Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy
One of the ways to solve the conditionally ill-posed problem for ranking quality criteria of the multiobjective system, whose regularization is made by applying the fractal formalism, is hereby considered. Ranking quality criteria by their importance is based on the definition of the area of self-si...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2018
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| Cite this: | Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy / V.I. Bolshakov, V.M. Volchuk, Yu.I. Dubrov // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 9. — С. 1165-1171. — Бібліогр.: 13 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1518572025-02-09T23:34:43Z Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy Регуляризация одной условно некорректно поставленной задачи металлургического извлечения (металлов из руд) Реґуляризація однієї умовно некоректно поставленої задачі металургійного вилучення (металів з руд) Bolshakov, V.I. Volchuk, V.M. Dubrov, Yu.I. Физико-технические основы эксперимента и диагностики One of the ways to solve the conditionally ill-posed problem for ranking quality criteria of the multiobjective system, whose regularization is made by applying the fractal formalism, is hereby considered. Ranking quality criteria by their importance is based on the definition of the area of self-similarity for partial criteria. Рассматривается один из путей решения условно некорректной задачи ранжирования критериев качества многокритериальной системы, регуляризация которой производится применением фрактального формализма. Ранжирование критериев качества по их значимости базируется на определении области самоподобия частных критериев. Розглядається один із шляхів вирішення умовно некоректної задачі ранжування критеріїв якости багатокритеріяльної системи, реґуляризація якої проводиться застосуванням фрактального формалізму. Ранжування критеріїв якости за значенням базується на визначенні области самоподібности окремих критеріїв. 2018 Article Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy / V.I. Bolshakov, V.M. Volchuk, Yu.I. Dubrov // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 9. — С. 1165-1171. — Бібліогр.: 13 назв. — англ. 1024-1809 PACS: 06.60.Mr, 61.43.Hv, 61.72.Ff, 81.05.Bx, 81.40.Ef, 81.40.Np, 81.70.Bt DOI:10.15407/mfint.40.09.1165 https://nasplib.isofts.kiev.ua/handle/123456789/151857 en Металлофизика и новейшие технологии application/pdf Інститут металофізики ім. Г.В. Курдюмова НАН України |
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Физико-технические основы эксперимента и диагностики Физико-технические основы эксперимента и диагностики |
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Физико-технические основы эксперимента и диагностики Физико-технические основы эксперимента и диагностики Bolshakov, V.I. Volchuk, V.M. Dubrov, Yu.I. Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy Металлофизика и новейшие технологии |
| description |
One of the ways to solve the conditionally ill-posed problem for ranking quality criteria of the multiobjective system, whose regularization is made by applying the fractal formalism, is hereby considered. Ranking quality criteria by their importance is based on the definition of the area of self-similarity for partial criteria. |
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Article |
| author |
Bolshakov, V.I. Volchuk, V.M. Dubrov, Yu.I. |
| author_facet |
Bolshakov, V.I. Volchuk, V.M. Dubrov, Yu.I. |
| author_sort |
Bolshakov, V.I. |
| title |
Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy |
| title_short |
Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy |
| title_full |
Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy |
| title_fullStr |
Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy |
| title_full_unstemmed |
Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy |
| title_sort |
regularization of one conditionally ill-posed problem of extractive metallurgy |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| publishDate |
2018 |
| topic_facet |
Физико-технические основы эксперимента и диагностики |
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https://nasplib.isofts.kiev.ua/handle/123456789/151857 |
| citation_txt |
Regularization of One Conditionally Ill-Posed Problem of Extractive Metallurgy / V.I. Bolshakov, V.M. Volchuk, Yu.I. Dubrov // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 9. — С. 1165-1171. — Бібліогр.: 13 назв. — англ. |
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Металлофизика и новейшие технологии |
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PACS numbers: 06.60.Mr, 61.43.Hv, 61.72.Ff, 81.05.Bx, 81.40.Ef, 81.40.Np, 81.70.Bt
Regularization of One Conditionally Ill-Posed Problem
of Extractive Metallurgy
V. I. Bolshakov, V. M. Volchuk, and Yu. I. Dubrov
Prydniprovs’ka State Academy of Civil Engineering and Architecture,
24a
Chernyshevs’ky Str.,
49600 Dnipro, Ukraine
One of the ways to solve the conditionally ill-posed problem for ranking quality
criteria of the multiobjective system, whose regularization is made by applying
the fractal formalism, is hereby considered. Ranking quality criteria by their
importance is based on the definition of the area of self-similarity for partial
criteria. At the first stage of research, the key parameter is identified under
the defined initial conditions by ranking quality criteria with the use of the
fractal formalism. Ranking the criteria by importance is based on the estima-
tion of the comparative value of areas of their self-similarity. The self-
similarity coefficients of these areas appear as the regularization parameter at
this stage of research. These areas are estimated by means of definitions of
their relative values as the ratio of the range for existence of each partial crite-
rion referred to its upper boundary. The second stage of research is initiated by
the need to determine the optimal combinations of the target product proper-
ties in the working area, depending on the customer conditions. For this pur-
pose, the sub-compromise regions are defined in the working area of partial
criteria under consideration (compromise criteria). The sub-compromise par-
tial regions are regions of the optimal combinations of properties. Thus, the
regularization at the second stage is made in co-ordination with the require-
ments of the customer. In this connection, the example of solving the condi-
tionally ill-posed problem for optimizing the multicriteria and multiparamet-
ric technologies of fabrication of cast iron rolls is given.
