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《发酵与生物工程手册》(英文版)4 Statistical Methods For Fermentation Optimization

A common problem for a biochemical engineer is to be handed a microorganism and be told he has six months to design a plant to produce the new fermentation product. Although this seems to be a formidable task, with the proper approach this task can be reduced to a manageable level. There are many ways to approach the problem of optimization and design of a fermentation process,
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Statistical methods For Fermentation Optimization Edwin O. Geiger 1.0 INTRODUCTION A common problem for a biochemical engineer is to be handed a microorganism and be told he has six months to design a plant to produce th new fermentation product. Although this seems to be a formidable task, with the proper approach this task can be reduced to a manageable level. There are many ways to approach the problem of optimization and design of a fermentation process. One could determine the nutritional requirements of the organism and design a medium based upon the optimum combination of each nutrient, i.e., glucose, amino acids, vitamins, minerals, etc. This approach has two drawbacks. First, it is very time-consuming to study each nutrient and determine its optimum level, let alone its interaction with other nutrients. Secondly, although knowledge of the optimal nutritional require- ments is useful in designing a media, this knowledge is difficult to apply when economics dictate the use of commercial substrates such as corn steep liquor, soy bean meal, etc, which are complex mixtures of many nutrie 2.0 TRADITIONAL ONE- VARIABLE-AT-A-TIME METHOD The traditional approach to the optimization problem is the one- variable-at-a-time method In this process, all variables but one are held constant and the optimum level for this variable is determined. Using this

Statistical Methods For Fermentation Optimization Edwin 0. Geiger 1.0 INTRODUCTION A common problem for a biochemical engineer is to be handed a microorganism and be told he has six months to design a plant to produce the new fermentation product. Although this seems to be a formidable task, with the proper approach this task can be reduced to a manageable level. There are many ways to approach the problem of optimization and design of a fermentation process, One could determine the nutritional requirements of the organism and design a medium based upon the optimum combination of each nutrient, i.e., glucose, amino acids, vitamins, minerals, etc. This approach has two drawbacks. First, it is very time-consuming to study each nutrient and determine its optimum level, let alone its interaction with other nutrients. Secondly, although knowledge of the optimal nutritional require￾ments is useful in designing amedia, this knowledge is difficult to apply when economics dictate the use of commercial substrates such as corn steep liquor, soy bean meal, etc., which are complex mixtures of many nutrients. 2.0 TRADITIONAL ONE-VARIABLE-AT-A-TIME METHOD The traditional approach to the optimization problem is the one￾variable-at-a-time method. In this process, all variables but one are held constant and the optimum level for this variable is determined. Using this 161

162 Fermentation and Biochemical Engineering Handbook optimum, the second variable's optimum is found, etc. This process works if, and only if, there is no interaction between variables. In the case shown in Fig. l, the optimum found using the one-variable-at-a-time approach was 85%, far from the real optimum of 90%. Because of the interaction between the two nutrients, the one-variable-at-a-time approach failed to find the true optimum. In order to find the optimum conditions, it would have been necessary to repeat the one-variable-at-a-time process at each step to verify that the true optimum was reached. This requires numerous sequential experimental runs, a time-consuming and ineffective strategy, especially when many variables need to be optimized. Because of the complexity of microbial metabolism, interaction between the variables is inevitable, espe- cially when using commercial substrates which are a complex mixture of many nutrients. Therefore, since it is both time-consuming and inefficient, the one-variable-at-a-time approach is not satisfactory for fermentation development. Fortunately, there are a number of statistical methods which will find the optimum quickly and efficiently. 3.0 EVOLUTIONARY OPTIMIZATION An alternative to the one-variable-at-a-time approach is the technique of evolutionary optimization. Evolutionary optimization(EVOP), also known as method of steepest ascent, is based upon the techniques developed by Spindle, et al. Q] The method is an iterative process in which a simplex figure is generated by running one more experiment than the number of variables to be optimized. It gets its name from the fact that the process slowly evolves toward the optimum. A simplex process is designed to find the slope, i.e., path with greatest increase in yield The procedure starts by the generation of a simplexfigure. The simplex figure is triangle whentwo variables are optimized, a tetrahedron when three ariables are optimized, increasing to an n+l polyhedron, where n is the number of variables to be optimized. The experimental point with the poorest response is eliminated and a new point generated by reflection of the eliminated point through the centroid of the simplex figure. This process is continued until an optimum is reached. In Fig. 2, experimental points 1, 2 and 3 form the vertices of the original simplex figure. Point I was found have the poorest yield, and therefore was eliminated from the simplex figure and a new point(B )generated. Point 3 was then eliminated and the new point ( C)generated. The process was continued until the optimum was reached The EVOP process is a systematic method of adjusting the variables until an optimum is reached

