A R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S Volume 5 7 Issue 3 E. ZIÓŁKOWSKI ALGORITHMS OF FURNACE CHARGE BURDEN OPTIMISATION IN FOUNDRIES ALGORYTMY OPTYMALIZACJI NAMIAROWANIA W SYSTEMACH ZAŁADUNKU PIECÓW ODLEWNICZYCH This rtile esries the methos helpful in hoosing n optimum hrge uren founry furnes with lssifition of those tehniques whih n e useful in respetive lultions. A si prt of ll methos re mthemtil moels esriing vrious situtions whih n hppen in the tehnologil proess when hrge uren is lulte. The primry prmeter in ll moels is the hemil omposition of hrge onstituents, whih n e efine in either eterministi or fuzzy m. The seonry prmeter opte in lultions re the urrently use unit pries of hrge mterils. The nlysis n synthesis of the methos of hrge lultion is omplete with n emple where the optimistion tsk of hrge uren lultion hs een solve using s tool the uthor s own omputer progrm. Keywors: uren lultions, fuzzy optimistion, founry furnes W rtykule przestwiono metoykę ooru optymlnego nmiru wsu l pieów olewnizyh wrz z klsyfikją meto mogąyh mieć zstosownie w olizenih. Integrlną zęśią meto są moele mtemtyzne ujmująe możliwe sytuje w proesie tehnologiznym nmirowni wsu. Prmetrem postwowym w poszzególnyh moelh jest skł hemizny skłników, który może yć zefiniowny w posti eterministyznej lu rozmytej. Prmetrmi rugorzęnymi, jkie przyjęto o olizeń, są eny jenostkowe zkupu mteriłów wsowyh. Dopełnieniem nlizy i syntezy metoyki wyznzni nmiru jest przykł olizeniowy, w którym znie optymlizji rozwiązno, stosują utorski progrm komputerowy.. Introution In n ttempt of optimising the sting proution proess it is neessry to use the most moern methos of plnning n ontrol t nerly ll stges of the tehnologil proess. One of the stges eing most importnt in the mnufture of stings is frition of liqui metl hrterise y stritly etermine prmeters. Most frequently these prmeters re itte y the type of the pplie tehnology of mnufture, inluing the quntity weight of hrge, hemil omposition s well s, n tpping n pouring temperture. A very importnt ftor in optimistion of the eonomi proution of stings is lso the ost of frition of mss volume of molten metl whih, uner given onitions, shoul e s low s possile. To stisfy the ove requirements, vrious ttempts hve een me to etermine optimum hrge uren, juste to the type of founry furne urrently use n to the opte poliy of mngement of the hrge mterils store in founry. The rtile esries the lssifition of the methos of hrge uren lultion, with selete mthemtil moels of tsks optimum hrge uren omputtion. The priniples of moelling the unertin fuzzy vlues, whih re use in the esription of inurte hemil omposition n/or unit prie of iniviul hrge mterils hve een esrie.. Clssifition of methos use to lulte the hrge uren It follows from vrious soures of inmtion known to the uthor tht the referene literture Polish n interntionl lks stuy whih woul isuss in omprehensive wy the theory of hrge uren lultion. Some funmentls re only ville, n they n serve s kgroun ginst whih theory like FOUNDRY MECHANISATION, AUTOMATION AND DESIGNING, FACULTY OF FOUNDRY ENGINEERING, AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, 3-59 KRAKÓW, 3 REYMONTA STR., POLAND
488 the one esrie in this rtile n e onstrute. The min element in this theory will e the metho of hrge uren lultion. The uthor proposes lssifition of the methos use so fr hrge uren lultion oring to the following riteri: I. Melting proess. Clultion of hrge uren to frite molten metl: Clultion of hrge uren n empty furne Clultion of hrge uren furne prtilly fille with: soli hrge melte hrge.. Clultion of hrge uren to orret the hemil omposition of molten metl in furne or in founry lle: simplifie methos, optimising methos. II. Type of metho use in hrge uren lultion. Digrmmti methos: grphil methos, geometri methos.. Anlytil methos 3. Numeril methos omputer-ie methos: liner lgery, optimistion methos of liner progrmming methos of squre progrmming III. Type of hrge n vlues of st lloy prmeters hemil omposition, unit prie:. Deterministi esription. Fuzzy esription: two-level, multi-level with finite numer of levels, ontinuous. IV. Type of optimistion:. Monoriteril tsks of hrge uren lultion. Multiriteril tsks of hrge uren lultion The presente lssifition llows the onventionl methos of hrge uren lultion, tking lso into onsiertion the most moern tehniques of fuzzy optimistion evelope y the uthor [-4]. 