Design of Experiments and Sequential Analysis used as an Optimization Technique of a Refrigerator Cabinet

Transcription

1 Desgn of Experments and Sequental Analyss used as an Optmzaton Technque of a Refrgerator Cabnet Axel J. Ramm Whrlpool Corporaton, Jonvlle, Santa Catarna, Brazl. Abstract Global compettve pressures are causng organzatons to fnd ways to better meet the needs of ther costumers, to reduce costs, and to ncrease productvty. Nowadays the applance ndustry, partcularly refrgeraton, s nserted n a very compettve scenaro and there s a bg effort to acheve cost reductons wthout causng qualty degradaton n order to survve n the core busness. Regardng varablty n manufacturng process, refrgeraton cabnets are the most affected components. Bascally they are bult of external sheet metal parts, plastc nner lners and the nternal volume s flled wth polyurethane foam. In ths paper t wll be presented some technques to better represent refrgerator cabnets fnte element modelng. Statstcal tools wll be used to help n numercal model calbraton and sources of varaton evaluaton. Once havng a calbrated model, an optmzaton analyss based on Desgn of Experments technque wll be carred out n ANSYS. Therefore Desgn of Experments and sequental analyss wll be drectly compared to subproblem approxmaton method of ANSYS optmzaton routne usng APDL (ANSYS Parametrc Desgn Language), whch s an essental step n the optmzaton process. The projected cost savngs covered by ths paper contents are about US$ ,00 per year and the generated knowledge wll be extremely useful for refrgerator cabnet desgn gudelnes. Introducton Reducng the tme and cost of product and process development s a key concern of today s compettve ndustral envronment. Snce there s ncreasng evdence n refrgerator cabnets cost reductons t s requred to plan a systematc and complete study to acheve ts structural optmzaton savng n the product cost whle not decreasng ts qualty. The man reason for guaranteeng cabnet robustness s that the lack of stffness can generate nsulaton loss due to door msalgnment and aesthetcs degradaton, affectng drectly the perceved qualty by the consumers. Concernng the cabnet robustness, the current producton cabnets are approved under a Product Test Method whch establshes a dstorton lmt for a fully assembled refrgerator cabnet at specfed angular door openngs of ts fully loaded door. In the same manner, there s a specfcaton lmt for door deflecton relatve to the cabnet (door drop). Once the door s loaded there occurs a cabnet dstorton and the door ends up movng downwards referentally to ts orgnal poston. In long perods of tme ths phenomenon can be ncreased by polyurethane foam vscoelastc behavor whose man consequence s the occurrence of creep, that s, an ncreasng deformaton under sustaned load and the rate of stran dependng on the stress. Therefore, the man focus was gven to the response varable door drop. In the other hand t s very complcated to evaluate an accurate value of cabnet deflecton or door drop snce there s a lot of varaton between products of the same model. The manufacturng process s surrounded of a great number of noses that affect drectly the cabnet stffness. In the same way, the product test condtons can sgnfcantly nfluence n the structural assessment results.

