In the paper is presented a part of the project for determining basically permissible the four – component coposition of the (4K) mixture of the polymer fibres on the basis of the numerical simulation. The mathematical models of the composition are developed on the basis of the linear equations. The computer solution of some variants of these models is performed by Gaus procedures in combination with the numerical method of Monte Carlo simulation

B. Davidović

B. Radulović

D. Letić

D. Z. Živković

I. Berković

V. Petrović

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The designing of the four – component composition of the blend of the polymer fibres on the basis of the numerical simulation

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251METALURGIJA 52 (2013) 2, 251-254B. DAVIDOVIĆ, D. LETIĆ, V. PETROVIĆ, I. BERKOVIĆ, B. RADULOVIĆ, D. Z. ŽIVKOVIĆTHE DESIGNING OF THE FOUR – COMPONENT COMPOSITION OF THE BLEND OF THE POLYMER FIBRES ON THE BASIS OF THE NUMERICAL SIMULATIONReceived – Prispjelo: 2012-05-23Accepted – Prihvaćeno: 2012-09-10Preliminary Note – Prethodno priopćenje ISSN 0543-5846METABK 52(2) 251-254 (2013)UDC – UDK 678.7:677.6=111B. Davidović, D. Letić, V. Petrović, I. Berković, B. Radulović, Technical Faculty ’’Mihajlo. Pupin’’, Zrenjanin, University of Novi Sad.D. Z. Živković, Technical High School, Belgrade, SerbiaIn the paper is presented a part of the project for determining basically permissible the four – component coposition of the (4K) mixture of the polymer fi bres on the basis of the numerical simulation. The mathematical models of the composition are developed on the basis of the linear equations . The computer solution of some variants of these models is performed by Gaus procedures in combination with the numerical method of Monte Carlo simulation.Key words: Polymers, Blending Matters Theory, Design, Monte Carlo SimulationINTRODUCTIONThe deﬁ ning of the composition of the determined substances in: chemistry [1], textile [2], metallurgy [3], food industry, pharmacy, thermodynamics, theoretical physics [4] etc., means ﬁ nding an optimal percentage-quantative relation of the com-ponents with best satis-fying charac-teristics in the mixture in which they take part. During combining of the mixture where partici-pate the ﬁ bres of various physical-chemical characte-ristics, one must take care about the established rule of the proportional limits of the ﬁ bres participating in the mixture. Each component taking part in the composi-tion of the mixture, participates in determing its prices, relatively to its participation. If, e.g., the price of one /kg/: of the A component Ca, of the component B: Cb, of the component C: Cc, ... of the com-ponent N: Cn, and their proportional participations in the mixture with x, y, z, ... , n,  then the price of the mixture Cm has been ob-tained as the linear combination of the following values in the currency units per piece (Cur, Unit/Piece):  (1)For the proportional participations of n-components is always important the relation:  (2)And the inequalities:  . The same is with the condition that the price of the mix-ture is higher than the cheapest one, and lower than the most expensive component, i.e.  (3)DETERMINIG THE COMPOSITION OF THE MIXTURE WITH FOUR COMPONENTSThe deﬁ ning problem of the various proportional par-ticipation of the components in the mixture was for many years a vexed question [2]. Namely, the calculation meth-od that is still the only one in the majority of our techno-logical systems, is known as “cross reckoning”. These calculation procedures give relatively quickly a solution, but only a restricted number of such solutions (2 up to 3). Very often with this calculation groups of two compo-nents each ought to participates equally in the blend. The mixture composing in this way doas not giva the capabil-ity of the analysis of various combinations and at the same time as well the ﬁ nding of some hidden savings. By solving two equations with four unknowns, this problem can be simply solved with the combination of the linear programming and simulation [5]. If in the mixture M whose price per one /kg/ is known, participate four com-ponents with the prices per the weight units Ca, Cb, Cc  and Cd with the supposition that:  and  (4)Determine percentage participations of x, y, z and w of the components A, B, C and D in the mixture M. Us-ing the task conditions, one obtains the system of the two linear equations with the four unknowns values x, y, z and w. this system has inﬁ nitely many solutions if we solve it on the basis two unknowns, e.g. per x and y: Given  (5)   252  METALURGIJA 52 (2013) 2, 251-254B. DAVIDOVIĆ et al.: THE DESIGNING OF THE FOUR – COMPONENT COMPOSITION OF THE BLENDwhere z and w take arbitrary values. Taking into consid-eration these solutions and conditions, this applies for every single variable: 0<x<1, 0<y<1, 0<z<1 and 0<w<1, equivalent as well to the found values x and y:   (6) Solving these in equations, one gets for w, for:    (7)  The system of the linear in equations can be solved as well by graphical method [6]. The group of the limi-tations in the ﬁ eld of the permissible solutions is pre-sented by the equations of the straight lines. Using gen-eral obtained forms, it is possible to present several characteristic cases of the graphic and table results of the composition of 4K mixture with the components: x, y, z and w.THE FIRST CASE: Ca<Cm<CbFor the chosen (selected) vector the price of the components and the mixture /CU/kg/ follows:  (8)and the function domain of the component z in the inter-val: z=0, 0,1…1, taking into consideration the solved system of the linear equations, follows the graphical and table interpretation in the Figure 1 and Figure 2.Determing of the marginal point Zmax of the z func-tion. Given  (9) The marginal points of the w(z) function are  (10)The ﬁ eld of the permissible solutions (the triangle 0, w3(z), Zmax) for the marginal values of the components z and w is presented in the Figure 2). The preparation for the graphic presentation can be made on the basis of the interval variable Z (with the increment of 0.009) and the function  :  and  (11)The numerical simulation of the values of the Z con-tents and W components in 4K mixture, is being per-formed on the basis of the random numbers [7], but only in the isolated ﬁ eld of the acceptable solutions. So, we get:• The simulated value: • The simulated value: The simulation of the contents X and Y components in the 4K mixture is being performed on the basis of solving the set equations of the mixtures:  (12) Where is: mat the matrix of the technical coeﬁ cients and vek limitation vector:  (13)On the basis of the previous, is being set the model for solving:   (14)So, one gets the remaining two values of the mixture composition:The simulated value  and .Figure 1  The graphic(al) interpretation of the solution in the domain of the independent variable zFigure 2  Some values of the independent variable z And dependently variable values w(z)253METALURGIJA 52 (2013) 2, 251-254B. DAVIDOVIĆ et al.: THE DESIGNING OF THE FOUR – COMPONENT COMPOSITION OF THE BLENDTHE PROGRAM VERIFICATION OF THE MODELThe matrix of the interval values of the participation of the components in 4K mixture, has been formed as the control matrix of the values of the least possible, simulated and maximal participations in the mixture composition:  (15)From the matrix can be formed a randon vector of the acceptable values as:  /%/ (16)The veriﬁ cation of the equation of 4K composition of the mixture on the basis of the second vector of the matrix column can be performed directly in programmed way [5]:  or  (17)The veriﬁ cation of the criterium of the price con-stancy of the mixture cost price is being performed on the basis of the multiplication produce of the prices vec-tor and the transposed random vector of the premissible 4K composition:  (18)It can be stated that both previous equations are satis-ﬁ ed on the basis of the obtained results, and that the vec-tor of the composition is an optimal vector of solution.THE SECOND CASE: Cb<Cm<CcFor the chosen vector the prices of the components and the mixture /CU/kg/:  (19)And the function domain of the component compo-sition z: z:=0, 0,1…1, the explicit form of the inequation of the mixture composition for the conditions  is being obtained as: (20)I ﬁ eld: II ﬁ eld: III ﬁ eld: IV ﬁ eld: Note: In the paper are being used as well the syn-taxes represented in Mathcad [7], which are similar to the usual mathematical syntaxes which often appear in literature.Determing of the marginal point Zmax  of Z func-tion.Given  (21)  Given  The marginal points of w(z) function The ﬁ eld of the permissible solutions (the polygon in Figure 3) for the marginal values of Z and W (Figu-re 4).    (22)The program solution in Mathcad in the form of the solution vector, is the next:The numerical simulation of the values of Z contents and W components in 4K mixture is being performed on the basis of the random numbers of the deﬁ ned ﬁ eld of the permissible solutions. So, one gets:Figure 3  The graphic interpretation of the fi eld of the acceptable solutionsFigure 4  The part of the limit values in the acceptable fi eld254  METALURGIJA 52 (2013) 2, 251-254B. DAVIDOVIĆ et al.: THE DESIGNING OF THE FOUR – COMPONENT COMPOSITION OF THE BLEND• The simulated value: • The simulated value: The simulation of the contents of X and Y compo-nents in 4K mixture is being performed on the basis of solving the set mixture equations:  (23) Where is: mat matrix of the technical coefﬁ cients and vek limitation vector:  (24)On the basis of the previous is set the model for solving:  (25)So, one gets on the basis of the computer simula-tion:The simulated value  and .THE PROGRAM VERIFICATION OF THE MODELThe control matrix of the interval values of the com-ponent participation in 4K mixture:  (26)On the basis of this is a random vector of the accept-able values:  /%/ (27)The veriﬁ cation of the composition equation of 4K mixture on the basis of the second vector of M matrix columin follows:  or  (28)The veriﬁ cation of the constancy criterium of the cost price of the mixture is being performed, also, on the basis of the multiplacation product of the prices vec-tor and transposed random vector of the permissible 4K mixture:  /CU/ (29)Note: CU – Currency Unit.It can be established that both equations are on the basis of the obtained results satisﬁ ed, and that the com-position vector is the second optimal solution vector.CONCLUSIONThe solutions that are being perceived in the set of diagrams and tables, present a set of points of the per-missible solutions that are among the restricting limit-ing straight lines: w1(z), w 2(z), w 3(z) and w4(z) and which present the coordinates within the watched poly-gon obtained on the basis of the cross – section of the points sets in the looked at ﬁ elds. These selectional ﬁ elds are the sole ones which present the basis for the simulation of the two values of the mixture composi-tion. The other two are easily obtained by solving the two equations with the two remained unknown values of the mixture composition. The character of these two calculated values is also random on account of the start-ing conditions and consequences. Very important com-ponne-nt of this model variants which are many and are not presented here, is the possibility of the program veriﬁ cation of the model. This, as a rule, presents the integral phase in the development of every simulation models. From the practical point of view the importance of the results of this project is great, because it can give certain savings, taking into consideration that the model based on the criterium of the composition optimum at the constant price of the mixture. The application of this model is universal, because it does not refer only to the deﬁ ning of the composition of the polymer ﬁ bres, but as well in the determing of the mixture of new materials for new technologies at the criterium of the constant price. Although in the paper are used the efﬁ cient com-bined methods of the linear progamming and numerical simulation, the ways for more effective solution of the mixture problem would be certainly in the multicriteria optimization. There would be set the conditions that would cover not only the criterion of the constant mix-ture price, but also very important matter, i.e. which components have the quantative and qualitative limita-tions in the mixture of different forms of the substance with the example of the polymer mixing in textile.REFERENCES[1] K. Michalek, J. Morávka, K. Gryc, Matematička analiza procesa homogenizacije u loncu s argonskim miješanjem, Metalurgija 48 (2009) 4, 219-222[2]  S. Šunjka, V. Petrović, The Technology of the Nonwoven Textile (fabric), Technical Faculty “M. Pupin”, Zrenjanin, (1995), p. 11[3] V. A. Grynkevych, I. Mamuzić, V. M. Danchenko, O razvoju metoda kompjuterske simulacije procesa obliko-vanja metala Metalurgija, 43 (2004) 3, 181-192[4]  J. Lebowitz and O. Penrose, Modern Ergodic Theory, Phy-sics Today, 26, (1973), 155-175.[5] D. Malindžák, M. Straka, P. Helo, J. Takala, Metodologije za dizajniranje simulacijskog modela logističkih sustava, Metalurgija 48 (2009) 4, 348-352[6]  D. Letić, B. Davidović, Operating and Project Manage-ment, Computer library, Belgrade, (2011), 511-537[7]  D. Letić, B. Davidović, I. Berković, T. Petrov, Mathemati-cs and Visualization in Mathcad 13, Computer Library, Belgrade, (2007), p. 759Note:  The responsible translator of English language is Srđan Šerer, Technical Faculty “M. Pupin”, Zrenjanin, University of Novi Sad, Serbia

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251METALURGIJA 52 (2013) 2, 251-254B. DAVIDOVIĆ, D. LETIĆ, V. PETROVIĆ, I. BERKOVIĆ, B. RADULOVIĆ, D. Z. ŽIVKOVIĆTHE DESIGNING OF THE FOUR – COMPONENT COMPOSITION OF THE BLEND OF THE POLYMER FIBRES ON THE BASIS OF THE NUMERICAL SIMULATIONReceived – Prispjelo: 2012-05-23Accepted – Prihvaćeno: 2012-09-10Preliminary Note – Prethodno priopćenje ISSN 0543-5846METABK 52(2) 251-254 (2013)UDC – UDK 678.7:677.6=111B. Davidović, D. Letić, V. Petrović, I. Berković, B. Radulović, Technical Faculty ’’Mihajlo. Pupin’’, Zrenjanin, University of Novi Sad.D. Z. Živković, Technical High School, Belgrade, SerbiaIn the paper is presented a part of the project for determining basically permissible the four – component coposition of the (4K) mixture of the polymer fi bres on the basis of the numerical simulation. The mathematical models of the composition are developed on the basis of the linear equations . The computer solution of some variants of these models is performed by Gaus procedures in combination with the numerical method of Monte Carlo simulation.Key words: Polymers, Blending Matters Theory, Design, Monte Carlo SimulationINTRODUCTIONThe defi ning of the composition of the determined substances in: chemistry [1], textile [2], metallurgy [3], food industry, pharmacy, thermodynamics, theoretical physics [4] etc., means fi nding an optimal percentage-quantative relation of the com-ponents with best satis-fying charac-teristics in the mixture in which they take part. During combining of the mixture where partici-pate the fi bres of various physical-chemical characte-ristics, one must take care about the established rule of the proportional limits of the fi bres participating in the mixture. Each component taking part in the composi-tion of the mixture, participates in determing its prices, relatively to its participation. If, e.g., the price of one /kg/: of the A component Ca, of the component B: Cb, of the component C: Cc, ... of the com-ponent N: Cn, and their proportional participations in the mixture with x, y, z, ... , n,  then the price of the mixture Cm has been ob-tained as the linear combination of the following values in the currency units per piece (Cur, Unit/Piece):  (1)For the proportional participations of n-components is always important the relation:  (2)And the inequalities:  . The same is with the condition that the price of the mix-ture is higher than the cheapest one, and lower than the most expensive component, i.e.  (3)DETERMINIG THE COMPOSITION OF THE MIXTURE WITH FOUR COMPONENTSThe defi ning problem of the various proportional par-ticipation of the components in the mixture was for many years a vexed question [2]. Namely, the calculation meth-od that is still the only one in the majority of our techno-logical systems, is known as “cross reckoning”. These calculation procedures give relatively quickly a solution, but only a restricted number of such solutions (2 up to 3). Very often with this calculation groups of two compo-nents each ought to participates equally in the blend. The mixture composing in this way doas not giva the capabil-ity of the analysis of various combinations and at the same time as well the fi nding of some hidden savings. By solving two equations with four unknowns, this problem can be simply solved with the combination of the linear programming and simulation [5]. If in the mixture M whose price per one /kg/ is known, participate four com-ponents with the prices per the weight units Ca, Cb, Cc  and Cd with the supposition that:  and  (4)Determine percentage participations of x, y, z and w of the components A, B, C and D in the mixture M. Us-ing the task conditions, one obtains the system of the two linear equations with the four unknowns values x, y, z and w. this system has infi nitely many solutions if we solve it on the basis two unknowns, e.g. per x and y: Given  (5)   252  METALURGIJA 52 (2013) 2, 251-254B. DAVIDOVIĆ et al.: THE DESIGNING OF THE FOUR – COMPONENT COMPOSITION OF THE BLENDwhere z and w take arbitrary values. Taking into consid-eration these solutions and conditions, this applies for every single variable: 0<x<1, 0<y<1, 0<z<1 and 0<w<1, equivalent as well to the found values x and y:   (6) Solving these in equations, one gets for w, for:    (7)  The system of the linear in equations can be solved as well by graphical method [6]. The