3. Stges of hrge uren lultion Stge I Determining, from tehnologil guielines or y lultions, the weight of the hrge unit. In most ses, the weight of the hrge epens on founry s proution emn n is restrite y the rte pity of melting instlltions. Only in the se of upols, the empiril mule re use, n they enle the weight of single hrge uren to e effetively etermine [, ]. Stge II Determining, y ssumption or lultion, the hemil omposition of hrge whih, given proes of melting, will ensure the orret hemil omposition of molten metl. Stge III Determining, y the selete metho, the hrge uren. This opertion inlues lultion of perent or mss frtion of hrge omponents inlue in the totl lulte hrge. The eisions of tehnologil nture tken t Stge II hve n importnt effet on finl output of the melting proess. This minly refers to the hemil omposition of molten metl whih my lso epen on ftors other thn the sole hemil omposition. The ftors tht re responsile the hemil omposition of molten metl inlue the type of hrge melting proess n the type of hrge melting instlltion. It is generlly ssume tht in eletri furnes inution, r n resistne, euse of melting losses, the ontent of hemil elements in molten metl is lower thn it is in the hrge. In upols this sitution is muh more omplite, minly euse the ontent of ron n sulphur tens to inrese, sine the metlli hrge is melte in the presene of soli fuel oke, ontining these oth elements. The hoie of metho to rry out Stge III epens on how omple the tsk of hrge uren lultion is, i.e. on the numer of hemil elements n hrge omponents inlue in lultions. The simplest tsks inluing one, two or three hemil elements n e solve y mens of selete igrmmti metho grphil or geometri or, if possile, n e reue to system of liner equtions n solve y one of the nlytil or numeril methos of the liner lger emple, Guss or Guss-Jorn elimintion metho. Further prt of this pulition esries in more omprehensive wy the mthemtil moels use in etermintion of optimum hrge uren. Clultion of hrge uren to eh type of the founry furne onsists of the three si stges:
489 4. A generlise eterministi moel of hrge uren lultion One of the first mthemtil moels of the tsk of hrge uren lultion ws esrie in []. The stuy esries the tsk of minimising, where the ojetive funtion is ssuming the following m: f = j j vetor of the serhe eision vriles; in this se it is the ontent of eh hrge omponent in % if m w = % or in kg if m w is epresse in kg, N numer of hrge mterils onsiere in lultions, j unit prie of j-th hrge mteril in prie unit per mss unit. The ojetive funtion serhes the hrge uren of the lowest totl ost. The tehnologil guielines this moel were efine s system of onstrints: i j j = i m w j = m w j j i=,,..., M;,,..., N i j ontent of i-th hemil element in j-th hrge omponent, %, i ontent of i-th hemil element in hrge, %, M numer of hemil elements onsiere in lultions, m w hrge weight in kg or equl to %, j upper limit of the ontent of j-th omponent in hrge in kg, or in %. In the quote referene literture [], five si hemil elements were tken into onsiertion, i.e. ron, silion, mngnese, phosphorus n sulphur C, Si, Mn, P n S. Aitionlly, the effet of onstrints on the ontent of iniviul mterils in the lulte hrge hs een ssume. The tsk of optimistion of the ojetive funtion with ontrints is the tsk of liner optimistion, sine oth funtion s well s the system of onstrints hve the m of liner funtions. In solving thus multe tsk one n use, e.g., the metho of simplees [8, 9]. The optimistion tsk my rete some prolems of stritly numeril nture. The system of onstrints oes not llow the lower limits in ontent of eh of the onsiere hrge mterils, so willingly use y proess engineers prepring the metl melting proess. If stritly etermine hemil omposition of hrge represente y vlue i is use, it my use errors when the vlues lulte with very high ury y omputer re roune, mking etermintion of the vlue of vetor impossile, whih the progrm reognises s lk of solution the optimistion tsk uner onsiertion. To eliminte the ove mentione inonvenienes, it woul e muh etter n esier to opt system of onstrints in the m [6]: i m w i j j i m w i=,,..., M j j j m w,,,..., N j = m w 3 i, i the lower n upper limit, respetively, in the ontent of i-th hemil element in hrge, %, j, j the lower n upper limit, respetively, in the ontent of j-th hrge mteril, epresse in the sme units s j. Then, the tsk of omputtion of the hrge uren onsists in serhing set of vlues of vetor suh tht will enle minimising the vlue of funtion uner onstrints 3. Yet, this tsk ontinues eing the tsk of liner optimistion, whih mens tht it is possile to use one of the methos of liner progrmming. 5. A generlise fuzzy moel of hrge uren lultion Applying the eterministi stritly etermine vlues of prmeters, suh s the hemil omposition or unit prie of hrge mterils is usully ue to: verging the vlues of the onsiere prmeters, ue to fuzzy lortory mesurements of the hemil omposition one on lrge volume of hrge mterils n simplifie t omprise in qulity ertifites, isregring some fetures hemil omposition, unit prie ommon to ifferent thes of the sme omponent, ollete in the sme in, onstrints impose y mthemtil moels use so fr to esrie the optimistion tsks, n iffiult to ess omputer progrms use solving of these tsks.
49 TABLE Compiltion of si polygonl memership funtions ompose of retiliner segments Nme of polygonl memership funtion Plotte grph Definition of memership funtion Asymmetri trpezoi or Left outsie Right outsie Retngulr or Symmetri tringle - + or Asymmetri tringle - or Symmetri trpezoi +- or 5.. Mthemtil esription of unertin fuzzy vlues In sitution when the preisely etermine vlues of prmeters use oversimplifitions onsierly restriting their prtil use, one n reur to esription of the investigte omin of relity using the theory of fuzziness. The unertinty of the vlues of the numeril vriles n e esrie y mens of fuzzy sets []. Aoring to [], y the nme of fuzzy set A, in numeril spe of onsiertions X, we enote set of pirs: A= { µ A, }, X 4
49 µ A memership funtion of fuzzy set A, whih to every element X sries the egree of its memershipµ A in fuzzy set A, whereµ A [; ]. The memership funtion sries to eh element of given vrile vlue from the omin [;]: µ A : X [; ], X 5 The vlue, lle the egree of memership, inms us to wht egree element elongs to fuzzy set A. In [] omprehensive review of fuzzy sets n the relte knowlege hve een presente. For the esription of unertin inurte, i.e. fuzzy, vlues of prmeters hrteristi of the hemil omposition n possily lso of the unit prie of hrge mterils, one n use the memership funtions of ifferent lsses. Tle elow gives ompiltion of the polygonl memership funtions uilt from retiliner segments. It hs een eie to esrie the fuzziness of the hemil omposition of hrge mterils y polygonl funtion in the m of uneven-rme trpezoi Fig.. The hoie hs een justifie in the following wy: the memership funtion in the m of symmetri trpezoi opertes t level, while the remining two funtions operte t level. In founry prtie, esriing the fuzziness of hemil omposition t more thn two levels of memership funtion signifintly inreses the imension of the optimistion tsk, whih fins no justifition in prtil pplition. It n e ssume tht level is pessimisti level, i.e. suh whih the rnge of the ontent of hemil elements will e the lest fvourle. On the other hn, level n e lle optimisti, i.e. the one whih the rnge of the ontent of hemil elements will e the nrrowest. Optimisti level Pessimisti level Fig.. Grphi interprettion of the ontent rnges of hemil elements t the levels:µ= pessimisti nµ= optimisti memership funtion in the m of symmetri trpezoi in terms of fuzzy esription of the hemil omposition of hrge omponents, the funtion of uneven-rme trpezoi reples ll other polygonl funtions of memership, the funtion is hrterise y four numers, whih mke the tehnologil ientifition very esy, n is very hny in olleting n proessing of inmtion in the tse of hrge mterils. The first numer enotes the lower ontent of hemil element t pessimisti level. The seon numer n the thir numer enote, respetively, the lower n upper ontents of this element t n optimisti level, while the lst, fourth numer enotes the upper ontent of hemil element t pessimisti level. The propose moel of esription of the fuzziness is lso pplile when esriing the unertin unit pries of hrge mterils llowe the lultions. 5.. Moel of hrge uren lultion fuzzy hemil omposition of hrge mterils n etermine unit pries The tsk of founry furne hrge uren lultion llowing fuzzy hemil omposition of hrge mterils will onsist in etermintion of suh frtions of j, whih the vlue of the ojetive funtion will e minimise to the m: min j j 6 uner the following onstrints: i jk j ik m w ā i jk j ik m w j j j m w j = m w i=,,..., M;,,..., N k=,,..., q 7 q numer of the memership funtion levels tken into onsiertion. For retngulr moel of the esription of fuzziness q=, the system of onstrints will ssume the m of:
49 i j j i m w ā i j j i m w j j j m w j = m w while trpezoil moel, the onstrints in optimistion tsk n e efine in the following wy: i j j i m w i j j i m w ā i j j i m w ā i j j i m w j j j m w j = m w 8 9 5.3. Moel of hrge uren lultion fuzzy hemil omposition of hrge mterils n fuzzy unit pries The hrge uren with stritly etermine unit prie of hrge mterils n e lulte when these mterils hve lrey een purhse y the founry, or when their purhse is plnne n it is known sure tht oth the terms of purhse s well s the prmeters shll not hnge [5]. In the sitution of long-term plnning of hrge mterils purhse, or when suen hnges of pries over short perios of time threten, very onvenient option my e the possiility of etermining hrge uren unertin pries of the hrge omponents. The fuzziness of the prie of hrge mterils n e simulte with the help of selete memership funtions in wy similr s it hppens in the se of fuzzy hemil omposition. If the prie of hrge omponent is epliitly etermine, then the ojetive funtion in optimistion tsk nnot e efine in orne with reltionship. In this se, eh level of the memership funtion esriing the prie fuzziness, seprte ojetive funtion shoul e pplie. If so, then there re mny ojetive funtions whih reue the optimistion tsk to multi-riteril liner progrmming uner the ssumption tht the system of onstrints is in m or in one of its vrints, e.g. 3 or 5. One of the possiilities to etermine the vlue of vetor, whih ll the liner ojetive funtions re to e minimise, is y efining the, so lle, ompromise ojetive funtion. In its generl m, this ojetive funtion n e esrie s: min q f k jk j k= + f k jk j, q mimum quntity of memership funtion egrees eh of the fuzzy numers llowe in the moels of fuzziness, f the vlue of ojetive funtion in n optimistion tsk in whih the ojetive funtion t k-th level k hs the m of min [ f k = N ] jk j uner onstrints f k the vlue of ojetive funtion in n optimistion tsk in whih[ the ojetive] funtion t k-th level hs the m of min f k = N jk j uner onstrints Thus multe ojetive funtion shoul e regre s n eft to etermine the vlue of vetor, i.e. hrge uren the totl ost of whih will pproh oth minimum n mimum vlues of unit pries of the iniviul hrge mterils. To efine the ojetive funtion, it is neessry to minimise, t eh k-th level, the totl ost of hrge, ssuming the minimum, first, n mimum, net, vlues of the ost of eh hrge omponent. This enles etermintion of the vlue of oeffiients f n f k k n sustituting them in reltionship. For retngulr moel of fuzziness, the ompromise ojetive funtion shll ssume the m of: where min f = min f j j j j n + f j j, f = min j j,
493 while the trpezoil moel of fuzziness it shll ssume the m of: f j j + min + f j j + f = min f = min f f j j +, 3 j j j j, f = min j j, 4 j j, f = min j j. 5 The ompromise ojetive funtions, n 3 re squre funtions, n theree, these funtions, solving n optimistion tsk of hrge uren lultion requires llowing system of onstrints, 3 or 5, respetively, n using metho of squre progrmming. Prtil pplition of the tsk of hrge uren lultion hrge mterils hrterise y fuzzy hemil omposition n unertin unit pries shoul e in the uthor s opinion limite to the se in whih the fuzzy prie is simulte y retngulr memership funtion. Assuming one intervl of hnges in the unit prie of eh hrge mteril seems to e suffiient in mngement of the hin of founry supplies. The use of more omple funtions to esrie unit pries onsierly epns the ompromise funtions n mkes progrmming of proeures reting these funtions utomtilly iffiult though not impossile. An emple of lultions given elow shows the possiility of using in optimistion tsk retngulr moel of fuzziness of the unit pries. 5.4. Emple of hrge uren lultion fuzzy hemil omposition trpezoil moel n fuzzy prie retngulr moel Tle gives fuzzy prmeters hemil omposition, unit prie of hrge mterils tken into onsiertion in this optimistion tsk. In the emple of lultion no itionl onstrints hve een ssume s regrs the ontent of eh hrge omponent. Beuse of fuzzy unit pries quote in this emple in the m of n intervl of their hnges in retngulr version of the memership funtion, it is neessry to solve, first, the tsk of minimising the ojetive funtion to m: f =.6 +.4 +. 3 + 5. 4 + 4. 5 + 4.7 6 6 n net the tsk of minimising the ojetive funtion to m: f =.7 + 3.4 +.9 3 + 5.6 4 + 4.6 5 + 5. 6 7 uner the following onstrints: TABLE Chemil omposition n unit prie of hrge mterils n ssume hemil omposition of hrge lultion of emple Chrge omponent Chemil omposition, % Prie C Si Mn PLN/kg.;.4;.6;.9.6;.9;.34;.4.5;.7;.;.5.6.7 3.4;3.5;3.6;3.75.6;.8;.9;..64;.75;.86;.98.4 3.4 3 3.;3.5;3.3;3.38.73;.84;.97;.5.35;.38;.39;.45..9 4.5;.;.;.5 64.;64.3;64.7;65..5;.3;.34;.35 5. 5.6 5 89.;9.;9.;9.5.;.;.;..;.;.;. 4. 4.6 6.4;.44;.5;.57.3;.4;.5;.6 8.;8.6;8.;8.5 4.7 5. CHARGE 3. 3..4.6.6.8
494. + 3.4 + 3. 3 +.5 4 + 89. 5 +.4 6 3..4 + 3.5 + 3.5 3 +. 4 + 9. 5 +.44 6 3..6 + 3.6 + 3.3 3 +. 4 + 9. 5 +.5 6 3..9 + 3.75 + 3.38 3 +.5 4 + 9.5 5 +.57 6 3..6 +.6 +.73 3 + 64. 4 + 5 +.3 6.4.9 +.8 +.84 3 + 64.3 4 + 5 +.4 6.4.34 +.9 +.97 3 + 64.7 4 + 5 +.5 6.6.4 +. +.5 3 + 65. 4 + 5 +.6 6.6.5 +.64 +.35 3 +.5 4 + 5 + 8. 6.6.7 +.75 +.38 3 +.3 4 + 5 + 8.6 6.6. +.86 +.39 3 +.34 4 + 5 + 8. 6.8.5 +.98 +.45 3 +.35 4 + 5 + 8.5 6.8 + + 3 + 4 = 8 A solution of the tsk of minimising the ojetive funtion 6 uner onstrints 8 is: = 38.73% =.% 3 = 59.8% 4 =.5% 5 =.4% 6 =.35% f = 4.6 9 The tsk of minimising funtion 7 with the system of onstrints 8 hs the following solution: = 94.49% =.% 3 =.% 4 =.88% 5 = 3.3% 6 =.5% f = 8.6 f= [4.6.6 +.4 +. 3 + +5. 4 + 4. 5 + 4.7 6 ] + + [8.6.7 + 3.4 +.9 3 + +5.6 4 + 4.6 5 + 5. 6 ]. The tsk of hrge uren lultion fuzzy prmeters of the hrge mterils hemil omposition n unit prie onsists in etermintion of the vlue of vetor, whih minimises the ompromise ojetive funtion uner onstrints 8. Hving introue the ove t to omputer progrm omputing the lgorithm of squre progrmming n hving perme the respetive omputtion proeure, the results shown in Figure re otine. For plotting ompromise ojetive funtion resulting from the reltionship, only the vlues of f from 9 n f from re neee. The frtions of iniviul hrge omponents given in reltionships 9 n re signifintly ifferent. This is importnt insmuh s in the se of solutions not iffering muh from eh other only unit prie hnges, plotting of ompromise funtion mkes no sense. The efinition of ompromise ojetive funtion is s follows: Fig.. Output of hrge uren lultions The lulte hrge uren is optiml the whole rnge of hnges in unit pries of the iniviul hrge mterils. Choosing this uren oring to Figure will e muh sfer thn it woul e if only verge unit pries of the iniviul hrge omponents were tken into onsiertion.