162 Fermentation and Biochemical Engineering Handbook optimum, the second variable's optimum is found, etc. This process works if, and only if, there is no interaction between variables. In the case shown in Fig. 1, the optimum found using the one-variable-at-a-time approach was 85%, far from the real optimum of 90%. Because of the interaction between the two nutrients, the one-variable-at-a-time approach failed to find the true optimum. In order to find the optimum conditions, it would have been necessary to repeat the one-variable-at-a-time process at each step to verify that the true optimum was reached. This requires numerous sequential experimental runs, a time-consuming and ineffective strategy, especially when many variables need to be optimized. Because of the complexity of microbial metabolism, interaction between the variables is inevitable, espe￾cially when using commercial substrates which are a complex mixture of many nutrients. Therefore, since it is both time-consuming and inefficient, the one-variable-at-a-time approach is not satisfactory for fermentation development. Fortunately, there are a number of statistical methods which will find the optimum quickly and efficiently. 3.0 EVOLUTIONARY OPTIMIZATION An alternative to the one-variable-at-a-time approach is the technique of evolutionary optimization. Evolutionary optimization (EVOP), also known as method of steepest ascent, is based upon the techniques developed by Spindley, et al.['] The method is an iterative process in which a simplex $figure is generated by running one more experiment than the number of variables to be optimized. It gets its name from the fact that the process slowly evolves toward the optimum. A simplex process is designed to find the optimum by ascending the reaction surface along the lines of the steepest slope, Le., path with greatest increase in yield. The procedure starts by the generation of a simplex figure. The simplex figure is atriangle when two variables are optimized, a tetrahedron when three variables are optimized, increasing to an n+l polyhedron, where n is the number ofvariables to be optimized. The experimental point with the poorest response is eliminated and a new point generated by reflection of the eliminated point through the centroid of the simplex figure. This process is continued until an optimum is reached. In Fig. 2, experimental points 1 , 2, and 3 form the vertices of the original simplex figure. Point 1 was found to have the poorest yield, and therefore was eliminated from the simplex figure and a new point (B) generated. Point 3 was then eliminated and the new point (C) generated. The process was continued until the optimum was reached. The EVOP process is a systematic method of adjusting the variables until an optimum is reached

5,89 2 3.75 2,5 Apparent optimum 9 1,25 1,25 2.59 3,75 5即 NUTRIENT 1 Figure 1. Example of one-variable-at-a-time approach. Contour plot of yield

Statistical Methods for Fermentation Optimization I63 i 0 (u a c( 0 c)

5,80 4 3,8 98 7 22,9 1,8 1,9 2,8 4,8 5, NUTRIENT 1 Figure 2. Example of evolutionary optimization contour plot of yield

164 Fermentation and Biochemical Engineering Handbook

Statistical Methods for Fermentation Optimization 165 Numerous modifications have been made to the original simplex method. One of the more important modifications was made by Nelder and Mead( 2 who modified the method to allow expansions in directions which are favorable and contractions in directions which are unfavorable. This modification increased the rate at which the optimum is found important modifications were made by Brissey 3] who describes a high algorithm, and Keefer 4 who describes a high speed algorithm and methods dealing with bounds on the independent variables Additional modifications were reported by Nelson, 51 Bruley, [61 Deming, 9]and Ryan I8 For reviews on the simplex methods see papers by Deming et al.[9]-[ll] evoP does have its limitations. first because of its iterative nature it is a slow process which can require many steps. Secondly, it provides only limited information about the effects of the variables. Upon completion of the EvoP process only a limited region of the reaction surface will have been explored and therefore, minimal information will be available about the effects of the variables and their interactions. This information is necessary to determine the ranges within which the variables must be controlled to insure optimal operation. Further, EVOP approaches the nearest optimum It is unknown whether this optimum is a local optimum or the optimum for the entire process Despite the limitations, EVoP is an extremely useful optimization technique. EVOP is robust, can handle many variables at the same time, and will always lead to an optimum. Also, because of its iterative nature, little needs to be known about the system before beginning the process. Most important, however, is the fact that it can be useful in plant optimization where the cost of running experiments using conditions that result in low ields or able product cannot be tolerated. In theory, the proce improves at each step of the optimization scheme, making it ideal for a production situation, For application of EvoP to plant scale operations, see Refs.12-14. The main difficulty with using EVOP in a plant environment is performing the initial experimental runs. Plant managers are reluctant to run at less than optimal conditions. Attempts to use process data as the initial experiments in the simplex is, in general, not successful because of confound- ng. Confounding occurs because critical variables are closely cor and therefore, the error in measuring the conditions and results tend to be greater than theeffect of the variables. Because of this, operating data usually gives a false perspective as to which variables are important and the changes to be made for the next step