2 The dspersal envronment make t only possble to dscern farly changes n cabnet stffnesses snce the dfferentaton wll effectvely occur f the data results for a modfed desgn retan robust from all sources of varaton. The ntal step of the proposed study wll be focused n sources of varaton apprasal n order to determne data dsperson (standard devaton from the mean results under a relablty degree) and to provde condtons to plan systematc laboratory tests to help n fnte element model calbraton. Consderng all the relevant structural components and fxatons, t was carred out a modelng of an overall refrgerator cabnet fnte element. In addton, t was modeled the refrgerator and freezer doors wth the purpose to provde proper loadng dstrbuton and serve as reference to evaluate the door drop due to cabnet dstorton. A proper fnte element model valdaton s essental to start the cabnet optmzaton study because knowledge about varaton has been ganed and now the numercal model structural behavor s quanttatvely n accordance wth the real condtons. The essental technque used to approach the study was the sequental strategy based on Desgn of Experments. The Desgn of Experments s a technque used to plan experments, consstng n a seres of tests of a system carred out by changng levels of factors and background varables and an observaton of the effect of that change on the response varable. The sequental nature of learnng should be consdered n plannng experments. As knowledge s ganed sequentally, experments wll be refned by usng new levels for the factors (nference space), addng some new factors or elmnatng the neglgble ones. To set up a factoral desgn, the nvestgator determnes the factors to be studed and the levels for each. A full factoral desgn conssts of all possble combnatons of the factors and levels, makng t possble to obtan the maxmum experment resoluton, snce there are not any alasng between man effects and hgher-order nteractons. The number of runs requred by a full factoral desgn ncreases geometrcally as the number of factors ncreases and an ncreasng amount of the data s used to estmate hgher-order nteractons. These nteractons are usually neglgble and therefore are of lttle nterest to the expermenter. Fractonal factoral desgns are an mportant class of expermental desgns that allow the sze of experments to be kept practcal whle stll enablng the estmaton of mportant effects. The negatve pont of usng fractonal factorals s the confoundng of factors and ts nteractons, dependng on the experment resoluton. All of the most mportant steps concernng Desgn of Experments sequental analyss are presented n ths paper snce fractonal factoral desgns and parameters selecton up to the fnal optmum soluton. Afterwards ANSYS tradtonal optmzaton technques specfcally Subproblem Approxmaton Method wll be used as a comparson of the study based on sequental analyss. Refrgerator Cabnet Geometry The product consdered n ths analyss s a 450 lters double door refrgerator whch has been n producton for over two years. The refrgerator cabnet constructon comprses bascally external sheet metal parts, polyurethane foam fllng and nternal plastc lners. The metal parts are: wrapper (wth roll-formed renforcements), back panel, bottom deck, ntermedary ral, front ral, compressor mountng plate, glder rals and hnges (upper, center and bottom). The cabnet nner lners are made of plastc materal adherng to the polyurethane foam wth outer layers and nternally formng the refrgerator and freezer compartments. Fgures 0 and 02 show some of the refrgerator cabnet most mportant parts.

3 Fgure 0. Cabnet Components Front Vew

4 Fgure 02. Cabnet and Doors Assembly Opened and Closed Doors Model Descrpton and Modelng Technques The fnte element model s composed of sold and shell elements. The polyurethane foam, EPS mullon, hnges and levelers were modeled wth sold tetrahedrons elements (Sold 45) and other parts, bascally composed of thn plates, were modeled wth Shell 8 elements. For all the connectons of clnch jonts and screws t was adopted the use of the element beam 8. All the materals propertes were consdered as lnear sotropc and the nput data was pcked from laboratory tests and suppler techncal reports. Partcularly for polyurethane foam, once t s susceptble to a hgh level of varaton (poston to poston n the cabnet and due to njecton process fluctuaton), the elastc propertes were evaluated n a laboratory test devce and elastcty modulus was ncluded as a factor n the analyss consderng the levels as the mean of ts mnmum and maxmum values. In order to adequately represent the real loadng condtons, t was evaluated the mass of all cabnet parts. All the evaluated masses were dstrbuted n ts proper poston and eventually the total mass and the gravty center of the real fully assembled cabnet was very smlar to the fnte element model one. On ts bottom base the cabnet was constraned smulatng a real operatng condton. Consderng that all cabnet supports were able to rotate, only translatons were restrcted at three base locatons. The fgure 03 shows the constrants applcaton.

5 Fgure 03. Boundary Condtons. The door attachments wth the hnges were modeled wth constrant equatons whch allowed controllng translatons and rotatons to be correctly transferred from the doors to the cabnet. Thus, t was possble to mpose door support only at ts bottom poston and release rotatons from hnge pns. Therefore t was necessary to set two addtonal constrants at the upper lner of each door to elmnate rgd body moton (fgure 03). Regardng the Product Test Method, the masses correspondng to the cabnet and doors loadng were dstrbuted at each own partcular locaton. The acceleraton due to gravty was appled as load boundary condton. The fgure 04 shows the masses dsposton n the cabnet and doors lners whch were appled to the fnte element model.

6 Fgure 04. Cabnet Loadng (Mass) Dstrbuton. For the door drop assessment t was consdered the analytcal model wth ts doors at the closed poston. The response varable was establshed as the subtracton of door drop and cabnet drop wth the purpose to consder only the relatve dsplacement between door and cabnet. The fgures 05 and 06 show the evaluaton scheme of cabnet door drop.