Key words: multiparametric technologies, regularization parameter, self-
similarity area, compromise criteria, mechanical properties, fractal formalism.
Corresponding author: Volodymyr Mykolayovych Volchuk
E-mail: volchuky@gmail.com
Citation: V. I. Bolshakov, V. M. Volchuk, and Yu. I. Dubrov, Regularization of One
Conditionally Ill-Posed Problem of Extractive Metallurgy, Metallofiz. Noveishie
Tekhnol., 40, No. 9: 1165–1171 (2018), DOI: 10.15407/mfint.40.09.1165.
Ìåòàëëîôèç. íîâåéøèå òåõíîë. / Metallofiz. Noveishie Tekhnol.
2018, т. 40, № 9, сс. 1165–1171 / DOI: 10.15407/mfint.40.09.1165
Îттèсêè äîстóïíû íåïîсðåäстâåííî îт èçäàтåëÿ
Ôîтîêîïèðîâàíèå ðàçðåøåíî тîëüêî
â сîîтâåтстâèè с ëèöåíçèåé
2018 ÈÌÔ (Èíстèтóт ìåтàëëîôèçèêè
èì. Ã. Â. Êóðäþìîâà ÍÀÍ Óêðàèíû)
Íàïå÷àтàíî â Óêðàèíå.
1165
https://doi.org/10.15407/mfint.40.09.1165
https://doi.org/10.15407/mfint.40.09.1165
1166 V. I. BOLSHAKOV, V. M. VOLCHUK, and Yu. I. DUBROV
Рîçгëÿäàєтüсÿ îäèí іç øëÿхіâ âèðіøåííÿ óìîâíî íåêîðåêтíîї çàäà÷і ðàí-
жóâàííÿ êðèтåðіїâ ÿêîстè бàгàтîêðèтåðіÿëüíîї сèстåìè, ðåґóëÿðèçàöіÿ
ÿêîї ïðîâîäèтüсÿ çàстîсóâàííÿì ôðàêтàëüíîгî ôîðìàëіçìó. Рàíжóâàííÿ
êðèтåðіїâ ÿêîстè çà çíà÷åííÿì бàçóєтüсÿ íà âèçíà÷åííі îбëàстè сàìîïîäі-
бíîстè îêðåìèх êðèтåðіїâ.
Ключові слова: бàгàтîïàðàìåтðè÷íà тåхíîëîгіÿ, ïàðàìåтåð ðåґóëÿðèçà-
öії, îбëàстü сàìîïîäібíîстè, îбëàстü êîìïðîìісó êðèтåðіїâ, ìåхàíі÷íі
âëàстèâîсті, ôðàêтàëüíèé ôîðìàëіçì.