Statistical Methods for Fermentation Optimization I65 Numerous modifications have been made to the original simplex method. One of the more important modifications was made by Nelder and Mead[?] who modifiedthe method to allow expansions in directions which are favorable and contractions in directions which are unfavorable. This modification increased the rate at which the optimum is found. Other important modifications were made by Bris~ey[~I who describes a high speed algorithm, and KeeferL4I who describes a high speed algorithm and methods dealing with bounds on the independent variables. Bruley,I6I Deming,ig] and Ryan.[8] For reviews on the simplex methods see papers by Deming et al.[9]-[11] EVOP does have its limitations. First, because of its iterative nature, it is a slow process which can require many steps. Secondly, it provides only limited information about the effects ofthe variables. Upon completion ofthe EVOP process only a limited region of the reaction surface will have been explored and therefore, minimal information will be available about the effects of the variables and their interactions. This information is necessary to determine the ranges within which the variables must be controlled to insure optimal operation. Further, EVOP approaches the nearest optimum. It is unknown whether this optimum is a local optimum or the optimum for the entire process Despite the limitations, EVOP is an extremely usefbl optimization technique. EVOP is robust, can handle many variables at the same time, and will always lead to an optimum. Also, because of its iterative nature, little needs to be known about the system before beginning the process. Most important, however, is the fact that it can be useful in plant optimization where the cost of running experiments using conditions that result in low yields or unusable product cannot be tolerated. In theory, the process improves at each step of the optimization scheme, making it ideal for a production situation. For application of EVOP to plant scale operations, see Refs. 12-14. The main difficulty with using EVOP in a plant environment is performing the initial experimental runs. Plant managers are reluctant to run at less than optimal conditions. Attempts to use process data as the initial experiments in the simplex is, in general, not successful because of confound￾ing. Confounding occurs because critical variables are closely controlled, and therefore, the error in measuring the conditions and results tend to be greater than the effect ofthe variables. Because ofthis, operating data usually gives a false perspective as to which variables are important and the changes to be made for the next step. Additional modifications were reported by

166 Fermentation and Biochemical Engineering Handbook The successful use ofEVoP depends heavily upon the choice for the initial experimental runs. If the initial points are far from the optimum and elatively close to one another, many iterations will be required. Reasonable step sizes must be chosen to insure that a significant effect of the variable is observed between the points, however, the step size should not be so great to encompass the optimum. a second factor to consider is magnitude effects If one variable is measured over a range of 0. 1 to 1.0 while another is measured over a range of l to 100 the magnitude difference between th variables can effect the simplex. Scaling factors should be used to keep all variables within the same order of magnitude 4.0 RESPONSE SURFACE METHODOLOGY The best method for process optimization is response surface method- ology(RSm). This process will not only determine optimum conditions, but also give the information necessary to design a process Response surface methodology(rSm) is a method of optimization using statistical techniques based upon the special factorial designs of Box and Behenkin(] and Box and Wilson. 51 It is a scientific approach to determining optimum conditions which combines special experimental de- signs with Taylor first and second order equations. The rsm process determines the surface of the Taylor expansion curve which describes the response(yield, impurity level, etc. )The Taylor equation, which is the heart of the rsm method, has the form Response=A+B XI+C: X2+.H X12+I X22+ MX1X2+NX1X3+ where A, B, C,.are the coefficients of the terms of the equation, and XI= linear term for variable 1 X2= linear term for variable 2 x1= nonlinear squared term for variable X22= nonlinear squared term for variable 2

