2 k Factorial Design Used to Optimize the Linear Mathematical Model through Active Experiment

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1 RECET J. (01), 54: k Factorial Design Used to Optimize the Linear Mathematical Model throgh Active Experiment Ioan MILOA Transilvania University of Brasov, Romania, milosan@nitv.ro Astract The data presented in this paper is part of a comprehensive stdy on mathematical modelling y active experiment, sing first-order designs in order to optimize linear mathematical models. In this respect, a k fll factorial design was sed. In the research, the experimentation plan was estalished to enale a higher nmer of experiments than the nmer of coefficients to e determined. There was estalished the matrix of the designed experiment and the calclation was performed according to the methodology specific to the 3 fll factorial experiments. The reslts otained y experimental design were employed in the stdy, sing a special Mo-i alloyed cast iron, heat treated, whose elongation vales were determined. Finally, a small nmer of determinations were sed to determine the specific technological parameters and the maximm hardness of the analysed material. Keywords DOE, k factorial design, cast iron, elongation 1. Introdction Where, as part of the scientific research, statistical methods are employed in all stages of an experiment (efore, dring, and after the experiment), the work is carried ot according to the following seqence: - determination of the nmer of experiments and the conditions for their condct (prior to the experiment); - processing of the reslts (dring the experiments); - determination of the lsions on the exection of ftre experiments (after the experiment). This specific manner of condcting research is called active experiment and involves the design of the experiment condcted y [1-9]: - estalishing the necessary and sfficient nmer of experiments and the conditions for their condct; - determination y statistical methods of the regression eqation, representing a certain degree of approximation, comptale, the model of the process; - determination of the conditions for otaining the optimal performance for the analysed process. The design of experiment (DOE) will e sed in this paper with k fll factorial experiments (FFE) to determine the direction of movement to an optimal range from a known point, calclation performed in the research regarding the optimization of the elongation vales of a heat treated special cast iron..research Ojectives The main ojective of this paper is to achieve mathematical modelling y active experiment, sing 3 fll factorial experiments, in order to optimize the linear mathematical model, determining the direction of movement to an optimal domain from a known point. In this case, of interest was the estalishment of technological parameters in order to maximize the elongation of a heat treated special cast iron. 37

2 3. teps for Applying Mathematical Modelling y Active Experiment Using the k Fll Factorial Design The designed experiment is solved according to the following steps [1-9]: (1) The nmer of factors (z i) taken into accont is determined and the vale of the factors expressed in natral nits and coded nits is correlated; () The aseline (z i0), the variation range (Δz i) and the pper (+1) and lower ( 1) levels of the analysed factors are determined; (3) The form of the mathematical model sed is estalished ased on the encoded vales and taking into accont the fact that it's important in optimizing the mathematical model y the design of experiment method, and in this case we can only determine the direction of movement to the optimal range (from a known point) and, therefore, only the linear part of the mathematical model is stdied, according to the [1, ] expression: y 0 1 x1 x 3 x3 i x i (1) (4 ) The matrix of the k fll factorial experiment design is ilt; (5) The nmer of parallel determinations (n i) to e performed is determined; (6) The coefficients of the ( i) mathematical model are calclated sing the expression [1, ]: xi y i i 1 () x i = process factors (normalized variale); y = arithmetic mean of process performance (arithmetic mean of parallel determinations); = nmer of experimental points (nmer of rows in the design matrix); (7) The dispersion of the parallel determinations (row dispersion) at each experimental point ( is calclated sing the expression [1, ]: m i1 y m 1 3 k y y k k y (4) y 1 m y k m k1 y k = state variale in parallel determinations; m = nmer of parallel determinations; () The homogeneity of the experimental dispersions is verified sing the Cochran criterion [1, ]: G C = calclated vale of the Cochran criterion; max G C max 1 = maximm vale (from the experiment design matrix) of the experimental dispersions; The calclated vale of the Cochran criterion (G C) is compared to the critical (talar) vale of the Cochran criterion (G T), whose expression is: G T G ; 1; (7) α = statistical coefficient of the confidence level sed, α = 0.05; ) (3) (5) (6)

