + (3).t. C ) + C ( ) + C ( ) + C ( ) r (4) ( 3 4 where mean value of the trength; tandard devaton of the trength; mean value of the tre; tandard devaton of the tre; C ( ) cot functon for mean trength; C ( ) cot functon for tandard devaton of trength; C ( ) cot functon for mean tre; and C ( ) cot functon for tandard devaton of tre. 3 B examnng model I and II, we can oberve that the objectve functon and the contrant are wtched and hare the ame decon varable. Both model addre the ame problem n a eparate manner. Therefore, the motvaton of th reearch to mnmze the total cot and maxmze the relablt n a multaneou manner. Proof. The proof for the total cot and relablt functon preented. The probablt dent functon for both the tre () and trength (), are gven below, repectvel: f ( ) exp π, 4 () (5) ( ) exp f, (6) π The relablt, R, equal to P(->0). B defnng a random varable -, normall dtrbuted random varable nce and are both normall dtrbuted. Hence, the mean and tandard devaton of are gven n the followng form, repectvel:, and + (7) Therefore, the Relablt functon gven below: R P(>0) exp d (8) 0 π Ung the tranformaton z, we can rewrte the relablt equaton n the followng form: R π z e + d Baed on the above equaton, we can clearl oberve that the relablt can be found ung normal table, where t hould be noted that the new varable z the tandard normal varable. We can alo ee that the relablt depend (9)

3 on the lower lmt of the ntegral. That, lowerng the lower lmt wll reult n hgher value of relablt. Thu, we can dentf the couplng equaton n the followng form: Z (0) + The value of Z ued to fnd the relablt from the cumulatve tandard normal table. The hgher the value of Z, the hgher the relablt. The total cot then gven b the ummaton of the ndvdual cot functon for the mean and tandard devaton for both tre and trength. Th can be wrtten n the followng form: TC C ) + C ( ) + C ( ) + C ( ) () ( The Propoed Model The man objectve of th reearch to develop a trength-tre nonlnear optmzaton model that addree cot and relablt n a multaneou manner, whch a multple repone optmzaton problem. Becaue the derablt functon approach one of the mot ueful method n dealng wth uch tpe of problem, we attempt to ue th approach for fndng the optmum oluton. For each repone Y (x), a derablt functon d ) ( Y agn number between 0 and to the poble value of Y, wth d ( ) 0 repreentng a completel underable value and d ( ) repreentng a completel derable Y or deal repone value. The ndvdual derablt functon are then combned ung the geometrc mean, whch gve the overall derablt D : Y D d Y d Y d k k Y / ( ( ) ( )... ( k )) () where k denote the number of repone. Notce that f an repone Y completel underable, then the overall derablt zero (.e. d ( Y ) 0 ). In our model we have two repone functon (.e. total cot, TC, and relablt, Z). The derablt functon for each of thee wll be d ( TC ) and d ( Z). The value of k n our cae. The method wll attempt to fnd optmum operatng condton that wll provde the "mot derable" repone value. The derablt approach cont of developng ndvdual derablt functon for each repone. To th end, an -tpe derablt functon for mnmzng the total cot combned wth an L-tpe derablt functon for maxmzng tem relablt. The ndvdual derablt functon for both the cot and relablt, repectvel, are hown below:.0, TC < T r TC U d ( TC ), T TC U (3) T U 0, TC > U 0, Z < L r Z L d ( Z ), L Z t (4) t L.0, Z > t where L lower value for the derable repone; U upper value for the derable repone; T a poble maxmum value for the repone; t a poble mnmum value for the repone; r a parameter to determne how trctl the target value of the total cot dered; and r a parameter to determne how trctl the target value of the relablt dered. Conequentl, the followng optmzaton model developed:

4 C max r ( ) + C( ) + C3( ) + C4( ) U. T U + t L L / r (5) 4. Numercal Example Informaton on the cot functon for the tattcal parameter of trength and tre requred n order to develop the fnal cot-relablt optmzaton model. A the trength of the materal ncreae, the cot aocated wth that value ncreae. Mantanng hgher average trength value requre controlled manufacturng procee envronment and better heat treatment procee, and hence reult n hgher cot value. mlarl, enurng le varablt reult n the trength value, nce t requre hgher machner precon reult n hgher cot. On the other hand, reducng the value of the mean and tandard devaton for the tre of materal generall provde hgher relablt value, whch requre addtonal cot. The Amercan ocet of Metal (AM, 969) ha developed varou table howng tattcal data for cot and properte of elected materal. Thee data have been alo ued b Kapur (974) and ftted polnomal functon were developed for the dcrete data wth a range from 30,000 to 75,000 p. The ftted functon for the dfferent cot functon are gven below:.35 c ( ) , 30,000 75, 000 p (6) c ( ) 800,,000 0, 000 p (7) 0.53 c 3 ( ) 8997, 0,000 68, 000 p (8) c 4( ) 366, 500 7, 500 p (9) The above reult gve the followng repone functon for the total cot: TC (0) Etablhng Lmt and Target Value for Repone Functon. The ue of the derablt functon n the preent cae requre knowledge about the lower, upper, and target value for both repone functon (.e. TC and Z). The target and upper value for the total cot were et b fndng the mnmum and maxmu m poble value of the TCfuncton baed on the varaton n the parameter lmt (.e. T $78.97, U $7.5). The target and lower value for the Z-functon were found n a mlar manner (.e. t 58.4, L.33). Baed on the reult obtaned above, therefore, the fnal optmzaton model can be wrtten n the followng form: r / r 7.5 ( ) + max () t. 30,000 75,000 (.a),000 0,000 (.b) 0,000 68,000 (.c) 500 7,500 (.d) 5. Reult The propoed model a nonlnear optmzaton model wth four decon varable, namel mean and tandard devaton of both trength and tre. In order to olve the optmzaton model, a numercal algorthm that

5 enumerate all poble value of the decon varable and ther correpondng optmum derablt-functon value wa developed and ued. A tated earler, r and r are exponent that determne how trctl the target value of t related repone dered, and the effect of varng thee value on the optmum oluton wa nvetgated whle olvng the model. For that purpoe, a new parameter that repreent that relatve mportance of each repone compared to the other wa ntroduced. There are certan expectaton related to change n the relatve-mportancerato (.e. RIR r r ), and thoe are clearl tated n the dcuon ecton of th paper. The algorthm wa ued to fnd optmum repone at dfferent value of RIR. The ample reult of the optmum repone at dfferent value of the derablt exponent are hown n Table. The effect of changng the RIR on the optmum total cot and optmum relablt are hown n Fgure and. From the fgure, t can be clearl een that ncreang the RIR value decreae both optmum value of the total cot and the relablt. The obtaned table and fgure are hown below: Table : ample optmum reult for dfferent value of the derablt exponent. r r r TC Z R r (p) (p) (p) (p) ($) Indcate optmum value. Total Cot.V. Relatve Importance Rato (r/r) Relablt.V. Relatve Importance Rato (r/r) Total Cot ($) RIR (r/r) Relablt r/r Fgure : The varaton of the optmum total cot a a functon of relatve mportance rato. Fgure : The varaton of the optmum relablt a a functon of relatve mportance rato. Fgure 3 how the varaton of the optmum value of the decon varable a a functon of the relatve mportance rato, RIR. It can be clearl een an ncreae n RIR wll reult n a decreae n the mean trength value, whle the mean tre and the tandard devaton of both trength and tre decreae.

6 OPtmum Decon Varable.V. RIR Optmum Decon Varable RIR (r/r) Mean() td() Mean() td() Fgure 3: The varaton of the optmum decon varable a a functon of relatve mportance rato. 6. Dcuon and Concluon Th paper ha provded a cot-effectve trength-tre relablt modelng approach n engneerng degn ung the concept of derablt functon. The propoed model enable degner to optmze both relablt and total cot n a multaneou manner ung normall dtrbuted functon for both tre and trength, whch addreed the dadvantage of prevou model dcued n the lterature. A we can ee n Fgure, the total cot decreae a the RIR ncreae, where the curve amptote to a certan value at farl large value of RIR. Large value of RIR mean we are gvng more mportance to the total cot functon and n a wa gnorng the effect of the relablt functon. A RIR approache a large enough value, our model become dentcal to that of Kapur (975) and gve almot the ame optmum reult (.e. TC $ 5.7, R0.99, 4050, 4405, 650, 455p). In th cae, the model provde optmum total cot at the lowet poble relablt level. On the other hand, havng a zero value for RIR repreent the cae where our man concern the relablt functon and the mportance of the total cot value neglected. Th wll reult n optmum relablt at a maxmum total cot (.e. TC $7.5, R.0), whch never the cae n a real-world applcaton. The ame argument apple to Fgure, where t can be een that ncreang RIR decreae the z-value and hence the relablt. everal extenon could be nvetgated n the future b conderng dfferent dtrbuton to overcome the lmtaton of normal dtrbuton, n addton to ncorporatng tme dependent trength-tre model. Bographcal ketch Mohammad T Khaawneh a PhD tudent n the Department of Indutral Engneerng at Clemon Unvert. He receved h B.. and M.. n Mechancal Engneerng from Jordan Unvert of cence and Technolog, Jordan. H reearch nteret are n the development of advanced technolog to olve nteretng human-machne tem degn problem, and modelng human n proce and qualt control tem. hannon R Bowlng a PhD tudent n the Department of Indutral Engneerng at Clemon Unvert. He receved h B.. n Electrcal Engneerng Technolog from Bluefeld tate College, WV, and M.. n Qualt Management from Eat Tenneee tate Unvert. H reearch focu n the ue of advanced technolog tem for mprovng human performance n the arcraft mantenance ndutr. ttcha Kaewkuekool a PhD tudent n the Department of Indutral Engneerng at Clemon Unvert. He receved h B.. n Producton Technolog Educaton from Kng Mongkut Inttute of Technolog, Thonbu, Thaland, and M.. n Indutral Engneerng from the Unvert of Mam, FL. H reearch focu n the ue of advanced technolog tem for mprovng human performance n qualt control tem. Bung Rae Cho an aocate profeor n the Department of Indutral Engneerng at Clemon Unvert. He receved h M.. n Indutral and tem Engneerng from Oho tate Unvert and h Ph.D. n Indutral Engneerng from the Unvert of Oklahoma. H pecalzaton qualt engneerng wth an empha n robut degn and tolerance nthe. Reference - Kapur, K. C., 975, "Optmzaton n Degn b Relablt", AIIE tranacton, pp Kapur, K. C., and Lamberon, L. R., 977, Relablt n Engneerng Degn, John Wle & on, Inc. 3- Cho, B. R., 00, Lecture note on Indutral Tetng and Qualt (IE 87), Clemon Unvert, C. 4- Amercan ocet for Metal (AM), 969, I, Properte and electon of Materal, II, Heat treatment, Cleanng and Fnhng, III, Machnng; all volume publhed b Amercan ocet for Metal, Metal Park, Oho.