166 Fermentation and Biochemical Engineering Handbook The successfid use of EVOP depends heavily upon the choice for the initial experimental runs. If the initial points are far from the optimum and relatively close to one another, many iterations will be required. Reasonable step sizes must be chosen to insure that a significant effect of the variable is observed between the points, however, the step size should not be so great as to encompass the optimum. A second factor to consider is magnitude effects. If one variable is measured over a range of 0.1 to 1.0 while another is measured over a range of 1 to 100 the magnitude difference between the variables can effect the simplex. Scaling factors should be used to keep all variables within the same order of magnitude. 4.0 RESPONSE SURFACE METHODOLOGY The best method for process optimization is response surface method￾ology (RSM). This process will not only determine optimum conditions, but also give the information necessary to design a process. Response surface methodology (RSM) is a method of optimization using statistical techniques based upon the special factorial designs of Box and Behenkir~[~~] andBox and Wilson.[ls] It is a scientific approach to determining optimum conditions which combines special experimental de￾signs with Taylor first and second order equations. The RSM process determines the surface of the Taylor expansion curve which describes the response (yield, impurity level, etc.) The Taylor equation, which is the heart of the RSM method, has the form: Response = A + B.X1 + CaX2 + . . . H-X12 + I.X22 + ... M*Xl*X2 +N*Xl*X3 + .,. where A,B,C,. . . are the coefficients of the terms of the equation, and X1 = linear term for variable 1 X2 = linear term for variable 2 Xl2 = nonlinear squared term for variable 1 X22 = nonlinear squared term for variable 2

Statistical Methods for Fermentation Optimization 167 X1X2 interaction term for variable l and variable Xl X3= interaction term for variable 1 and variable 3 The Taylor equation is named after the English mathematician Brook Taylor who proposed that any continuous function can be approximated by a power series. It is used in mathematics for approximating a wide variety of continuous functions. The RSM protocol, therefore, uses the Taylor equation to approximate the function which describes the response in nature, coupled with the special experimental designs for determining the coefficients of the Taylor equation The use of RSM requires that certain criteria must be met. These are 1. The factors which are critical for the process are known RSM programs are limited in the number of variables that they are designed to handle. As the number of variables increases the number of experiments required by the designs increases exponentially. Therefore, most RSM programs are limited to 4 to 5 variables. Fortunately for the scale up of most fermentations thenumber of variables to be optimized are limited. Some of the more important variables are listed in Table 1 Table 1. Typical Variables in a Fermentation Aeration rate Agitation rate Carbon/Nitrogen ratio Phosphate level Magnesium level Back Carbon Source Nitrogen source Dissolved oxygen level Power input

Statistical Methods for Fermentation Optimization I67 X1-X2 = interaction term for variable 1 and variable 2 XleX3 = interaction term for variable 1 and variable 3 The Taylor equation is named after the English mathematician Brook Taylor who proposed that any continuous function can be approximated by a power series. It is used in mathematics for approximating a wide variety of continuous functions. The RSM protocol, therefore, uses the Taylor equation to approximate the function which describes the response in nature, coupled with the special experimental designs for determining the coefficients of the Taylor equation. The use of RSM requires that certain criteria must be met. These are: 1. The factors which are critical for the process are known. RSM programs are limited in the number ofvariables that they are designed to handle. As the number of variables increases the number of experiments required by the designs increases exponentially. Therefore, most RSM programs are limited to 4 to 5 variables. Fortunately for the scale up of most fermentations the number of variables to be optimized are limited. Some of the more important variables are listed in Table 1. Table 1. Typical Variables in a Fermentation Aeration rate Agitation rate Temperature CarbodNitrogen ratio Phosphate level Magnesium level Back pressure Sulhr level Carbon Source Nitrogen source PH Dissolved oxygen level Power input