Рàссìàтðèâàåтсÿ îäèí èç ïóтåé ðåøåíèÿ óсëîâíî íåêîððåêтíîé çàäà÷è
ðàíжèðîâàíèÿ êðèтåðèåâ êà÷åстâà ìíîгîêðèтåðèàëüíîé сèстåìû, ðåгó-
ëÿðèçàöèÿ êîтîðîé ïðîèçâîäèтсÿ ïðèìåíåíèåì ôðàêтàëüíîгî ôîðìàëèç-
ìà. Рàíжèðîâàíèå êðèтåðèåâ êà÷åстâà ïî èх çíà÷èìîстè бàçèðóåтсÿ íà
îïðåäåëåíèè îбëàстè сàìîïîäîбèÿ ÷àстíûх êðèтåðèåâ.
Ключевые слова: ìíîгîïàðàìåтðè÷åсêàÿ тåхíîëîгèÿ, ïàðàìåтð ðåгóëÿ-
ðèçàöèè, îбëàстü сàìîïîäîбèÿ, îбëàстü êîìïðîìèссà êðèтåðèåâ, ìåхàíè-
÷åсêèå сâîéстâà, ôðàêтàëüíûé ôîðìàëèçì.
(Received March 13, 2018)
1. PROBLEM FORMULATION
The ranking of criteria by their importance for the multi-criteria sys-
tem should ensure its stable operation within the specified limits, ac-
cording to statutory documents [1]. The defined limits determine the
working area of parameters. When identifying the operation of such
systems, one often has to deal with complexities dictated by the influ-
ence of a number of factors that guarantee their stability. As a rule,
such problems are often solved by methods of applied mathematics [2–
4]. At the same time, the majority of the applied problems of extractive
metallurgy can be considered as conditionally ill-posed ones [5] that
initiate the introduction of additional stipulations for their more ade-
quate representation. Such stipulations may include the definition of
the ways to optimize the multicriteria and multiparameter technolo-
gies in the working area by finding areas with the best combination of
partial indicators (criteria). From this point of view, the problem of
ranking quality criteria of the multiparameter technologies, examined
earlier [6], taking into account the fractal formalism [7–9], seems to be
incomplete and can be transformed as the conditionally ill-posed one,
whose solution, depending on various initial conditions, can vary. Ac-
cording to J. Hadamard [10], the problem is deemed to be ill-posed, if:
it has no solution (in our concerned class), it has many solutions (from
two or more), and the procedure for finding the solution is unstable (i.e.
with the smallest measurement error or small perturbations of the ini-
tial data [11], the solution obtained can substantially differ from the
REGULARIZATION OF ILL-POSED PROBLEM OF EXTRACTIVE METALLURGY 1167
exact one).
Failure to fulfil, at least, one of the above-listed stipulations for the
problem under consideration confirms its ill-posedness.
In spite of the fact that the first item of stipulations is fulfilled,
which points to the existence of the solution for the problem of ranking
quality criteria, the second and third items are not fulfilled for this
problem. The second item cannot be fulfilled subject to possible chang-
es in the customer’s requirements as for the significance of partial cri-
teria. In this regard, the ranking of criteria can be carried out accord-
ing to the customer’s requirements, by the degree of influence of the
chemical composition elements, and by the detectable value of the area
of self-similarity for each of partial criteria [6] determining the stabil-
ity of the technology operation. The third item of ill-posedness stipula-
tions cannot be fulfilled as well, since the procedure for finding one or
another solution of this conditionally ill-posed problem can vary de-
pending on periodically changing parameters in the working area
(chemical composition, cooling conditions, etc.).
Proceeding from the peculiarities of the above-formulated stipula-
tions, the solution of this conditionally ill-posed problem concerning
the optimization of the multiparameter technologies is proposed to be
carried out in two stages. At the first stage of the research, the key pa-
rameter is identified under the defined initial conditions by ranking
quality criteria with the use of the fractal formalism. The ranking of
criteria by importance is based on the estimation of the comparative
value of areas of their self-similarity. The self-similarity coefficients
of these areas appear as the regularization parameter at this stage of
the research. These areas are estimated by means of definitions of their
relative values as the ratio of the range for existence of each partial
criterion referred to its upper boundary.
The second stage of research is initiated by the need to determine the
optimal combinations of the target product properties in the working
area, depending on the customer conditions. For this purpose, the sub-
compromise regions are defined in the working area of partial criteria
under consideration (compromise criteria). The sub-compromise partial
regions are regions of the optimal combinations of properties [12].