495 6. Summry The methos of hrge uren lultion esrie here, onsierly eten our knowlege founry prolems in prtiulr, ut lso siene in generl out optimising of proution proesses. Through moel pproh to the tsks oresponing to vrious situtions ourring in tehnologil proess, the opertion of hrge uren lultion n e ie y omputers n inlue in the system of utomti ontrol of evies whih prepre hrge uren founry furnes. In integrte system of proution plnning n preprtion, these methos my prove to e useful in serhes est strtegy to mnge the hin of supplies of new hrge mterils. REFERENCES [] C. P o r z u k i, C. K l t, Metlurgi i olewnitwo żeliw. Wy., Wywnitwo Śląsk, Ktowie 976. [] C. P o r z u k i, E. Z i ó ł k o w s k i, Computer- Aie Optimiztion of Cupol Buren Determintion. Zesz. Nuk. AGH Metlurgi i Olewnitwo. Krków, 7,, 93 99. [3] W.M. S k w, T. W h e l k o, Oliznie nmirów żeliwikowyh. Skrypty l szkół wyższyh. Szkoł Inżyniersk w Częstohowie. PWN, Łóź-Krków 955. [4] E. Z i ó ł k o w s k i, Zstosownie meto progrmowni mtemtyznego w optymlizji wytopu w pieh olewnizyh. Pr ziorow po rekją Jnusz Kprzyk i Jn Węglrz pt.: Bni operyjne i systemowe woe wyzwń XXI wieku. Moelownie i Optymlizj. Metoy i zstosowni. Akemik Ofiyn Wywniz EXIT, s. II- II-3, Wrszw. [5] E. Z i ó ł k o w s k i, Optymlizj nmiru wsu o pieów olewnizyh z zstosowniem mteriłów wsowyh o niepewnyh enh. Arhiwum Tehnologii Mszyn i Automtyzji. Komisj Buowy Mszyn PAN O/Poznń. 4, 3 spejlny, 3 4. [6] E. Z i ó ł k o w s k i, Zstosownie lgorytmów optymlizji rozmytej o określni nmiru wsu l pieów olewnizyh. Pr ziorow po rekją Romn Kulikowskiego, Jnusz Kprzyk i Romn Słowińskiego p.t.: Bni operyjne i systemowe 4. Poejmownie eyzji. Postwy metoyzne i zstosowni. Wy. EXIT, s. 9 3, Wrszw 4. [7] E. Z i ó ł k o w s k i, Postwy teoretyzne lgorytmu ilnsowni prmetrów wsu zestwinego z mteriłów wsowyh o niepreyzyjnym skłzie hemiznym. Arhiwum Olewnitw. PAN O/Ktowie, 6, 9, 443 448 6. [8] S.I. G s s, Progrmownie liniowe. Metoy i zstosowni. Wy. 4, PWN, Wrszw 98. [9] I. N y k o w s k i, Progrmownie liniowe. Wy., PWE, Wrszw 984. [] A. P i e g t, Moelownie i sterownie rozmyte. Akemik Ofiyn Wywniz EXIT, Wrszw 999. Reeive: 8 June 7.