168 Fermentation and Biochemical Engineering Handbook 2. The factors must vary continuously over the experimental ange tested. For example, the variables of pH, aeration rate, and agitation rate are continuous and can be used in an RSM model. Variables such as carbon source(potato starch vs com syrup)or nitrogen source(cotton seed meal vs soy bean meal) are noncontinuous and cannot be optimized by rSm. However, level of com syrup or level of soy bean meal are continuous and can be optimized 3. There exists a mathematical function which relates the For reviews on the RSM process see Henika[] or Giovanni. [8For details on the calculation methods see Cochran and Cox. [19] or Box. [20] The difficult and time-consuming nature of these calculations have inhibited the wide spread use of RSM. Fortunately, numerous computer programs are available to perform this chore. They range from the expensive and sophisticated, such as SASTM, to inexpensive, PC based programs, SPSS XTM, E-ChipTM, and X STATTM 2I] The availability of these programs however, has led to a"black box"approach to RSM. This approach can lead to many problems if the user does not have a thorough understanding of the process or the meaning of the results 5.0 ADVANTAGES OF RSM The response surface methodology approach has many advantages over other optimization procedures. These are listed in Table 2 Table 2. Advantages and Disadvantages of RSM Advantages of RSM 1. Greatest amount of information from experiments 2. Forces you to plan 3. Know how long project will take 4. Gives information about the interaction between variables 5. Multiple responses at the same time 6. Gives information necessary for design and optimization of a process Disadvantages of RSM 1. Tells what happens, not why 2. Notoriously poor for predicting outside the range of study

I68 Fermentation and Biochemical Engineering Handbook 2. The factors must vary continuously over the experimental range tested. For example, the variables of pH, aeration rate, and agitation rate are continuous and can be used in an RSM model. Variables such as carbon source (potato starch vs corn syrup) or nitrogen source (cotton seed meal vs soy bean meal) are noncontinuous and cannot be optimized by RSM. However, level of corn syrup or level of soy bean meal are continuous and can be optimized. 3. There exists a mathematical hnction which relates the response to the factors. For reviews on the RSM process see He~~ka[~’l or Giovanni.[’*] For details on the calculation methods see Cochran and or The difficult and time-consuming nature of these calculations have inhibited the wide spread use of RSM. Fortunately, numerous computer programs are available to perform this chore. They range from the expensive and sophisticated, such as SASTM, to inexpensive, PC based programs, SPSS￾Xm , E-Chipm, and X STATTM.[*l] The availability of these programs, however, has led to a “black box” approach to RSM. This approach can lead to many problems if the user does not have a thorough understanding of the process or the meaning of the results. 5.0 ADVANTAGES OF RSM The response surface methodology approach has many advantages over other optimization procedures. These are listed in Table 2. Table 2. Advantages and Disadvantages of RSM Advantages of RSM 1. Greatest amount of information from experiments. 2. Forces you to plan. 3. Know how long project will take. 4. Gives information about the interaction between variables. 5. Multiple responses at the same time. 6. Gives information necessary for design and optimization of a process. 1. Tells what happens, not why. 2. Notoriously poor for predicting outside the range of study. Disadvantages of RSM

Statistical Methods for Fermentation Optimization 169 5.1 Maximum Information from Experiments RSM yields the maximum amount of information from the m mount of work. For example, in the one-variable-at-a-time approach, shown Fig. 1, ten experiments were run only to find the suboptimum conditions However, using RSM and thirteen properly designed experiments not only would the true optimum have been found, but also the information necessary to design the process would have been made available. Secondly, since all of the experiments can be run simultaneously, the results could be obtained quickly. This is the power of response surface methodology RSM is a very efficient procedure. It utilizes partial factorial designs, such as central composite or star designs, and therefore, the number of experimental points required are a minimum table 3). A full factorial thr level design would require n experiments; while a full factorial five level design would require n experiments, where n is the number of variables to be optimized Response surface protocols, being a partial factorial design require fewer experiments. For example, if one were to examine five variables at five different levels, a full factorial design approach would require 3125 experiments. Response Surface Methodology, on the other hand, requires only 48 experiments, clearly a large savings in time, effort, and expense Table 3. Experimental Efficiency of RSM Number Number of Number of Variables Combinations Actual Experiments NARROW THREE LEVEL DESIGN 2 27 234 BROAD FIVE LEVEL EXPLORATORY DESIGN 20 3125 48