Thus, the regularization at the second stage is made in co-ordination
with the requirements of the customer.
In this connection, the example of solving the conditionally ill-posed
problem for optimizing the multicriteria and multiparameter technol-
ogies of production of cast iron rolls is given below.
2. RESEARCH MATERIAL
Investigated were the section-mill cast iron rolls with the globular
shape of graphite, chromium and nickel alloyed. The rolls were not
1168 V. I. BOLSHAKOV, V. M. VOLCHUK, and Yu. I. DUBROV
subjected to the heat treatment and had the smooth surface of barrels.
The structure of the rolls in the working area (up to ≅ 50 mm) had the
following shape (Fig. 1).
Dimensions of the globular graphite inclusions (their diameter) on
the grinding surface varied from 35 µm to 59 µm, which corresponds to
point 6 according to the standard ISO 945-75. The globular graphite in
the working layer of the barrel at the depth of ≅ 5–15 mm is much less
common to find in comparison with carbides, including ledeburite,
which is explained by its content up to ≅ 3% (the standard is ШÃ4 ac-
cording to ÃÎСТ 3443). The globular shape of graphite is caused by the
addition of a small amount of magnesium—up to 0.054% by weight.
Pearlite has the different dispersion (average distance between two ad-
jacent cementite plates): from 0.79 µm to 2.2 µm. The area occupied by
cementite averaged 20–30% according to calculations with the use of
the linear method of A. Rosival, which corresponds to the score of Ц25.
Mechanical properties of the roll-foundry iron (ultimate strength
σВ, ultimate bending strength σ, impact strength KС, Shore hardness
HSD) were determined according to ÃÎСТ 27208-87 with the use of
standard techniques. The billets, from which samples were prepared
for full-scale tests, have been selected from the cast samples. Samples
were cut from the working area of smooth metal barrels of rolls in the
tangential direction, as well as manufactured from the samples sepa-
rately cast in castings. The impact strength of cast iron was deter-
mined without the notch on the samples of 10×10×55 mm size. To as-
sess the ultimate bending strength, the samples of 10×10×90 mm were
used, and in tensile tests—the samples of 25 mm diameter, with the
gauge length of 50 mm.
To improve the reliability of the results obtained, the data on the
a b
Fig. 1. Structure of the working layer of cast iron roll barrels: globular graph-
ite (a), colonies of ledeburite, graphite eutectic, pearlite matrix, HNO3 etch-
ing (b).
REGULARIZATION OF ILL-POSED PROBLEM OF EXTRACTIVE METALLURGY 1169
full-scale tests of tangential samples were used (the sampling for 283
metal fusions of the gross weight of ≅ 800 tons) [4].
3. EXPERIMENTAL RESULTS AND DISCUSSION
To implement the first stage of the research, the working area of cast
iron quality criteria (compromise criteria), depending on their chemical
composition, is determined according to the current ТÓÓ 14-2-1188-97
within the following limits: σВ ≈ 240–460 MPa, σ ≈ 330–940 MPa,
KС ≈ 9–33 kJ/m2, HSD ≈ 40–60, taking into account the whole range of
applicable cooling conditions for rolls in the metal shape.
In order to determine the working area of stable indicators for quali-
ty criteria of the multiparameter technologies, their ranking by im-
portance was carried out. The criteria ranking methodology is more par-
ticularly described in the paper [6]. The criteria ranking procedure was
carried out on the basis of the analysis concerning their area of self-
similarity. In this case, the regularization parameter for the stable oper-
ation of the technology is represented by the self-similarity coefficient.
The area of self-similarity of the normalized quality criteria for the
ultimate strength varied from 240 to 460 MPa, i.e., by 220 units (with-
in the limits of (220/460)⋅100% ≈ 48%). We therewith assume that the
self-similarity coefficient
B
Kσ is equal to 0.48. The self-similarity in-
dicators for the remaining partial criteria were similarly calculated
and made up for Kσ = 0.65, KKС = 0.73 and KHSD = 0.33.
