An Application of LFP Method for Sintering Ore Ratio

Transcription

1 An Applcaton of LFP Method for Snterng Ore Rato X Cheng, Kalng Pan, and Yunfeng Ma School of Management, Wuhan Unversty of Scence and Technology, P.R.Chna, 408 suxn49@63.com Abstract. The proper rato decson of snterng burden s a sgnfcant secton for both of decreasng snterng costs and ncreasng qualty of ron. At present most company n Chna take the Fxed-rato method and lnear programmng (LP) model to calculate the proper rato for snterng. The former s the performance apprasal method for producton cost management of ron. The latter s to use maths method to mprove the computaton process. Ths paper brngs up a lnear fractonal programmng (LFP) model combnng the advantages of both methods to compute the proper rato to mnmze the ron cost per ton for snterng. Next based on the producton data from some steel company ths paper takes use of MATLAB to solve the problem. Then comparng the solutons wth the orgnal method, the tradtonal LP model and LFP model the conclusons are revealed n the end. Keywords: Lnear fractonal programmng, Lnear programmng, Optmzaton models, Snterng ore blendng. Introducton Snterng frstly took use of Lnear programmng (LP) models to calculate proper rato only wth chemcal proportons of ndustral raw materals as optmzaton objects[].wth the research content contnuously developng physcal and some metallurgcal propertes also lst n the object type of optmzaton models. And wth more and more varables and constrans, the problem become larger and larger. Smultaneously dffculty of solvng the problem s enlarged. LP model s substantal to cope wth the emprcal procedure method by approxmately changng the orgnal lnear fractonal object type nto lnear one and changng the chemcal composton of ore blender nstead of snter as constrant condtons. Factually neglect the metal loss n snterng process. The optmal soluton of LP s actually the mnmzed cost of an approxmate soluton. The reason s that the cost s not calculated accurately. Producton runnng take emprcal procedure method whch s also called Fxed-rato method[2] wth regard to burns. The formula of workng out the ron cost per ton s that total cost equals to the sum of plots of the rato and the correspondng prce of the materal dvded the sum of burned materals. Obvously based on the Fxed-rato method, the lnear fractonal programmng (LFP) method whose object s a fracton can smplfy the computaton of proper rato and ron cost n snterng. Y. Tan, Y. Sh, and K.C. Tan (Eds.): ICSI 200, Part II, LNCS 646, pp , 200. Sprnger-Verlag Berln Hedelberg 200

2 24 X. Cheng, K. Pan, and Y. Ma 2 Constructon of LFP Model Accordng to producton practce, only raw materals whch conform to ndcators of physcal and chemcal propertes can put nto the furnace. And the requrement of ore dressng s that proporton and content of chemcal s unformly dstrbuted and stable. Suppose that the partcle sze, chemcal composton of raw materals and other ndcators act n full complance wth sntered metallurgcal performance requrements. The same knd fuel from the same orgn s assumed wth the same chemcal composton. Suppose that proporton and content of chemcal s unformly dstrbuted, stable and well-mxed. I s a set of names of raw materals. J s a set of names of chemcals. Other symbols are defned as followng. aj s the j'th chemcal composton percentage of the 'th raw materals, where the unt s %, for any belonged to I and any j belonged to J. x s the 'th raw materals rato, whch s also the decson varable, where the unt s %, for any belonged to I. p s the prce of the 'th raw materals, where the unt s yuan per ton, for any belonged to I. E s the lower lmt of j'th chemcal composton percentage, where the unt s %, j for any j belonged to J. E s the upper lmt of j'th chemcal composton percentage, j where the unt s %, for any j belonged to J. M s the lower lmt of the 'th raw materals rato, where the unt s %, for any belonged to I. M s the upper lmt of the 'th raw materals rato, where the unt s %, for any belonged to I. R s the upper lmt of snter bascty, R s the lower lmt of snter bascty. The snter bascty s the value of quotent of the sum of composton percentage of CaO dvded the sum of composton percentage of SO2, where there s no unt. r s the upper lmt of fuel percentage, where the unt s %. r s the lower lmt fuel fuel of fuel percentage, where the unt s %. S s the supply of the 'th raw materals wthn plan perod, where the unt s ton, for any belonged to I. rf s the sum of foregn ore percentage, where the unt s %.ë2 s ron recovery rate coeffcent whch s a constant. Qs s the output of snter, where the unt s ton. The LFP model s made up of decson varables, object functon and constrans as follows. Mn f(x)= λ 2 = = px a,5 x 00 ()