7 Fgure 05. Analyzed Fnte Element Model: Cabnet Dstorton due to Door Loadng. Fgure 06. Cabnet Door Drop Evaluaton.

8 Sources of Varaton Evaluaton Only when a numercal model s calbrated t can be used to perform a correct assessment of ts structural performance and afterwards run an optmzaton analyss. The frst step of a model calbraton was the defnton of some possble sources of varaton whch could generate errors to test data collecton. From the Test Process Map t was elected four factors consdered as the most nfluencng ones: dfferent products of the same model, test setup, door openng velocty, tme to read data and dfferent measurements. The Sample Three Dagram s presented n fgure 07. Fgure 07. Experment Sample Tree Dagram. The total sum of cabnet lateral dsplacement (sway X) was consdered to be the response varable. Regardng the test condtons, ntally the doors were closed (zero poston), thus the sgnal starts to be saved and hence the doors were opened. The fgure 08 shows an example a sample of sgnal collected from one experment run. The sum modulus of the sgnal peak and valley s consdered to be the cabnet Sway X. Fgure 08. Sgnal Collected from a Laboratory Test. In the Sample Three Dagram t can be realzed that the data collecton started wth the random choce of four products of the same model. Each product was tested under two dfferent setups (put the cabnet at the test poston), two door openng veloctes (fast and slow) and the tme for readng the cabnets deflecton was computed at 5 seconds and 3 mnutes respectvely. Each confguraton was measured three tmes, totalng a number of 96 measures.

9 For the data collecton analyss t was used an mportant statstcal tool: Control Charts. Control Charts for Averages and Ranges are a smple and effectve way to present data for a use as a bass for process stablty evaluaton, makng t possble to track both the process level and process varaton at the same tme, as well as detect the presence of specal causes. The fgure 09 shows the Sample Mean (Averages) and Sample Range chart for dsplacement measurement at the cabnet upper lateral corner. Xbar-R Chart of Sway X 5,0 Sample Mean 4,8 4,6 4,4 4, Sample UC L=4,6283 _ X=4,5094 LC L=4,3904 0,3 UC L=0,2993 Sample Range 0,2 0, _ R=0,62 0,0 LC L= Sample Fgure 09. Sample Means Average and Sample Range Charts. The process s sad to be SPC (stable, predctable and consstent) f all the ponts exstng n the Range Chart are nsde the control lmts. Analyzng the Sample Range Chart n fgure 09 t can be checked out the process stablty. An mportant ssue to carry out s the MSE (Measurement System Evaluaton). The measurement system s sad to have enough dscrmnaton f there are as much as requred levels of resoluton n the Range chart. The capacty to evaluate measure between subgroups occurs f there are more than 50% of the ponts out of the control lmts n the Sample Mean chart. Lookng at Sample Mean Chart n fgure 09 t can be realzed that these requrements are also verfed for the measurement process. Fgure 0 shows a very mportant graph - Varablty Chart. Varablty Chart plots the mean for each level sde by sde. Along wth the data, you can vew the mean, range, and standard devaton of the data n each category, seeng how they change across the categores. Ths graph s also very helpful to look for systematc effects between factors and levels. A very nterestng thng to observe s that always at the tme of 3 mnutes the measures are greater than at the tme of 5 seconds t can be explaned by the creep effect of the polyurethane foam nsde the cabnet.

10 Fgure 0. Varablty Chart and Components of Varaton (COV) Calculaton. To quantfy the contrbuton of each factor t s employed a varance components analyss (COV). Performng a COV study t could be checked that the varaton between products s the most sgnfcant over the other selected factors. Ths ssue ndcates that no matter levels of setup, velocty, tme and measure levels are confgured, always the product-by-product varaton wll stand out among them. Check that the bottom part of fgure 0 dsplays that 7,7% of total varaton n the response varable s due to dfferences among products (manufacturng). Wth the purpose to assure better relablty n data for numercal model calbraton t was decded that the number of samples for dfferent products of the same model had to be ncreased. Therefore 6 products were randomly selected from the manufacturng lne and tested consderng only the factors: products and measures. The fgure shows the control charts for the assgned experment.

11 Xbar-R Chart of X Sample Mean 4,8 4,6 4,4 4,2 4,0 UCL=4,569 _ X=4,442 LC L=4, Sample 3 5 0,3 UC L=0,32 Sample Range 0,2 0, _ R=0,247 0,0 LC L= Sample 3 5 Fgure. Sample Means Average and Sample Range Charts. The fgure 2 shows the data dstrbuton from the collected samples wth normal ft dsposal. 2 0 Hstogram of X Normal Mean 4,442 StDev 0,2842 N 48 Frequency ,9 4,2 4,5 X 4,8 5, Fgure 2. Normal Ft Dstrbuton for the Collected Samples. The calculated mean and standard devaton were 4,442 mm and 0,2842 mm respectvely. Or better, Sway X = 4,442 ± 0,2842.