Statistical Methods for Fermentation Optimization I69 5.1 Maximum Information from Experiments RSM yields the maximum amount of information from the minimum amount ofwork. For example, in the one-variable-at-a-time approach, shown in Fig. 1, ten experiments were run only to hd the suboptimum conditions. However, using RSM and thirteen properly designed experiments not only would the true optimum have been found, but also the information necessary to design the process would have been made available. Secondly, since all of the experiments can be run simultaneously, the results could be obtained quickly. This is the power of response surface methodology. RSM is a very efficient procedure. It utilizes partial factorial designs, such as central composite or star designs, and therefore, the number of experimental points required are a minimum (Table 3). A full factorial three level design would require n3 experiments; while a full factorial five level design would require n5 experiments, where n is the number of variables to be optimized. Response surface protocols, being a partial factorial design, require fewer experiments. For example, if one were to examine five variables at five different levels, a full factorial design approach would require 3 125 experiments. Response Surface Methodology, on the other hand, requires only 48 experiments, clearly a large savings in time, effort, and expense. Table 3. Experimental Efficiency of RSM Number Number of Number of Variables Combinations Actual Experiments NARROW THREE LEVEL DESIGN 2 9 3 27 4 81 5 234 BROAD FIVE LEVEL EXPLORATORY DESIGN 2 25 3 125 4 625 5 3 125 13 15 27 46 13 20 31 48

170 Fermentation and Biochemical Engineering Handbook 5.2 Forces One To plan Thesuccessful use of an RSM protocol requires careful planning on the part of the experimenter before beginning the protocol. The ranges over which the variables are to be tested must be chosen with care. Choosing a range which is too narrow can result in a variable being discarded as not significant, not because the variable did not have an effect, but rather because the effect of the variable over the range evaluated was small in comparison to the experimentalerror. The range must be large enough so that the variable has a significant effect over the range evaluated On the other hand, choosing a range which is too large can also result in a variable being discarded as not significant, not because the variable did not have an effect, but rather because the Taylor equation could not adequately explain the effect of the variable. It must be remembered that rsm does not determine the function which describes the results, but rather determines the Taylor expansion equation which best fits the data. Over a limited range, the Taylor equation will approximate the function which describes the results. The wider the range chosen the less likely a Taylor expansion equation which meaningfully explains the data will be obtained. Therefore, ranges which include extreme minimums and maximums for a variable should be avoided. Further. the experimenter needs to have an approximation as to where the optima exists It is a sad state of affairs to have completed the rsm protocol only to find that the optimum conditions were outside of the range evaluated. RSM i notorious for its inability to predict outside the range evaluated. It is strongly advised that preliminary experiments be done to determine the ranges over which the variables are to be evaluated 5.3 Know How Long Project Will Take a distinct advantage of the rsm procedure is that one knows how many experiments and the time frame needed to complete the process. This is especially helpful for budgetary purposes and the allocation of scarce scientific resources. Using RSM, the experimenter has the information necessary to determine whether a project is worth undertaking 5.4 Interaction Between Variables With the one-variable-at-a-time approach, it is difficult to determine the amount ofinteraction between variables. Response surface methodology since it looks at all the variables at the same time can calculate the interaction

I70 Fermentation and Biochemical Engineering Handbook 5.2 Forces One To Plan The successful use ofan RSM protocol requires careful planning on the part of the experimenter before beginning the protocol. The ranges over which the variables are to be tested must be chosen with care. Choosing a range which is too narrow can result in a variable being discarded as not significant, not because the variable did not have an effect, but rather because the effect of the variable over the range evaluated was small in comparison to the experimental error. The range must be large enough so that the variable has a significant effect over the range evaluated. On the other hand, choosing a range which is too large can also result in a variable being discarded as not significant, not because the variable did not have an effect, but rather because the Taylor equation could not adequately explain the effect of the variable. It must be remembered that RSM does not determine the function which describes the results, but rather determines the Taylor expansion equation which best fits the data. Over a limited range, the Taylor equation will approximate the function which describes the results. The wider the range chosen the less likely a Taylor expansion equation which meaningfully explains the data will be obtained. Therefore, ranges which include extreme minimums and maximums for a variable should be avoided. Further, the experimenter needs to have an approximation as to where the optima exists. It is a sad state of affairs to have completed the RSM protocol only to find that the optimum conditions were outside of the range evaluated. RSM is notorious for its inability to predict outside the range evaluated. It is strongly advised that preliminary experiments be done to determine the ranges over which the variables are to be evaluated. 5.3 Know How Long Project Will Take A distinct advantage of the RSM procedure is that one knows how many experiments and the time frame needed to complete the process. This is especially helpful for budgetary purposes and the allocation of scarce scientific resources. Using RSM, the experimenter has the information necessary to determine whether a project is worth undertaking. 5.4 Interaction Between Variables With the one-variable-at-a-time approach, it is difficult to determine the amount of interaction between variables. Response surface methodology, since it looks at all the variables at the same time, can calculate the interaction

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