The key parameter among quality criteria under consideration is the
impact strength having the highest self-similarity coefficient
KKС = 0.73. Accordingly, indicators of the impact strength for cast iron
with the structure of the globular graphite are more sensitive to
changes in the in-house technology than other properties of σВ, σ and
HSD are. Since it is assumed that, for the regularization of the tech-
nology operation with respect to the key parameter KС, it is necessary
to maintain indicators of the chemical composition and conditions for
cooling of rolls within the limits prescribed by statutory documents. In
case of violating the in-house technology, changes in the areas of self-
similarity of partial criteria are possible, and consequently—changes
in their ranking.
To implement the second stage of the research, the compromise cri-
teria (Fig. 2), consisting in the normalized representation of variables,
whose value is given in percentage [12, 13], were determined. Since
this stage is aimed at finding the ways to optimize the multiparameter
technologies by identifying areas with the best combination of criteria,
from the point of view of the customer, the working areas of criteria
represent, from these positions, their compromise criteria. According-
ly, the regularization of the technology at this stage will be to identify
areas with the best combination of criteria. Regarding the key parame-
1170 V. I. BOLSHAKOV, V. M. VOLCHUK, and Yu. I. DUBROV
ter, the following segments are obtained: 1 (KС–HSD–σ), 2 (KС–σ), 3
(KС–HSD), 4 (KС–σВ) (Fig. 2), which makes it possible to consider
these segments from the perspective of optimizing the multi-
parameter technologies.
Based on the customer preference, combinations of partial criteria are
selected from the compromise criteria with the possibility to forecast
the range of the chemical composition of the roll, which gives the oppor-
tunity for the regularization of the technology to produce cast iron rolls.
4. CONCLUSIONS
Therefore, the given regularization of the conditionally ill-posed prob-
lem using the fractal formalism contributes to optimization of the
multiparameter technologies by choosing combinations of partial qual-
ity criteria according to the customer’s preference under different ini-
tial conditions.
REFERENCES
1. Yu. Dubrov, V. Bol’shakov, and V. Volchuk, Puti Identifikatsii Periodicheskikh
Mnogokriterialnykh Tekhnologiy [Paths to Identification of Periodic
Multicriteria Technologies] (Saarbrucken: Palmarium Academic Publishing:
2015) (in Russian).
2. V. I. Bol’shakov, Substrukturnoe Uprochnenie Konstruktsionnykh Staley
[Substructural Hardening of Structural Steels] (Toronto: Basilian Press: 1998)
(in Russian).
Fig. 2. The compromise criteria of the mechanical properties of cast iron as
function of the chemical composition.
REGULARIZATION OF ILL-POSED PROBLEM OF EXTRACTIVE METALLURGY 1171
3. T. S. Skoblo, N. M. Vorontsov, S. I. Rudyuk, N. A. Budag’yants, and
V. A. Voronina, Prokatnye Valki iz Vysokouglerodistykh Splavov [Forming
Rolls from High-Carbon Alloys] (Moscow: Metallurgiya: 1994) (in Russian).
4. A. Ye. Krivosheyev, Litye Valki [Cast Rolls] (Moscow: Metallurgizdat: 1957)
(in Russian).
5. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (New York:
Winston: 1977).
6. V. M. Volchuk, Metallofiz. Noveishie Tekhnol., 39, No. 7: 949 (2017)
(in Russian).
7. B. B. Mandelbrot, The Fractal Geometry of Nature (New York–San Francisco:
Freeman: 1982).
8. O. Eh. Zasimchuk, Yu. G. Gordienko, and R. G. Gontareva,
Metallofiz. Noveishie Tekhnol., 22, No. 4: 71 (2000) (in Russian).
9. V. Bolshakov, V. Volchuk, and Yu. Dubrov, Fractals and Properties of
Materials (Saarbrucken: Lambert Academic Publishing: 2016).
10. J. Hadamard, Princeton University Bulletin, 13: 49 (1902) (in French).
11. S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications
(De Gruyter: 2011).
12. V. I. Bol’shakov, V. M. Volchuk, and Yu. I. Dubrov, Visn. Nac. Akad. Nauk
Ukr., No. 8: 66 (2013) (in Ukrainian).
13. V. I. Bol’shakov, V. N. Volchuk, and Yu. I. Dubrov, Dopov. Nac. Akad. Nauk
Ukr., No. 11: 77 (2014) (in Russian).
https://doi.org/10.15407/mfint.39.07.0949
https://doi.org/10.1515/9783110224016
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/HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|