Available online at ScienceDirect. Procedia Engineering 69 ( 2014 ) Ivo Malakov, Velizar Zaharinov*

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1 Avbe onne t wwwscencedrectcom ScenceDrect Proced Engneerng 69 ( 04 ) th DAAAM Internton Symposum on Integent Mnucturng nd Automton, 03 Computer Aded Determnton o Crter Prorty or Structur Optmzton o Technc Systems Ivo Mov, Vezr Zhrnov* TU-So, bvd Kment Ohrds 8, So 000, Bugr Abstrct In the present pper choosng o methods or determnton o crter prorty (sgncnce) or mutcrter optmzton o the structure o technc systems hs been done For ths purpose, nown methods re nyzed nd evuted gnst combnton o chosen ctors o nuence Agorthms nd sotwre modues re deveoped, bsed on nown methods or determnton o the weght coecents' vector, whch modues re mpemented n dog system or structur optmzton Resuts re shown rom the ppcton o the sotwre or sovng prtcur test probem usng the consdered methods 04 The Authors Pubshed by Esever Ltd 04 The Authors Pubshed by Esever Ltd Open ccess under CC BY-NC-ND cense Seecton nd peer-revew under responsbty o DAAAM Internton Venn Seecton nd peer-revew under responsbty o DAAAM Internton Venn Keywords: technc systems; mutcrter optmzton; structur vrnt; prorty; weght coecents; gorthms; sotwre Introducton Durng the desgn process o technc systems t s necessry to sove mutpe tmes the probem or choosng optm (rton, eectve) structur vrnt [, ] The re condtons necesstte ths choce to be mde ccordng to combnton o crter, whch crter re very oten contrdctory Thereore, the probem or choosng n optm vrnt o technc system represents mutcrter optmzton probem In common cse the chosen crter or evuton o the terntve vrnts, coud hve derent retve mportnce (vue, prorty, sgncnce) dependng on the condtons o the prtcur probem Sovng o mutcrter optmzton probem retes to number o dcutes One o the mn dcutes s denng prorty or the objectve unctons [, 4, 5, 0] Speczed terture [5-0] nyss shows, tht n * Correspondng uthor Te: E-m ddress: vzhrnov@yhoocom The Authors Pubshed by Esever Ltd Open ccess under CC BY-NC-ND cense Seecton nd peer-revew under responsbty o DAAAM Internton Venn do: 006/jproeng

2 736 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) unmbguous choce o sutbe methods or determnton o prorty when sovng certn group o probems cnnot be mde Aso or number o nown methods or determnng the mportnce o crter, there s c o gorthmc nd sotwre deveopment, whch mes dcut ther use n prctce On the other hnd, t s nown, tht s mny more soutons o the MOP re produced, whe tng nto ccount derent mportnce o the crter, s much ncreses the possbty or ndng souton, whch stses n the best wy possbe the requrements o the decson mer The purpose o the present pper s deveopment o gorthms nd sotwre modues o chosen methods or determnton o objectve unctons' prorty, whch modues w be ntegrted n dog system or structur optmzton o technc systems Nomencture А squre mtrx wth bnry comprsons α prncp rght egenvector n mx mxmum rght egenvue number o compred crter α -th component o vector α j component o row, j -th coumn, beongng to mtrx wth bnry comprsons А Σ vector derved rom (4) CI consstency ndex CR consstency rto RCI rndom consstency rto w weght coecent or -th objectve uncton w j weght coecent or j -th objectve uncton B rectngur mtrx derved rom (9) W weght vector b vector derved rom (9) -th objectve uncton j j -th objectve uncton A Σ vector derved rom () ᾱ vector wth components ᾱ ᾱ components o vector ᾱ dened n (3) R rn o mportnce IF mportnce ctor PF prt uncton K set o crter ndexes Choosng o method or determnton o crter prorty The probem or comprng vue (sgncnce) o the crter s one o the mn probems when sovng mutcrter optmzton probems [, 4, 5, 0] Ths probem cn be ormuted n the oowng wy - nd method or denng prorty mthemtcy nd ts eve o nuence on the choce o optm structur vrnt o technc system Ths s compex ts, becuse or ts souton quntttve evuton o sgncnty derng objects (the crter or choosng o optm structur vrnt) hs to be done, nd whch crter re bstrct concepts, chrcterzng the quty o terntve vrnts From the nown wys o ssgnng crter mportnce, through prorty order, prorty vector nd weght vector (weght coecents vector), the tter s the most commony used When ssgnng weght vector, or every crteron, K, weght coecent s ssgned (sgncnce coecent) w, K The coecent w s re postve number Ths number w denes the retve "weght", "mportnce", "vue" o the th crteron n

3 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) reton to the other crter The weght vector W w }, K, represents dmenson vector, dened n snge hypercube - W { w : w [0, ], w } K { In speczed terture re descrbed scores o expert methods or evuton o objectve unctons' prorty, but u comprtve nyss o methods s mpossbe to me I there ws snge unvers method, such comprson coud hve been mde, but such method does not exst Moreover there re no commony ccepted crter or evuton o the methods For sovng ths probem, nown terture sources [4-0] were nyzed nd the oowng mn ctors were cssed, hvng nuence over the choosng o method or evuton o the crter prorty: compexty o conductng the expert evuton nd bor-consumpton or obtnng the expert normton; vbty o ddton normton (or nstnce, normton or boundry vues o the crter n the esbe set, etc); degree o greement o the experts' opnons; ppcbty (here under ppcbty s understood the crter count, or whch the ppcton o gven method s ecent); bor-consumpton or processng o the normton gven by the experts; retonshp between the wy o denng crter prorty nd the used mutcrter optmzton methods Ater conductng nyss o over 70 nown methods [] or determnton o crter prorty, use o bnry comprsons methods s proposed Nevertheess ther bor-consumpton, these methods re the smpest nd the most justed rom psychoogc pont o vew For the experts t s most convenent n ech step to evute quttvey ony two crter The expert cn compre these crter nd to te whch one o them s more mportnt, to gve n evuton o the type "equ mportnce", "sght superorty o one crteron over other", "strong superorty", etc, but the expert cnnot nswer how much one crteron s more mportnt thn other In humn thnng usuy mges nd words re used, not numbers Thereore requestng n nswer rom the expert n the orm o prtcur number or vue representng quntttve equvent o the eve o superorty o one crteron over others, w be very dcut ts or the expert The bnry comprsons methods re chrcterzed wth retvey hgh eve o consstency nd rebty o the obtned resuts For ths purpose n some o them re provded correspondng procedures The so ced u or doube comprson s pped The bnry comprsons o the crter re mde twce For exmpe, t the begnnng the rst nd second crter re compred, the thrd nd ourth crter re compred nd so orth unt the st crteron Ater tht the sme procedure s crred out bcwrds Ths provdes mens or vodng ccdent errors Moreover, wth these methods the n resuts or the weght coecents coud be obtned by ppyng the method or consecutve pproxmton Accordng to ths method, or every oowng pproxmton, resuts rom the prevous pproxmton re used s coecents o sgncnce or the experts' rtocntons When certn condtons re met, ths process s convergent The normzed weght coecents pproch some constnt vues, the tter strcty represent retons between the crter or gven nput dt The mn dsdvntge o the chosen group o methods s tht they re bor-consumng, whch s reted to the necessty or compex ccuton procedures For overcomng ths probem, gorthms nd sotwre modues were deveoped, bsed on nown bnry comprsons methods, whch re ntegrted n the sotwre system or mutcrter optmzton PoyOptmzer [3] 3 Agorthms o methods or determnton o crter prorty In ths prgrph re shown gorthms o nown methods or determnton o weght coecents 3 Sty's method [7] Wth Sty's method souton o the mtrx equton () [7] s sought A nmx, where n mx, () where the oowng gorthm s used:

4 738 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) Step The components o the prncp rght egenvector (coumn mtrx) re dened rom () j j () Step The vector s normzed by the sum o ts components (3) (3) The components o the normzed vector re the ccuted weght coecents Step 3 For determnton o n mx rst the components o vector A (row mtrx) re ound rom (4) A, (4) Step 4 The vector A s mutped by the vector The resut s equ to n mx Step 5 The consstency ndex CI nd the consstency rto CR re ccuted rom (5) [7] nmx CI, CR CI RCI (5) where RCI s gven n Tbe Step 6 The crteron or cceptng the gven mtrx wth bnry comprsons s (6) [7] CR 0 (6) I CR s greter thn 0 t s recommended tht reconsderton o the evutons o the comprsons s to be done, unt condton (6) s stsed Tbe Vues o the rndom consstency ndex RCI [7] RCI Prdos nd Mnn's method [8] The method s so nown s the humn rtonty pproch [8] Step The mtrx o bnry comprsons A s composed, hvng the orm (7) [8] 3 A, 3 3 (7)