12 Numercal Model Calbraton For numercal model calbraton t was selected the product wth the value of the response varable (sway X) nearest to the samples mean. The product support at the rear corner of the base on the hnge sde was removed because ths locaton lfts off the floor endng up not carryng any load onto the floor. From ths step on the response varable consdered n the analyss was Door Drop, as shown n fgures 05 and 06. Intally the cabnet was set on the test platform and dsplacement transducers were postoned at the specfed locatons. An addtonal gauge was placed on the cabnet upper poston wth the purpose to measure cabnet door drop. In order to smulate the numercal model loadng boundary condton t was removed all the loaded shelves from the refrgerator, the doors were closed and all the dsplacement gauges were put at zero reference. The acquston data system was set to start the sgnal recordng. The refrgerator doors were open and the shelves wth dstrbuted mass were carefully placed at ther orgnal postons. Thus, the doors were closed and the sgnal was stopped to be recorded after 20 seconds of measurement. The fgure 3 shows the output data sgnal: n blue t s represented the lateral dsplacement and n red the back dsplacement. Fgure 3. Data Sgnal from a Laboratory Calbraton Test. All the measurements were repeated fve tmes n order to make sure the acqured data was relable. These values ndcated a good correlaton wth the fnte element model data - see fgure 4.

13 Fgure 4. Correlaton Between Real Data and Numercal Model. Desgn of Experment and Sequental Analyss Optmzaton The frst optmzaton approach was to execute sequental vrtual DOEs (Desgn of Experments). The factors and levels selecton were based on 6-Sgma tools. 6-Sgma s a framework based on QFD (Qualty Functon Deployment) method whose phlosophy s the use of engneerng knowledge and crtcal thnkng to evaluate the desgn factors that could possbly affect the response varable. The goal was to determne the effects of the cabnet components desgn on the door drop. These relatonshps are genercally termed Y = f(x) at Whrlpool Corporaton. The Y beng the dependent varable (.e. door drop n ths case) and the x's beng the ndependent varables,.e. component geometry and thckness (stffnesses). It was also used a specalzed software n statstcal tools (Mntab) to calculate the sgnfcance of the assgned factors and the nteractons among them. All the fnte element analyss executed durng ths procedure were performed n ANSYS envronment takng advantage of APDL batch mode to run the DOE s treatments. The desgn varables consdered n the analyss were: SCR (screw fxaton between ntermedary ral and wrapper front flange), BP (back panel stffness), IR (ntermedary ral thckness), CMP (compressor mountng plate thckness), GR (glder ral thckness), GRL (glder ral lateral rbs), BD (bottom deck thckness) and FR (front ral thckness). The state or response varable used to control the cabnet stffness was door drop under statc load applcaton and the cost functon to be mnmzed was cabnet sheet metal parts mass. Assumng that the most promnent factor responsble for products manufacturng varaton was polyurethane foam stffness, another factor, PU (polyurethane foam stffness) was consdered n the analyss. The ntal desgn was a fractonal factoral desgn wth 0 factors (desgn varables) and 6 runs, resultng a resoluton III experment - man effects confounded wth second order effects. The FRD (Factors Relatonshp Dagram) s shown n fgure 5.

14 Fgure 5. Factors Relatonshp Dagram for the Frst DOE. The graph presented n fgure 6 s called Normal Probablty Plot of the effects. The ponts that do not fall near the lne usually sgnal mportant effects. Important effects are larger and further from the ftted lne than unmportant effects. Unmportant effects tend to be smaller and centered around zero. Fgure 6. Normal Probablty Plot for the Frst DOE.

15 The Pareto chart (fgure 7) allows you to look at both the magntude and the mportance of an effect. Ths chart dsplays the absolute value of the effects, and draws a reference lne on the chart. Any effect that extends past ths reference lne s potentally mportant. Fgure 7. Pareto Plot for the Frst DOE. A Man Effects Plot s a plot of the means at each level of a factor. Mntab plots the means at each level of the factor and connects them wth a lne. A man effect occurs when the mean response changes across the levels of a factor. You can use man effects plots to compare the relatve strength of the effects across factors. Fgure 8. Man Effects Plot for the Frst DOE.