3 An Applcaton of LFP Method for Snterng Ore Rato 25 st.. j = E E, j J j j a,5 λ 2 = = ax x 00 Qx s S, I a,5 x 00 (2) (3) 2 x = r f (4) = R = = a x,3 a x,5 R (5) M x M, I (6) r x x r fuel 29 + fuel (7) 23 x = 00 (8) = The object functon formula () s a lnear fracton whch numerator and denomnator are sum of lnear relatonshp. The numerator s the mxed prce of each knd of materal. And the denomnator s the sum composton of TFe of mxed materals after snterng.ë2 s ron recovery rate coeffcent that s the percentage of melted ron become pure ron. Obvously the burned loss a,5 s ncludng. f(x) s total prce dvded output of TFe. In other words the object s to mnmze the cost of ron per ton. The constrans are composed of chemcals restrctons, supply constrans, foregn ore lmt, snter bascty boundary as follows. Formula (2) s the chemcal composton percentage restrctons, ncludng TFe, CaO, MgO, SO2, Al2O3, S, P and Ig. Ig s the burnng loss of snterng. In formula (2), x s varable and other coeffcent s known. Formula (2) can equvalent transformed to nequalty wth lnear relatonshp as follows. Both sdes of formula (2) multply the denomnator, whch s a,5 a,5 x E j a x E x j j = 00 = = 00 Then transpose. The result s a,5 E 0 j a x, j = 00 j J (9)

4 26 X. Cheng, K. Pan, and Y. Ma a,5 a E 0 j j x, = 00 j J (0) The lower lmt of j'th chemcal composton percentage s transformed to formula (9). And the upper transformed to formula (0). After that the condton (2) can be transformed to lnear constrans. Formula (3) s the supply constrans, n the same argument can be transformed to a,5 S λ 0 2 x Q x s = 00, I () Formula (4) s foregn ore lmt, whch s equalty. Formula (5) s snter bascty boundary, whch can be transformed to [ ] R a a x 0 (2),5,3 = [ ] a R a x 0 (3),3,5 = Formula (6) (7) (8) are all lnear relatonshp equatons. All constrants become lnear relatonshp. Accordng to numercal optmzaton theory[3], lnear search method can help the teratve calculaton to solve the model of whch algorthm effectveness has been proofed by reference[4]. Many knds of mathematcal software have developed optmzaton toolbox to solve the problem drectly such as MATLAB. 3 Calculaton and Solvng Though the optmzaton toolbox of MATLAB has a GUI nterface whch can drectly nput the smple maths model, the LP model has too many varables and parameters to use the convenent GUI panel. It s nevtable to wrte m fles to defne the functons to solve the problem. Reference[5] recommends how to wrte MATLAB programmng code to nput the model nto the optmzaton toolbox and save the program as m fles n detal. When nput the sentence n the command wndow to call the functons from m fles, the program automatcally choose the sequental quadratc programmng (SQP) method and the lnear search method named quas-newton whch s also called varable metrc algorthm[3] for teraton to search the optmal. Gettng the optmal of the model, the ron cost per ton can be calculated. Data should all nput nto a mat fle n the same drectory wth the m fles. The content of the chemcal composton of raw materals matrx, supply and other known values are shown n Table. Upper and lower lmts of each chemcal composton of ore blender are shown n Table 2. From Table and Table 2, I = and J =8 s known. The value of rf s 65. The snter bascty boundary value s between.75 and Table data s from producton database of some steel company.

5 An Applcaton of LFP Method for Snterng Ore Rato 27 Table. Prces and percentages of snter raw materals table ID Content (%) Upper Lower Supply Prce TFe CaO MgO SO2 Al2O3 S P Ig % % t yuan/t

6 28 X. Cheng, K. Pan, and Y. Ma Table 2. The chemcal composton restrctons and qualty of the optmal table TFe CaO MgO SO2 Al2O3 S P Ig Upper% lower% Snter ore % Table 3. Iteratve calculaton report Max Lne search Drectonal Frst-order f(x) Iter F-count constrant steplength dervatve optmalty e e e Table 4. The summary of calculatons table Cost Cost savngs per ton Percentage of cost savng per ton (yuan/t) (yuan/t) Fxed-rato method LP model % LFP model % Cost Fxed-rato method LP model 2 LFP 3model Fg.. Bar s the cost of Fxed-rato method. Bar 2 s the calculated cost of LP model. And bar 3 s the ron cost per ton of LFP model. The bar chart takes use data from Table 4.