5 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) w where j, j, j 9,8,7,,,,,,,,,, j [8] w j j Step The mtrx equton (8) [8] s composed BW b, (8) Where [8]: w w B , W, b (9) 0 w 0 0 0, Step 3 The ner est squres probem (0) [8] s composed ccordng to (9) [8]: x b BW (0) Step 4 The system o ner equtons () s composed w x w x x w 0, 0 0 () Step 5 Ater sovng the ner equtons system () the components o vector W re ound, whch components re the sought weght coecents 33 Vochns nd Inson's method [9] For ths method the denton o prorty or the crter s done by one expert, who consecutvey mes quttve evuton bout the retve mportnce o prs o crter by usng the symbos: " " - more mportnt, " " - wth equ mportnce, " " - ess mportnt The resuts re wrtten down n preerence mtrx A Ater tht the expert must dene how mny tmes the mportnce o the compred crter ders j Derent cses re possbe: strongy derng crter, modertey derng crter nd wey derng crter [9] Determnton o the weght coecents w s done n the oowng order:

6 740 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) Step The symbos,, re repced n the preerence mtrx wth ther correspondng vues Step The sum j j wrtten s coumn vector () s ccuted or every row o the preerence mtrx A j The resuts re A j j () Step 3 The vues o the coecents o sgncnce re ccuted or crter These vues re eements o A by the coumn vector coumn vector, produced by mutpyng the preerence mtrx A j, e the vue o j j j j j j s ccuted rom (3) (3) j Step 4 The vues o the normzed weght coecents o the crter re ccuted usng (4) j (4) 34 Snge comprsons method [6] Step The crter re ordered by decresng mportnce (cortege) The most mportnt crteron or the evuton, styng n the rst pce o the order, s evuted s 00 nd wth rn o R Step Every crteron s compred to the precedng by mportnce nd vue or the bnry comprson between the two crter s dened (5) R R j 0, 0,,0, j, R ;, (5) Step 3 The sgncnce coecent or every crteron s ccuted (6) IF R ; R R j (6) Step 4 The sgncnce coecents re normzed nd so the weght coecents re obtned

7 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) Sotwre deveopment On the bss o the deveoped gorthms nd sotwre modues or determnton o crter prorty or mutcrter optmzton probems reted to technc systems, grphc user nterce hs been deveoped, whch s ntegrted n the sotwre system PoyOptmzer [3] The user nterce provdes the decson mer, wth n esy nd cer wy to dene nd nput hs/her subjectve preerence towrds gven trget uncton (crteron) The m durng deveopment o the grphc user nterce or the prorty denton modues, hs been or mxmum bstrcton o the user rom purey quntyng the prorty, e concrete vue, nsted to the user s presented vsu wy, through whch he/she cn express hs/her preerence Ths s cheved through nterctve sces nd symbos Emntng the determnstc nture o the prorty denton, ds the decson mer when mng choce, becuse n most cses the user hs dcutes when sed to gve prtcur vue or the prorty o one crteron over other (or over the rest crter) Sty's method, so nown s method or herrchy nyss, cn be used or denng crter prorty or probems wth three to ten objectve unctons For nput o the bnry comprson between two objectve unctons horzont sces re used On the two sdes o ech sce there re correspondng objectve unctons The poston o the sce's ndctor cn be chnged The menng o the ndctor's poston s s oows: s cose to gven objectve uncton the ndctor s postoned, s much tht objectve uncton s preerred over the other objectve uncton correspondng to the gven sce Sty's method checs or consstency o the bnry comprsons' mtrx When consstency s cng, e there s contrdcton n the nput normton, Sty's method proposes chnge o certn prortes, so tht the probem becomes consstent Ater the chnge, the newy entered normton must be reccuted The user nterce or congurng nput dt or Sty's method s shown on Fg Fg User nterce or Sty's method The Prdos nd Mnn's method cn be used or the sme number o unctons, s Sty's method The user nterce s dentc wth tht used or Sty's method nd the worng prncpe s the sme Frst the Sty method s used or removng ny nconsstences n the nput normton, nd ter tht, the weght coecents re ccuted The Prdos nd Mnn's method s chrcterzed by the nterpretton, tht when mnmzng regret, osses or rsng ncome, the decson mer s mng or error mnmzton when evutng bnry comprsons Vochns nd Inson's method oers to the user possbty or evuton by two ndctons: retve mportnce nd derence n the sgncnce o the crter The method s chrcterzed by ts smpcty nd the possbty or tng n ccount the degree o mportnce between the compred crter On Fg s shown the user nterce when usng ths method For nput o the retve mportnce the user uses the button " " When ths button s pressed menu ppers, rom whch the retve mportnce between the two crter cn be chosen The symboc ssgnments re s oows: "<" - the objectve uncton rom the et sde o the symbo s ess mportnt rom the uncton tht s on the rght sde o the symbo; " " - the two objectve unctons re equy mportnt; ">" - the objectve uncton rom the et sde o the symbo s more mportnt rom the uncton tht s on the rght sde o the symbo

8 74 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) The derence n the sgncnce o the compred trget unctons s ssgned through the vertc sces The down poston o the ndctor mens "sghty derng crter"; postonng o the ndctor n the mdde o the sce mens "modertey derng crter"; the up poston o the ndctor mens "sgncnty derng crter" Fg User nterce or Vochns nd Inson's method () nd or the snge comprsons method (b) For the snge comprsons method the decson mer rrnges the objectve unctons n cortege, n whch the most mportnt uncton s rst, oowed by the second most mportnt nd so orth Then ech trget uncton s compred wth the prevous one n the cortege nd bnry comprsons re ormed The user nterce when usng the snge comprsons method s shown on Fg b 5 Exmpe For gven dt concernng crter mportnce nd the orderng 4 3, determne the weght coecents vues by usng the exmned methods, nd nd the correspondng structur vrnts o the TS, whch re Preto optm or the oowng mutcrter probem: mn x ( xn ), mn x ( xn ), mn 3 x 3 ( xn ), mn 4 x 4 ( xn ) n n n n (7) stsyng the constrnts (8) x 05, g x, g x 64 g (8) 9 3 Vues or the objectve unctons nd the constrnts re gven n tbur orm (Tbe ) Wth F n, n 8 re mred the prt unctons o the technc system (tbe rows), whch the system must reze Ech prt n uncton s executed rom dente set o terntve devces xn X n { xn, xn,, xn } [3] For exmpe the rst prt uncton F s executed rom three terntve devces, the second F nd thrd F 3 one re executed so rom three devces, the ourth F 4 rom our devces nd etc In ech ce o the tbe re gven seven vues, representng the vues nd prmeters o the correspondng eementry devce x n, or whch prmeters optm vues re sought ( xn ), 4 or there re mposed constrnts g m ( xn ), m 3 There re no constrnts concernng comptbty between the eementry devces Thereore the number o possbe vrnts or budng the system s

9 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) Tbe Vues o the objectve unctons nd the constrnts PF X X X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 0 F 78;9;89 9;4;46 30;66;03 64;79 48;64 39; ;0 49;7 003;6 F 0;8;9 5;45;35 76;0;40 38;055 79;05 48; ;5 40;0 790;7 F 3 96;06;96 36;0;66 7;98;6 69;74 30;044 78;0 34;8 553;7 607;9 F 4 33;4;09 ;06;7 86;07;69 7;89;7 56;087 03;040 5;05 43;0 87;9 39;7 089;9 634;9 F 5 84;77;7 3;77;49 7;8;46 4;08;88 48;79 0;07 7; 90;07 065;7 94;9 594;9 098;7 F 6 3;9;88 49;04;35 5;45;78 9;36;80 46;4;3 36;4;6 45;6;3 4;89 55;09 0;6 9;8 6;48 45;65 7;47 083;5 0857;9 675;6 57;9 955;6 387;6 05;0 F 7 0;63;84 3;3;7 5;;34 5;53;0 48;60;34 5;8;85 8;86;33 43;60;54 49;0;7 93;06;7 56;89 50;7 6;075 4;07 68;3 39;08 8;55 98;03 76;9 ;7 754;0 045;8 878;6 87;0 67;7 0789;8 085;9 45;6 ;0 0;7 F 8 97;0;7 0;95;76 46;04;6 93;73;77 78;6;6 49;84; 6;43;77 47;84; 4;0;9 46;005 99;08 87;9 ;65 79;08 87;074 3;7 86;056 73;7 370;7 56;0 833;9 0487;6 0347;5 0360;0 063;0 970;8 56;9 The ccuted weght coecents, 4 or ech crteron re shown n Tbe 3 Tbe 3 Prorty vectors ccuted by usng the deveoped sotwre modues Method 3 4 Sty Prdos nd Mnn Vochns nd Inson Snge comprsons The obtned resuts show tht the coecents vues ccuted by the methods used re derng, though nsgncnty, whe the cortege (prorty orderng) s preserved or the objectve unctons These derences re expned by the derent sovng pproches used n ech method Ths gves n ddton possbty or thorough nd precse study o the possbe soutons o mutcrter optmzton probems, becuse sm chnges n the weght coecents vues ed to choosng o derent Preto optm structur vrnts o the TS [8] As conrmton o ths sttement, probem (7) nd (8) s soved wth the obtned weght coecents The resuts re shown on Fg 3, where w, w, w3 nd w4 re retve devtons rom the optmum o ech crteron when sovng probem (7) nd (8) For comprson the st souton (Compromse) s obtned or equ mportnce (wthout

10 744 Ivo Mov nd Vezr Zhrnov / Proced Engneerng 69 ( 04 ) prorty) o compred crter Determnton o the weght coecents s o sgncnce or ndng n optm structur vrnt o TS 6 Concuson Fg 3 Soutons o the probem wth derent vues or the weght coecents Fctors nuencng the choce o method or determnton o crter prorty, when sovng mutcrter optmzton probems, hve been systemtzed Tng nto ccount these ctors, group o methods hs been chosen whch group s o methods or bnry comprsons They re rezed under the orm o sotwre modues, mpemented n the dog system or mutcrter optmzton PoyOptmzer The deveoped sotwre modues provde the possbty or deep nd precse exmnton o the possbe soutons o mutcrter optmzton probems, by vryng the weght coecents vues In tht wy, the chnces or ndng souton, whch stses n the best possbe wy the requrements o the decson mer, re ncresed The resuts rom ths deveopment w d the desgners o compex technc systems n ther wor, when they re choosng n optm vrnt by decresng the tme needed or decson mng The uture deveopment o the dog system w be n terms o mpementng ddton modues or determnton o weght coecents through ppcton o rnng methods, budng o the Preto ront wthout the need o ccutng possbe combntons (brute orce pproch) nd sovng o probems or choosng o optm structur vrnt under the condtons o ncompete normton Reerences [] Boydjev, I, I Mov Choce o n optm structure opton o technc system n condtons o ncompete normton Anns o DAAAM or 00 & Proceedngs o the 3th Internton DAAAM Symposum, ISBN , Venn, Austr, 36000, Pubshed by DAAAM Internton, Venn, Austr 00, pp [] Mov, I Methodoogy or choce o optm ssemby structur vrnt (n Bugrn) Reserch wor quyng to receve the tte o Proessor, So, 009 [3] Mov, I & Zhrnov, V Interctve Sotwre System or Mutcrter Choosng o the Structur Vrnt o Compex Technc Systems, Anns o DAAAM or 0 & Proceedngs o the 3rd Internton DAAAM Symposum, ISBN , ISSN , pp , Ktnc, B (Ed), Pubshed by DAAAM Internton, Venn, Austr 0 [4] Mhevch, V & Voovch V (98) Computton methods or study nd desgn o compex systems (n Russn) Nu, Moscow [5] Rbce, M, Dneson M & Eenberg L Stte-o-the-Art Prescrptve Crter Weght Ectton Advnces n Decson Scences, vo 0, Artce ID 76584, 4 pges, 0, do:055/0/76584 [6] Rnz, P, H Schmtz Nutzwert-Kosten Anyse VDI Verg GmbH, Duessedor, 977 [7] Sty, TL Egenvector, nd ogrthmc est squres Eur J Oper Res 990 V 48,, pp 5660 [8] Trntphyou, E Mut-crter decson mng methods A comprtve study Kuwer Acdemc Pubshers, Dodrecht, 000 [9] Vochns, AM, E J Inson Technoogy o products or mesurng equpment (n Russn) Lenngrd, Mshnostroene, 988 [0] Zcheno, UP Opertons study (n Russn) Kev, Vush sho, 988