16 Regardng to ncrease the data relablty, t was decded to ncrease the resoluton, performng a fold-over (mrror) of the experment. Therefore, analyzng both experments (orgnal, fgure 5 and fold-over, fgure 9) together, the output resoluton was ncreased to IV, provdng only alasng of man effects and three way nteractons. Fgure 9. Factors Relatonshp Dagram for the Fold-Over Experment.

17 Fgure 20. Normal Probablty Plot for the Fold-Over Experment. As the experment resoluton has been ncreased, the most sgnfcant dfference was that the factor K (confoundng structure K = AB = CE = DH = FG) whch appeared as sgnfcant n the frst DOE, n fact, was revealed to be the nteracton AB n the fold-over experment (resoluton IV). Fgure 2. Pareto Plot for the Fold-Over Experment.

18 Fgure 22. Man Effects Plot for the Fold-Over Experment. Consderng a mean value for PU (polyurethane foam stffness) and settng asde rrelevant factors for the response varable (door drop) such as compressor mountng plate thckness (CMP), glder ral thckness (GR), glder ral lateral rbs (GRL), bottom deck thckness (BD) and front ral thckness (FR), t was decded to run a full factoral experment (resoluton ). Fgure 23. Factors Relatonshp Dagram for the Full Factoral Experment.

19 Fgure 24. Normal Probablty Plot for the Full Factoral Experment. Fgure 25. Pareto Plot for the Full Factoral Experment.

20 Fgure 26. Man Effects Plot for the Full Factoral Experment. The chart presented n fgure 27 s called Interactons Plot. An Interactons Plot s a plot of means for each level of a factor wth the level of a second factor held constant. An nteracton between factors occurs when the change n response from the low level to the hgh level of one factor s not the same as the change n response at the same two levels of a second factor. That s, the effect of one factor s dependent upon a second factor. You can use nteractons plots to compare the relatve strength of the effects across factors. Fgure 27. Interactons Plot for the Full Factoral Experment.

21 The most senstve factor regardng mass (cost) s the cabnet wrapper. There s an ncrease of 5,2% n cabnet lateral dsplacement and 3,4% n door drop f t s decded to reduce the thckness from 0,6mm to 0,55mm (8,3%). The presence of a screw connector between ntermedary ral and wrapper was the most sgnfcant factor to retan the door drop at a reasonable level. To compensate the ncrease of door drop due to wrapper thckness reducton (3,4%) t was studed the possblty of usng two screw connectors between the wrapper front flanges and the ntermedary ral. Ths desgn change generated a reducton of 2% n door drop, makng up for the wrapper stffness loss. APDL Optmzaton Technques The ANSYS program offers two optmzaton methods to accommodate a wde range of optmzaton problems. The subproblem approxmaton and the frst order method. The subproblem approxmaton method can be descrbed as an advanced zero-order method n that t requres only the values of the dependent varables (objectve functon and state varables), and not ther dervatves. There are two concepts that play a key role n the subproblem approxmaton method: the use of approxmatons for the objectve functon and state varables, and the converson of the constraned optmzaton problem to an unconstraned problem. The converson s done by addng penaltes to the objectve functon approxmaton to account for the mposed constrants. For ths method, the program establshes the relatonshp between the objectve functon and the DVs by curve fttng. Ths s done by calculatng the objectve functon for several sets of DV values (that s, for several desgns) and performng a least squares ft between the data ponts. The resultng curve (or surface) s called an approxmaton. Each optmzaton loop generates a new data pont, and the objectve functon approxmaton s updated. It s ths approxmaton that s mnmzed nstead of the actual objectve functon. State varables are handled n the same manner. An approxmaton s generated for each state varable and updated at the end of each loop. The search for a mnmum of the unconstraned objectve functon approxmaton s then carred out by applyng a Sequental Unconstraned Mnmzaton Technque (SUMT) at each teraton. The frst step n mnmzng the constraned problem s to represent each dependent varable by an approxmaton, represented by the ^ notaton. For the objectve functon, and smlarly for the state varables, ^ ( f x) f ( x) + error ^ ( g x) g( x) + error ^ ( h x) h( x) + error ^ ( w x) w( x) + error The most complex form that the approxmatons can take on s a fully quadratc representaton wth cross terms. Usng the example of the objectve functon, ^ f = a0 + n a x + n n j b x x j j