7 An Applcaton of LFP Method for Snterng Ore Rato 29.43% 99.57% LP model Fg. 2. Pe s the ron cost of snterng ore computed by the LP model. Pe 2 s the cost savng percentage from the Fxed-rato method. 4.28% 95.72% LFP model Fg. 3. Pe s the ron cost per ton calculated by the LFP model. Pe 2 s the cost savng percentage from the Fxed-rato method. Takng the vector of zeros, X0, X as start pont to compute can obtan the same optmal X*. Calculaton report whch takng X as the start pont shown n Table 3. X0=(8,3,0,2,3,0,5,0,2,2,0,0,0,7,0,0,0,0,8,0,3,2,5,0,0,0,0,0,5,0)T, X=(8,2,0,2,5,0,5,0,,2,0,0,0,7,0,0,0,0,8,0,3,2,5,0,0,0,0,0,5,0)T, X*=(8,3,0,2,4,0,5,0,,2,0,0,0,7,0,0,0,0,8,0,3,2,5,7.2042,0,8.895,0,0,5,0)T. Step length and teratons ndcate that convergence of the quas-newton search algorthm perform well. The effectvty of the SQP algorthm to solve the quadratc form s already testfed n reference[4]. The result shows that the selecton of start pont can mpact the convergence rate to some degree but acqure the same soluton. The fnal optmzaton result s not affected. LFP model calculatons need not select the feasble rato as the ntal teraton pont. Ths method s more advanced than the Fxed-rato method. All content of chemcal composton of the optmal are wthn the boundary as shown n Table 2. That content of MgO and TFe reached the boundary reveals that these chemcal composton restrctons have effect on the optmal. Table 4 shows the result of calculatons of the tradtonal LP model, the LFP model and the Fxed-rato method. Accordng to Table 4, the ron cost per ton of Fxed-rato method s yuan per ton. The cost of LP model s yuan per ton. And the ron cost per ton of LFP model s yuan per ton. So LP model can save 7.89 yuan per ton. That s.43% of Fxed-rato method. LFP model can save 53.4

8 220 X. Cheng, K. Pan, and Y. Ma yuan. That s 4.28% of Fxed-rato method. Apparently the LFP model can save much more money for snterng. 4 Concluson The result of LFP model argues the followng conclusons. () Iron cost per ton of optmzaton models s smaller than Fxed-rato method, ndcatng that the model applcaton of scentfc and effectve. Optmzaton model program compute the rato of raw materals for snterng greatly smplfes the Fxedrato method whch takes use of tradtonal manual spreadsheet. (2) The results depct that the applcaton of LFP to calculate snterng ore rato s feasble and operatng well. (3) The optmal soluton obtaned by LP s only an approxmate soluton of Fxedrato method. From the perspectve of the assessng of cost savngs, the effect s less than the LFP model. (4) In addton of the LFP model based on constrant set methods, some tght constrants can be used as one of the goals to establsh another type of model based on fuzzy set theory, lke mult-objectve LFP model. For example the pure ron content only reached the lowest lmt constrants n the LFP model. It s natural to consder maxmzng TFe content as one of the goals. Acknowledgments. Ths research was supported by school of management of WUST and Xangtan ron and steel company. We wsh to thank the referees for ther very useful suggestons on the project. References. Wang, D.-q.: Applcaton of Lnear Programmng n Producton of Mxng Materals to Snte. Chna Metallurgcal. J. 5(8), 9 22 (2005); 线性规划在烧结矿配料中的应用. 中国冶金 2. Na, S.-r.: Iron Calculaton, pp Metallurgy Industry Press, Bejng (2005); 炼铁计算 3. Sun, W.-y., Xu, C.-x., Zhu, D.-t.: Optmzaton Method, pp. 5, Hgher Educaton Press, Bejng (2004); 最优化方法 4. Benson, H.P.: Fractonal programmng wth convex quadratc forms and functons. European Journal of Operatonal Research 73(2), (2006) 5. Gong, C.: Profcent n Matlab calculaton, pp Electroncs Industry Press, Bejng (2009); 精通 MATLAB 最优化计算