RACKGROlJND STUDY
I. Recognition of and statement orthe problem
3. Selection of the Response Variable
In selecting the response variable, the experimenter should be certain that this vanable really provides useful information about the process under study. Most often, the average or standard deviation (or both) of the measured eharactenstic will be the response variable.
Multiple response are not unusual. Gab>ecapability (or measurement error) is also an Important factor. If gage capability is poor. thcn only relatively large faelor elfects WIllbe dctcctcd by the experiment or additional replicatlun WIllbe required.
4 ChOIceof the experimental Design:
It the first three steps are done correctly, this step is relatively easy. ChOice of design
involves thc consideration of sample sizc (number of replicates), the selection of a suitable
run order for the esperimeotal trials, and the detenninatlOn of whether or not blocking or
other randomization restrictions are lOvolved.In selecting the design, it ISImportant to keep
the expenmenlal objectIves In mind. In many engineering experiments, some of the factors levels will result in different valucs for the rcsponse, 11is important to ident1fy the those lilctors which causes thIs deference and in estimating the magnitude of the responsc change,
5, Performmg the Expenmenl.
Whcn running the experiment, it is vital to monitor the process care1i.tl1yto ensure that evel)1hing is heing done according to plan, Errors in experimcntal procedure at this stagc will usually destroy experimental validity. Up-front planning is crucial to success. It is ClIsy to underestimate thc logistical and planning aspects of running a desired experiment in a
complex manufacturing or rcsearch and development environment,
6. Data Analysis:
Statistical methods should be used to analyze the data so that results and conclusions are.
ohJective mther than judgmental in nature. If the experiment nas been desib"'ed CllITe\.-1lyand if 1t has been performed according to the design, then the stillsca! methods required are not elaboratc, There are many excellent softwarc pac)::agc dcsigned to assist m data
Illtcrprctation. Residual analysts and model adequacy chceking arc also important analysis techniqucs,
7 Conclusion and Recommendations:
Once the data have been analyzed, the experimenter musl draw practical conclusions about the results and recommend a course of action, Graphical methods arc often usdul in this stage, particularly in presentmg the results to others, Follow-up runs and conllrmatlon testmg
should also be performed to validate the conclusions from he experiment.
, Throughout this entire process, it is important to keep in mind that experimentation 1San
important part of the learning process, where we tentatively formulate hypotheses about a
system, perform expenments to investigate these hypotheses and on the basis of the results formulate new hypotheses and so on.
A successful experiment requires knowledge of the important factors, the ranges over which these factors should be varied, the appropTlate number of levels to usc and the proper units of measurement l'or these variables,
~'8ctori81Uesian:
Many expenments involve the study of the effects of two or more factors, In general, factorial deSigns arc most clliclerrt for this type of experiment. By a factorial desIgn, we mean that in each complcte trial or rephcatlon of the experiment all possible combinations of the levels of the factors are investigated. When factors are arranged in a factorial desi~,'n,they arc ollen said to be crossed.
The effect of a factor is defined to be lhe change in response produced by a change III the level of the factor. This is frequently called a main effect because It refers to the primary
factors of lIlterest in the experiment.
In some experiments, we many find that the ditlbcnce IIIresponse between thc levels of one factur is not thc same al all levels of the other factors. When this occurs, there is an interactlon between the factors, When an interaction is large, the corresponding main effects have hltle practical meaning.
The sImplest types or factorial designs involve only two factors or SelS or treatments, There are a levels offactor A and B levels of factor 13~ and these arc arranged IS a factorial design;
that is, each replicate of the experiment contains all ab treatment combinations In general, there are n rephcates.
The factorial designs have several advantage s. They arc more efficient than one-factor-al.a- time expenments. l'urthermorc, a factorial desIgn I, necessary when mteractlOns may be
present to avoid misleading conclusions. l'inally, factorial designs allow the effects of a factor to be estimated at several levels of the other factors, yieldmg eoncluslons that are valid
over a range of experimental conditions.
The
ZK"llctorial Design:
factonal desIgn arc widely used in experiments involving several factors where It is neees'Kiry to study the joint effect of the factor, on a response. Proportional Data method,
Approximate mcthod~. The exact method ctc arc used for the analysis of factorial designs.
Howcver there are several special cases of the general factorial design that arc important because they arc widely used in research work and also because they form the basIS of other deSlb'llSof considerable practical value.
The most Important of these special cases IS that ofK factors, each at only two levels. These level may be quantitative, such as two values of temperature, pressure, time or tbey may be quahtative, pressure, time or they may be quahtative, such as two machines, two operators the 'high' and "low' levels of a factor or perhaps the presence and absence of a factor. A complete replicate of such a design requires, 2 x 2 x 2 x x 2 ~ 2K observalJons and is called a
i-:
factorial design.To analYSIS of this llScful series of designs, It lS assume that the factors may iixed or the designs are completely randomized or the usual normahty assumptions arc sallsfied.
The 2Kdesign is particularly useful in the early stagcs of the experimental work, whco there are likely to be many factors to be investigated. It provides the smallest number of runs with which K facto4rs can be studied in a complete factorial design, Because there are ooly two level, foreach factor, we must ll..%umethat the response is approximately hnear over the
range of the factor levels chosen. Similarly for K factors each of whIch three levels can be rcpre,ent by 3K
2.4 RESPONSE SURFACE METHODOLOGY (RSM):
Response surface methodology or RSM is a collection of mathematical and statistical techniques that are useful for the modehng and analysis of problems in which a response of interest is internal by several variables and the objective is to optlmi~e thIS response,
In most RSM prohlems, the form of the relationship between the response and the independent vanahles is unknown. Thus, the first step 1TIRSM IS to find a suitable
approximation for the true functional relationshIp bet,,"een response and the set of independent variables. Usually, a low-order polynomial m some region of the independent
variables is employed. If the response is well modeled by a linear function of the mdependent variables, then the approximating function is the first order model.
If there is curvature is the system, then a polynomial of hIgher degree must be used, such as the !.econd-order model.
Almost all RSM problems utilize one or both of these approximating polynnmials. Of course, it is unlikely that a polynomial model will be a Tea,onable approximation of the true functional relationship over the entire space of the independent variables hut for a relatively small region they usually work qulte well,
The method of least squares is used to eS\lmate the parameTers In the approximating polynomials. The response surface analysis then done in terms of the fitted surface. If the fitted surface is an adequate approximation of the true response fUn,,'tlOn,then analysis of the fitted surface ""ill be approximately equivalent to analysis of the actual system. The model
parameters can be estimated most effectively if proper experimental designs arc used to coIled data.
The eventual objective of RSM is to determine the optimum operating conditions for the system or to determine a region of the factor space in whICh operating ,~irications arc satisried RSM IS not used primarily to gain understanding of the physical mechanism of the sy,tem. although RSM may assist 10the gaining of such knowledge. Furthermore, note that nptllnum in RSM IS used in a special sense. The hill chiming procedures ofRSM guarantee convergence to a local optimum only.