APPLICATION OF MULTIVARIATE LOSS FUNCTION FOR ASSESSMENT OF THE QUALITY OF TECHNOLOGICAL PROCESS MANAGEMENT
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1 , Brno, Czech Republc, EU APPLICATION OF MULTIVARIATE LOSS FUNCTION FOR ASSESSMENT OF THE QUALITY OF TECHNOLOGICAL PROCESS MANAGEMENT Abstract Josef TOŠENOVSKÝ ) Lenka MONSPORTOVÁ ) Flp TOŠENOVSKÝ ) )VŠB-Techncká unverzta Ostrava, )OPF SU Karvná josef.tosenovsky@vsb.cz The artcle s devoted to the problem of assessment of the qualty of technologcal process management, usng a multvarate loss functon. An attached example also llustrates the calculaton of the TSL crteron and the possblty of an optmal set-up of a process. Key words: loss functon, regresson analyss, optmzaton INTRODUCTION In prncple, the same problems to be resolved occur n most technologcal processes. The problems are related to nputs, outputs and the course of the process. Frst, n order for the process to run successfully, ts key nputs must be set approprately. Ths settng, however, must be based on the knowledge of requred outputs for whch one defnes qualty characterstcs Y and ther requred levels, or addtonally boundares wthn whch Y s must fall. Also, one tres to answer the queston of whch nputs X have the most sgnfcant nfluence on the output characterstcs Y before settng them up. At the same tme, montorng of both the nputs and outputs must be n place to ensure the process runs ts predetermned course. The course of the process s assessed so that the process can be further mproved or ts potental mperfectons corrected. In terms of organzaton, such problems may be consdered as part of the pre-producton stage (nputs selecton and ther set-up), producton stage or postproducton stage. A varety of statstcal methods s used for each of these stages. Nowadays, the desgn of experments (DOE) and control charts are the most effectve tools for the pre-producton plannng. The latter s used for process output montorng as well. In the post-producton stages, the so-called process capablty assessment s carred out, usng a varety of capablty ndces. Many statstcal methods are used n the descrbed stages. The least attenton s probably pad to the statstcal assessment of outputs, for whch t s usually only stated whether the characterstc Y falls wthn the prescrbed bounds. At the customers request, capablty ndces are farly often calculated as well. See [toš], for example, for further detals on these measures. A less known alternatve to capablty ndces s the concept of loss functon ntroduced by dr. Gench Taguch. Ths approach s nterestng for two reasons. It expresses the level of the process by a fnancal loss resultng from not achevng the pre-defned values of Y s. It also enables one to calculate the optmal set-up of the process under very general condtons nvolvng an arbtrary number of nput parameters X and an arbtrary number of outputs Y, whch are potentally functons of the nputs, Y = f(x,...,x k ). The outputs may also be mutually dependent. Ths stuaton s what ths artcle deals wth. UNIVARIATE LOSS FUNCTION Loss functon s based on the smple prncple that the customers have a certan dea about what they expect from the suppler, and the farther the result s from ths dea, the less the customers are satsfed. Formal
2 , Brno, Czech Republc, EU descrpton of ths prncple s embedded n the loss functon [] whch quantfes customer s dssatsfacton by a fnancal loss accordng to the equaton L( Y) T k( Y ), () T = target value of the qualty characterstc (the customer s dea), Y = trully acheved value of the qualty characterstc, Y-T = dfference between the realty and the dea, k = constant, L(Y) = fnancal loss of the customer due to not achevng T. The calculaton of L(Y) s not nterpreted only as the ablty to meet customer s demands, but also as the ablty of the supplers to set up ther processes correctly. Hgher L(Y) means lower qualty of the management, as values near zero confrm a perfect producton set-up. The constant k n equaton () follows the relaton A k, () d A = constant expressng costs (a usual loss resultng from exceedng the tolerance), d = tolerance A more nterestng measure than the ndvdual loss calculated for each separate product s the average loss resultng from exceedng the tolerance T. The average loss satsfes the equaton A EL( Y) s d s k., (3) s = varance of the observed qualty characterstc EL(Y) = average value of L(Y) Snce fndng the constant A mght be a problem, a standardzed loss functon SL(Y), see [], was ntroduced by the equaton Y T ( Y). USL LSL SL. () Ths functon doesn t requre the value of A. The resultng value of SL(Y) has no physcal dmenson, so that, for nstance, SL(Y) =,7 may be nterpreted as a 7% loss caused by the dfference Y T. MULTIVARIATE LOSS FUNCTION Dmensonless standardzed loss functon enables to sum values of dfferent qualty characterstcs, and thus enables to defne a multvarate loss functon for ndependent qualty characterstcs, formng a vector y = (Y,...,Y p ), and ther target values T, summarzed by the vector t = (T,...,T p ). The functon descrbed n [] has the form
3 , Brno, Czech Republc, EU p Y T TSL( y). USL LSL (5) If the Y s depend on the vector of varables x = (X,...,X m ), the relaton (5) can be rewrtten as p Y T TSL( y), USL LSL (6) Y (x) s the functon of the form Y = f(x,...,x m ). If the qualty characterstcs Y,...,Y p are mutually dependent varables, equaton (6) can be generalzed as n [3],.e. LY ( y t) T C( y t), (7) y(x) = vector of the qualty characterstcs Y whch depend on the vector of nputs x = (X,...,X m ), so that Y = f (X,...,X m ), t = vector of the target values, C = costs matrx (analogy to the cost constant A n relaton ()). The coeffcents c on the man dagonal of the matrx c c C (8) c c are calculated accordng to the equaton c ( USL LSL ) (9) If the varables Y and Y are ndependent, the matrx C s dagonal and c j = c j =. Example (data from [] are used here) Suppose the process has two outputs Y,Y and three nputs X,X,X 3, through whch the process s managed. Tab. Process specfcaton Inputs LSL Actual USL 5 X X 5 X3 Outputs LSL Target USL Y Y TSL 9,7689 Target values T and boundares not to be exceeded, the lower bounds LSL and the upper bounds USL, are prescrbed for the outputs. For the nputs X, only the lower and upper bounds LSL a USL are defned, and target values are replaced wth the currently used levels at whch the nputs are set up (tab.). These data are called the process specfcaton. The level of the settngs s assessed wth multvarate loss functon (6). Its use, however, requres us to fnd regresson functons that would descrbe the dependence of the outputs Y on the nputs X,...,X 3. These regresson functons are []
4 , Brno, Czech Republc, EU Y ( x x x) 9,639 6,679x,563x,5x3,3x3,875x, 5 Y ( x x x) 58,6,8357x,365x, 5 Now, substtutng these nto (6) for Y (x),y (x) and settng X =,X = 6 and X 3 = 5 gves Y (x) = 58,96 and Y (x) = 8,59. For the specfcaton T = 65, T = 8, LSL = 65, USL = 635, LSL = 75, USL = 85, the resultng TSL s TSL (58,96 65) (8,59 8).. (635 65) (85 75) Y T. USL LSL Y T. USL LSL Ths calculaton can also be performed wth matrces accordng to (7). In ths case, t follows that TSL Y T, Y T USL LSL 3 USL LSL , Y T Y T ,7689 The matrx form s more convenent as t enables to generalze the calculaton for the case when outputs are dependent,.e. when the matrx C contans non-zero elements c j. Ther calcuaton s descrbed, for nstance, n [3]. In case of non-zero coeffcents c c, the matrx calculaton has the form TSL so that USL LSL Y T, Y T c USL LSL c Y T Y T Y T Y T TSL.. c( Y T )( Y T ) USL LSL USL LSL Based on TSL, we may now compare the qualty of dfferent process settngs, and also solve the followng optmzaton problem: what settng of the nputs X,X a X 3 mnmzes TSL and causes the outputs Y to be close to the values T. TSL s very senstve to parameter changes: for nstance, settng x 3 = and leavng the other nputs wthout changes results n TSL = 5.3, or x = 5 gves TSL = 55,97. If the process was assessed wth capablty ndces, such as the most frequently used C pk, there would be no connecton wth the regulated nputs and no possblty to calculate the optmal settng.,
5 , Brno, Czech Republc, EU Tab. Optmzaton of nputs Inputs LSL Optmal USL 5 7,8 X 5 5, 75 X 6,9 X3 Outputs LSL Target USL Y Y CONCLUSION Usng loss functons to assess the manageral qualty turns out to be very sutable because not only does t assess the state of the management, but t also enables to calculate the optmal settngs. TSL s usable for an arbtrary fnte number of nputs and outputs, regardless of whether ndependence of the outputs s requred or not. At the same tme, assessng the manageral qualty wth the TSL calculatons s very smple, although optmzaton requres a specal software. TSL,9. - AKNOWLEDGEMENT Ths paper was elaborated wthn the frame of the specfc research project No. SP/ whch has been solved at the faculty of Metallurgy and Materal Engneerng, VŠB-TU Ostrava wth the support of the Mnstry of Educaton, Youth and Sports n the Czech republc. LITERATURE [] Kapur,K.C., Cho,B.R. Economcs Desgn of the Specfcaton Regon for Multple Qualty Characterstcs, IIE Trans.,996, 8(3), [] Suhr,R., Batson, R.G. Constraned Multvarate Loss Functon Mnmzaton. Qualty Engneerng,, 3(3), [3] Pgnatello,J.J.Jr. Strateges for Robust Multresponse Qualty Engneerng. IIE Trans,993, 5(3), 5-5. [] Taguch,G. Introducton to Qualty Engneerng. Asan Productvty Organzaton. Inc. Deaborn986,MI. [5] Tošenovský,J. Ekonomcké a technologcké hodnocení způsoblost procesů: Algortmy a řešené úlohy. Program CapaDemo. Ostrava: Dům technky 7, 3 s.
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