22 You can control curve fttng for the optmzaton approxmatons. You can request a lnear ft, quadratc ft, or quadratc plus cross terms ft. By default, a quadratc plus cross terms ft s used for the objectve functon, and a quadratc ft s used for the SVs. A weghted least squares technque s used to determne the coeffcent, a and b j. Wth functon approxmatons avalable, the constraned mnmzaton problem s recast as follows: ^ ( Mnmze f = f x) Subject to _ x x x ^ g ^ x) g _ ( =,2,3,,n) ( + α ( =,2,3,,m ) ^ h β h ( x) ( =,2,3,,m 2 ) w ^ _ γ w x) w + ( γ ( =,2,3,,m 3 ) The next step s the converson from a constraned problem to an unconstraned one. Ths s accomplshed by means of penalty functons, leadng to the followng subproblem statement. Mnmze n m ^ m ^ m F( x, pk ) = f + f0 pk X ( x ) G( g = ) H ( h ) 2 3 = = = ^ W ( w ) n whch X s the penalty functon used to enforce desgn varable constrants; and G, H, and W are penalty functons for state varable constrants. The reference objectve functon value, f 0, s ntroduced n order to acheve consstent unts. A sequental unconstraned mnmzaton technque (SUMT) s used to solve the equaton above each desgn teraton. The subscrpt k above reflects the use of sub teratons performed durng the subproblem soluton, whereby the response surface parameter s ncreased n value (p < p 2 < p 3 etc.) n order to acheve accurate, converged results. At the end of each loop, a check for convergence (or termnaton) s made. The problem s sad to be converged f the current, prevous, or best desgn s feasble and any of the followng condtons are satsfed: The change n objectve functon from the best feasble desgn to the current desgn s less than the objectve functon tolerance; The change n objectve functon between the last two desgns s less than the objectve functon tolerance; The changes n all desgn varables from the current desgn to the best feasble desgn are less then ther respectve tolerances; The changes n all desgn varables between the last two desgns are less than ther respectve tolerances. Convergence does not necessarly ndcate that a true global mnmum has been obtaned. It only means that one of the four crtera mentoned above has been satsfed. Therefore, t s your responsblty to determne f the desgn has been suffcently optmzed. If not, you can perform addtonal optmzaton analyses.

23 Subproblem Approxmaton Method Procedure The same problem prevously assgned by Desgn of Experments and Sequental Analyss was submtted to ANSYS Subproblem Approxmaton Optmzaton. The Desgn Varables consdered n the analyss were: SCR (screw fxaton between ntermedary ral and wrapper front flange), BP (back panel stffness), IR (ntermedary ral thckness), CMP (compressor mountng plate thckness), GR (glder ral thckness), GRL (glder ral lateral rbs), BD (bottom deck thckness) and FR (front ral thckness). The State Varable was Door Drop and the Cost Functon to be mnmzed was Cabnet Mass. The results of ANSYS Subproblem Approxmaton technque are lsted n fgure 28 n comparson of the results obtaned by Desgn of Experments and Sequental Analyss. Fgure 28. Comparson of DOE Optmzaton Technque and Subproblem Approxmaton Method. Conclusons The study startup was drected to evaluate sources of varaton from manufacturng and structural tests whch could nfluence n the response varables. As knowledge about varable was obtaned, t was carred out refrgerator cabnet fnte element modelng assocated wth real calbraton data. Desgn of Experments technque s a very powerful tool to learn about the response varable senstvtes n a low cost manner. The Sequental Analyss makes t possble to retan nformaton about factor effects and nteractons over the experments. Ths feature s crucal to apprase elmnaton of non-sgnfcant factors, robust desgn guded by factors nteractons and ncrease n experments resoluton. The results gven by the two optmzaton approaches were very smlar, as depcted n fgure 28. However each method was characterzed of ndvdual advantages. The fnal optmzed desgn provded 5% of mass savng per product, resultng n a cost reducton of about US$ ,00 per year.

24 References [] WHEELER, D. J., CHAMBERS D. S., Understandng Statstcal Process Control, Second Edton, SPS Press, Knoxvlle, Tennessee, 992; [2] MINITAB, Mntab Statstcal Software, Release 3.3 for Wndows, 2000; [3] MONTGOMERY, D. C., Desgn and Analyss of Experments, John Wley & Sons, Inc New York, 997; [4] Manual ANSYS, Desgn Optmzaton, ANSYS Inc., 2005; [5] ARORA, J. S., Introducton to Optmum Desgn, McGraw-Hll, 989; [6] VANDERPLAATS, G. N., Numercal Optmzaton Technques for Engneerng Desgn wth Applcatons, McGraw-Hll, 984.