3 ν1, ν = degrees of freedom; ν1 = m 1; ν = ; = nmer of experimental points (nmer of rows in the design matrix). Two sitations arise in this case: a) G C<G T, it follows that the experimental dispersions are homogeneos; ) G C>G T, it follows that the experimental dispersions are not homogeneos; If the G C<G T rle is complied with, the calclation of the experimental error (reprodciility dispersion), is performed next according to the expression [1, ]: m 1 = dispersion of parallel determinations at each experimental point; m = nmer of parallel determinations; (9) The significance of each coefficient i with a confidence interval Δ i is verified according to the calclation method of the tdent criterion. The confidence interval Δ i is calclated according to the expressions [1, ]: i t v 0 i () ; ; (9) i i ; (10) = mean sqare deviation of i factors; i i = i factors dispersion. i 0 39, (11) Two sitations arise in this case as well [1-9]: a) if i i, the i coefficients are part of the relevant mathematical model; ) if i i, the i coefficients cannot e part of the relevant mathematical model. The calclation is stopped and the coefficient(s) that does (do) not verify the tdent criterion is (are) removed; (9) The correlation etween the mathematical model and the experimental data is verified sing the Fischer criterion. The possiility of sing this mathematical model for optimizing the process is verified on this occasion. This verification is employed to determine whether the calclated approximation of the linear dependence y = f(x i) is sfficiently precise in relation to the research accracy. The verification is carried ot sing the following expressions [1-9]: 0 Fc, (1) F c = calclated vale of the Fischer criterion; = ordance dispersion (error de to the mathematical model); m i1 y ν = nmer of degrees of freedom sed to calclate l = nmer of mathematical model coefficients; i, ; ν = l; (13)

4 and i y is determined y the expression: y i y y, (14) y = arithmetic mean of process performance (arithmetic mean of parallel determinations); y is the calclated state variale (mathematical model vale) for each experimental point (for each row in the design matrix); The calclated vale of the Fischer criterion (F c) will e compared with the critical (talar) vale of the Fischer criterion (F T) whose expression is: FT F; v; v0, (15) α = statistical coefficient of the confidence level sed, α = 0.05; ν ; ν 0 = nmer of degrees of freedom sed to calclate the critical (talar) vale of the Fischer criterion (F T);ν = l and ν 0 = (m 1), [1-9], = nmer of experimental points (nmer of rows in the design matrix); l = nmer of coefficients in the mathematical model; l = 4 ( 0, 1,, 3); n i = nmer of parallel determinations. Two sitations arise in this case as well: a) if F c F T, this mathematical model matches the experimental data and, therefore, it can e sed to determine the direction of movement to an optimal range from a known point; ) if F c F T, this mathematical model does not match the experimental data and, therefore, it is not a linear model and it cannot e sed in this case as it does not meet the ojective of the research. 4. Experimental Procedre The stdied material was a C-i cast iron with the following composition (% in weight): 3.63 %C;. %i; 0.45 %Mn; 0.01 %P; %; %Mg; 0.4 %C and 0.40 %i. The data presented in this paper is part of a comprehensive stdy on heat treated G cast iron hardness. For this prpose, eight specimens, Ø30 5 mm, were sed. The parameters specific to the thermal treatment applied are as follows: - the astenitizing temperatre, t A[C]; - the holding time at the astenitizing temperatre, A [min]; - the temperatre at isothermal level, t iz [C]; - the holding time at the isothermal level, iz [min]. All these experimental specimens, were performed at isothermal maintenance in salt-ath (55% KO 3+45% ao 3), and the cooling after the isothermal holding was done in air. 5. olving the Designed Experiment The designed experiment is solved according to the steps presented aove [1-9]. It was determined as the nmer of the z i= 3 factors analysed, i.e. t A; t iz; iz. Tale 1 shows the factors analysed alongside ase levels and variation ranges. Tale 1. Determination of the factors analysed, of ase levels and of variation ranges Factors z 1 z z 3 Code x 1 x x 3 Base level, (z i0) Variation range, (Δz i) Upper level, (+1) Lower level, (-1)

5 Tale shows the 3 fll factorial experiment design matrix with the reslts otained according to the design. Tale. Matrix of the 3 fll factorial experiment design o.exp. The order of the x ivariales Process performance, y k experiments x 0 x 1 x x 3 y 1 y y The expressions (1) present the calclation of the coefficients of the mathematical model ( i): ; 0.15 ; ;.375 ; Tale 3 shows the calclation of the dispersion of parallel determinations ( experimental point. Tale 3. Calclation of the dispersion of parallel determinations ( y 1 y y ) in each ) in each experimental point o.exp. y y 3 ν Σ Σ(Δ 3-1= / = y ) The homogeneity of the experimental dispersions is verified sing the Cochran criterion, according to the expression (6): G C 0.03 (16) The critical (talar) vale of the Cochran criterion (G T) is determined from the criterion-specific tales [1, ] and the following vale is otained: G T = G α, ν, ;G T = G 0.05; ; = ince G C<G T (0.03 < ), it follows that the experimental dispersions are homogeneos and