group of the limi-tations in the fi eld of the permissible solutions is pre-sented by the equations of the straight lines. Using gen-eral obtained forms, it is possible to present several characteristic cases of the graphic and table results of the composition of 4K mixture with the components: x, y, z and w.THE FIRST CASE: Ca<Cm<CbFor the chosen (selected) vector the price of the components and the mixture /CU/kg/ follows:  (8)and the function domain of the component z in the inter-val: z=0, 0,1…1, taking into consideration the solved system of the linear equations, follows the graphical and table interpretation in the Figure 1 and Figure 2.Determing of the marginal point Zmax of the z func-tion. Given  (9) The marginal points of the w(z) function are  (10)The fi eld of the permissible solutions (the triangle 0, w3(z), Zmax) for the marginal values of the components z and w is presented in the Figure 2). The preparation for the graphic presentation can be made on the basis of the interval variable Z (with the increment of 0.009) and the function  :  and  (11)The numerical simulation of the values of the Z con-tents and W components in 4K mixture, is being per-formed on the basis of the random numbers [7], but only in the isolated fi eld of the acceptable solutions. So, we get:• The simulated value: • The simulated value: The simulation of the contents X and Y components in the 4K mixture is being performed on the basis of solving the set equations of the mixtures:  (12) Where is: mat the matrix of the technical coefi cients and vek limitation vector:  (13)On the basis of the previous, is being set the model for solving:   (14)So, one gets the remaining two values of the mixture composition:The simulated value  and .Figure 1  The graphic(al) interpretation of the solution in the domain of the independent variable zFigure 2  Some values of the independent variable z And dependently variable values w(z)253METALURGIJA 52 (2013) 2, 251-254B. DAVIDOVIĆ et al.: THE DESIGNING OF THE FOUR – COMPONENT COMPOSITION OF THE BLENDTHE PROGRAM VERIFICATION OF THE MODELThe matrix of the interval values of the participation of the components in 4K mixture, has been formed as the control matrix of the values of the least possible, simulated and maximal participations in the mixture composition:  (15)From the matrix can be formed a randon vector of the acceptable values as:  /%/ (16)The verifi cation of the equation of 4K composition of the mixture on the basis of the second vector of the matrix column can be performed directly in programmed way [5]:  or  (17)The verifi cation of the criterium of the price con-stancy of the mixture cost price is being performed on the basis of the multiplication produce of the prices vec-tor and the transposed random vector of the premissible 4K composition:  (18)It can be stated that both previous equations are satis-fi ed on the basis of the obtained results, and that the vec-tor of the composition is an optimal vector of solution.THE SECOND CASE: Cb<Cm<CcFor the chosen vector the prices of the components and the mixture /CU/kg/:  (19)And the function domain of the component compo-sition z: z:=0, 0,1…1, the explicit form of the inequation of the mixture composition for the conditions  is being obtained as: (20)I fi eld: II fi eld: III fi eld: IV fi eld: Note: In the paper are being used as well the syn-taxes represented in Mathcad [7], which are similar to the usual mathematical syntaxes which often appear in literature.Determing of the marginal point Zmax  of Z func-tion.Given  (21)  Given  The marginal points of w(z) function The fi eld of the permissible solutions (the polygon in Figure 3) for the marginal values of Z and W (Figu-re 4).    (22)The program solution in Mathcad in the form of the solution vector, is the next:The numerical simulation of the values of Z contents and W components in 4K mixture is being performed on the basis of the random numbers of the defi ned fi eld of the permissible solutions. So, one gets:Figure 3  The graphic interpretation of the fi eld of the acceptable solutionsFigure 4  The part of the limit values in the acceptable fi eld254  METALURGIJA 52 (2013) 2, 251-254B. DAVIDOVIĆ et al.: THE DESIGNING OF THE FOUR – COMPONENT COMPOSITION OF THE BLEND• The simulated value: • The simulated value: The simulation of the contents of X and Y compo-nents in 4K mixture is being performed on the basis of solving the set mixture equations:  (23) Where is: mat matrix of the technical coeffi cients and vek limitation vector:  (24)On the basis of the previous is set the model for solving:  (25)So, one gets on the basis of the computer simula-tion:The simulated value  and .THE PROGRAM VERIFICATION OF THE MODELThe control matrix of the interval values of the com-ponent participation in 4K mixture:  (26)On the basis of this is a random vector of the accept-able values:  /%/ (27)The verifi cation of the composition equation of 4K mixture on the basis of the second vector of M matrix columin follows:  or  (28)The verifi cation of the constancy criterium of the cost price of the mixture is being performed, also, on the basis of the multiplacation product of the prices vec-tor and transposed random vector of the permissible 4K mixture:  /CU/ (29)Note: CU – Currency Unit.It can be established that both equations are on the basis of the obtained results satisfi ed, and that the com-position vector is the second optimal solution vector.CONCLUSIONThe solutions that are being perceived in the set of diagrams and tables, present a set of points of the per-missible solutions that are among the restricting limit-ing straight lines: w1(z), w 2(z), w 3(z) and w4(z) and which present the coordinates within the watched poly-gon obtained on the basis of the cross – section of the points sets in the looked at fi elds. These selectional fi elds are the sole ones which present the basis for the simulation of the two values of the mixture composi-tion. The other two are easily obtained by solving the two equations with the two remained unknown values of the mixture composition. The character of these two calculated values is also random on account of the start-ing conditions and consequences. Very important com-ponne-nt of this model variants which are many and are not presented here, is the possibility of the program verifi cation of the model. This, as a rule, presents the integral phase in the development of every simulation models. From the practical point of view the importance of the results of this project is great, because it can give certain savings, taking into consideration that the model based on the criterium of the composition optimum at the constant price of the mixture. The application of this model is universal, because it does not refer only to the defi ning of the composition of the polymer fi bres, but as well in the determing of the mixture of new materials for new technologies at the criterium of the constant price. Although in the paper are used the effi cient com-bined methods of the linear progamming and numerical simulation, the ways for more effective solution of the mixture problem would be certainly in the multicriteria optimization. There would be set the conditions that would cover not only the criterion of the constant mix-ture price, but also very important matter, i.e. which components have the quantative and qualitative limita-tions in the mixture of different forms of the substance with the example of the polymer mixing in textile.REFERENCES[1] K. Michalek, J. Morávka, K. Gryc, Matematička analiza procesa homogenizacije u loncu s argonskim miješanjem, Metalurgija 48 (2009) 4, 219-222[2]  S. Šunjka, V. Petrović, The Technology of the Nonwoven Textile (fabric), Technical Faculty “M. Pupin”, Zrenjanin, (1995), p. 11[3] V. A. Grynkevych, I. Mamuzić, V. M. Danchenko, O razvoju metoda kompjuterske simulacije procesa obliko-vanja metala Metalurgija, 43 (2004) 3, 181-192[4]  J. Lebowitz and O. Penrose, Modern Ergodic Theory, Phy-sics Today, 26, (1973), 155-175.[5] D. Malindžák, M. Straka, P. Helo, J. Takala, Metodologije za dizajniranje simulacijskog modela logističkih sustava, Metalurgija 48 (2009) 4, 348-352[6]  D. Letić, B. Davidović, Operating and Project Manage-ment, Computer library, Belgrade, (2011), 511-537[7]  D. Letić, B. Davidović, I. Berković, T. Petrov, Mathemati-cs and Visualization in Mathcad 13, Computer Library, Belgrade, (2007), p. 759Note:  The responsible translator of English language is Srđan Šerer, Technical Faculty “M. Pupin”, Zrenjanin, University of Novi Sad, Serbia

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The designing of the four – component composition of the blend of the polymer fibres on the basis of the numerical simulation

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