6 the experimental error (the reprodciility dispersion),, is calclated next according to the expression (): The calclation of the dispersion in determining the coefficients of the mathematical model, calclated sing the tdent criterion, according to the expressions (9)-(11): i ; i ; t t0.05;. 306 []; it follows that: Δ i = = As i i : (for 0); (for 1); (for ); (for 3); i coefficients are part of the mathematical model otained. The mathematical model specific to the parameters previosly estalished and according to the expression (1) will take the form: ỹ = x x.375 x 3. The correlation etween the mathematical model and the experimental data is verified sing the Fischer criterion, according to expressions (1) (15), as shown in Tale 4. Tale 4. The correlation etween the mathematical model and the experimental data is verified according to the Fischer criterion o.exp. Σ(Δ y y 4 0 y y ) y ν 1-4= ; F c The critical (talar) vale of the Fischer criterion (F T) is determined from the criterion-specific tales [1, ] and the following vale is otained: F T = F(α; ν 1; ν ) = F(0.05; 4; ) = As F c F T ( ), this mathematical model matches the experimental data and, therefore, it can e sed to determine the direction of movement to an optimal range from a known point. Following the analysis of the vales otained in the case of the 3 fll experiment design, presented in Tale 4, it is noted that an optimal (maximm) vale of the hardness of the analysed material ( HB) was otained in the case of the designed experiment no.7, which had the following factors of the analysed process: x 1 = at the pper level (+1) = 900 C, corresponding to the astenitizing holding temperatre; x = at the lower level (-1) = 300 C, corresponding to the isothermal stage holding temperatre; x 3 = at the lower level (-1) = 10 min., corresponding to the isothermal stage holding time. 6. Conclsion The analysis of all data taken into accont leads to the following lsions: a) the eqation of the determined mathematical model shows that, in the variation ranges chosen for the analysed factors, their inflences on the stdied process are different, as follows: - the factor z 1 ( 1 = 0.15) corresponding to the astenitizing holding temperatre has the

7 strongest inflence on the process; - the factor z 3 ( 3 = -.375) corresponding to the isothermal stage holding time inflences the process to a lower extent; - the factor z ( = ) corresponding to the isothermal stage holding temperatre has the weakest inflence on the process; () the experimental dispersions, verified sing the Cochran criterion, are homogeneos; (c) statistically, all i coefficients of the mathematical model otained differ from zero and, ths, they are part of the mathematical model otained, and the verification was performed sing the tdent criterion; (d) the optimal (maximm) hardness of the analysed material ( HB) was otained for the designed experiment no.7; (e) the correlation etween the experimental data and the calclated mathematical model was verified and the verification was ased on the Fischer criterion. References 1.Taloi D., Brat C., Florian E., Bercean E. (193): Optimizarea proceselor metalrgice (Optimization of metallrgical processes). Editra Didactiă şi Pedagogică, Bcreşti, p Taloi D. (197): Optimizarea proceselor tehnologice. Aplicaţii în metalrgie (Optimization of technological processes. Applications in metallrgy).editra Academiei Române, Bcreşti, p Lndstedt T., eifert E., Aramo L., Thelin B., yström Å., Pettersen J., Bergman R. (199): Experimental design and optimization. Chemometrics and Intelligent Laoratory ystems, I: , Vol. 4, o. 1-, p. 3-40, 4. Bingham D.R., Li W. (00): A class of optimal rost parameter designs.jornal of Qality Technology, I: , Vol. 34, o. 3, p , 5. van de Ven P.M. (00): On the eqivalence of three estimators for dispersion effects in nreplicated two-level factorial designs. Jornal of tatistical Planning and Inference, I: , Vol. 13, o. 1, p. 1-9, DOI: /j.jspi Kkreja A., Chopra P., Aggarwal A., Khanna P. (011): Application of Fll Factorial Design for Optimization of Feed Rate of tationary Hook Hopper. International Jornal of Modeling and Optimization, I: , Vol. 1, o. 3, p , DOI: /IJMO.011.V ***: Accessed ***: Accessed ***: Accessed: