Rapid Development of New Polymer Blends: The Optimal Selection of

May 25, 2006 - A data-based approach to the development of industrial polymer blends with specified final properties is presented. There are basically...
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Ind. Eng. Chem. Res. 2006, 45, 4653-4660

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Rapid Development of New Polymer Blends: The Optimal Selection of Materials and Blend Ratios Koji Muteki and John F. MacGregor* Department of Chemical Engineering, McMaster UniVersity, Hamilton, Ontario L8S 4L7, Canada

Toshihiro Ueda Mitsubishi Chemical Corporation, Yokkaichi, Mie, Japan

A data-based approach to the development of industrial polymer blends with specified final properties is presented. There are basically three major degrees of freedom to control the final product properties: the selection of raw materials, the selection of the ratios in which to blend them, and the selection of process conditions used to manufacture them. In this paper, we present a new optimization approach that simultaneously addresses all of these degrees of freedom, but the primary focus will be on the selection of the materials and their ratios. The approach involves building partial least-squares (PLS) models that combine databases on previously made blends and databases on the properties of the component materials used in these blends. The resulting models are then used in an optimization framework to select raw materials from much larger databases (including materials never previously used) and to select the ratios in which to blend them in order to yield a blend product with specified end properties at a minimum cost. The methodology is applied to two industrial polymer blending problems that involve the replacement of raw materials while keeping the same product properties and minimizing total raw material cost. 1. Introduction Competition in chemical process industries is increasingly being based on market differentiation and value-added products. In polymer products, customer demands are diversified and the lifetimes of products are becoming shorter. Therefore, rapidly developing new products required by customers with minimum costs is an increasingly important problem. A new data-based modeling and optimization strategy that simultaneously takes into account both the selection of raw materials and the ratios in which to blend them is presented. The final goal is to speedily achieve target products having specified properties with minimum experiments and minimum total material costs. The databases generally available for this problem are depicted in Figure 1 for the case of blends of one class of raw materials (the industrial example for the case of multiple classes of raw materials will be described in the section 3). The databases consist of a raw material property database (XDB), raw material cost data (C), and data from previous blending operations. The latter consists of a (M × N) R matrix containing the ratios of all the materials used in the formation of previous blends, the process blending conditions (Z) used during these blending experiments, and an (M × L) Y matrix containing L properties measured on the final blends. M is the number of previous blends, and N is the number of all the raw materials used in the blends. The data in R contains the fraction of each N raw material used in the blend (0 eri,j e1.0 and ∑j)1 ri,j ) 1.0). The (NN × K) XDB matrix contains the information on all the available raw materials including both those used and not used in the past. K is the number of raw material properties, and NN is the number of all the available raw materials. Most data on the XDB matrix can be often obtained from suppliers of * To whom correspondence should be addressed. Tel.: 905-5259140 ext. 24951. Fax: 905-521-1350. E-mail: [email protected].

the raw materials, while some additional measurements may have to be measured inside the blending company. In some uses, the data on the prior blends (Z, R, and Y) may not be sufficiently rich in information required to achieve the optimal blending solution proposed in this paper. However, we will proceed to present our methodology assuming the databases contain sufficient information. The design of experiments to generate adequate data matrices (Z, R, and Y) for this work is discussed in ref 1. The problem of finding the optimal selection of raw materials and their mixture ratios is something that has not been treated very efficiently in most industrial applications. Traditional approaches tend to treat the two problems (material selection and blend ratios) as separate steps. A set of raw materials is selected usually based on the experimenters’ best guess, and then, a set of blending experiments (perhaps designed or trial and error) are run to see if the target properties can be achieved. If the results are not entirely acceptable, another set of raw materials is selected and the process is repeated. Such approaches lead to many blending experiments, a very inadequate investigation of the large number of possible materials that could be used, and a very long development time. It is the objective of this paper to present a methodology that can greatly reduce the development time for new products. The methodology simultaneously treats the problem of the selection of raw materials and their ratios by effectively exploiting the readily available data on past blending runs and on raw material properties (Figure 1). The quality of the databases is crucial in the success of the approach. However, in many cases, with a little effort, the databases required to implement this methodology can be readily assembled. The paper is organized as follows. In section 2, a methodology that simultaneously optimizes the selection of raw materials, their blend ratios, and the process conditions is presented. It consists of combining all the databases shown in Figure 1 through building a new mixture partial least-squares (PLS)

10.1021/ie050953b CCC: $33.50 © 2006 American Chemical Society Published on Web 05/25/2006

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Figure 1. Database available in industrial mixture design.

Figure 2. Combined data structure.

modeling approach and, then, solving an optimization problem to achieve the desired properties at minimum effort. In section 3, two industrial examples are used to demonstrate the effectiveness of the new methodology. The industrial examples concern the replacement of certain raw materials (rubbers) in the blend formulation while keeping the same target product properties and minimizing the total raw material cost. A sensitivity study is also performed to find alternative blends with similar costs and similar properties. 2. Methodology In this section, the new approaches are presented as generalized methodologies, using the databases shown at Figure 1. They consist of combining the databases, using mixture rules, building a new PLS mixture model taking into account the raw material properties, and then formulating the optimization. 2.1. Multiblock Nature of the Databases. The databases available for general mixture design already have been shown in Figure 1. The raw material property information and experimental blending information are now aligned for modeling as shown at Figure 2. The (K × N) XT matrix consists of the raw material property data that corresponds to the subset of materials used in past experimental blends R. The XT matrix is a subset of the much larger raw material database matrix XDB. There are several features of this data structure. The first is that there are no common dimensions over all the data matrices. The matrices Z, R, and Y have one dimension in common (i.e., the blending material sample direction). XT has one dimension in common with the R matrix but no dimension common to Y. That is, XT has only an indirect relationship to Y through R. This kind of L-shape data structure can occur in situations when auxiliary data such as XT are obtained from a different source. A second feature is that raw material property matrices (XT,

XDB) will usually contain missing data. This arises because different suppliers of the raw materials often provide measurements on different variables and omit measurements of others. This is often done by the supplier to emphasize the features of their own specific materials. Even the same suppliers often provide measurements on different properties for different product grades. For this missing data problem in the raw material property data, Muteki et al.2 present two effective approaches to impute the missing data. Both approaches incorporate the auxiliary mixture data information in R and Y to help in the estimation of the missing data. 2.2. Mixture Rules to Combine the Databases. In the paper, the concept of the “ideal mixing rule” (Grassman et al.3) is employed to combine the raw material property data matrix XT and the mixture ratio matrix R. Muteki and MacGregor4 present a more general multiblock PLS approach for combining these matrices but show that under assumptions that generally will hold in polymer blending problems, it essentially leads to this simpler blending rule approach used here. The main goal of this paper is to provide a viable methodology for using these databases to optimize the selection of materials and their blend ratios and to present some successful industrial applications of this methodology. The mixing effect of the raw material properties on blends xmix(m,k) (m ) 1, 2, ..., M; k ) 1, 2, ..., K) based on the ideal mixing rule can be expressed as a sum of the raw material property x(n,k) (n ) 1, 2, ..., N; k ) 1, 2, ..., K) times the mass fraction of the raw materials r(m,n) (m ) 1, 2, ..., M; n ) 1, 2, ..., N), as follows:

xmix(m,k) ) r(m,1)‚x(1,k) + r(m,2)‚x(2,k) + N

... r(m,N)‚x(N,k) )

r(m,j)‚x(j,k) ∑ j)1

(1)

That is, the mixing raw material property matrix Xmix (M × K) is simply expressed as

Xmix ) R‚X

(2)

For instance, consider the case where a given raw material property of the rubber A has a value of 4.0 and that of the rubber B has a value of 10.0 and the blend consists of 60% of the rubber A and 40% of the rubber B. Then, the mixing raw material property is obtained as 6.4 () 4 × 60/100 + 10 × 40/100). Clearly, properties such as the compound content (e.g., styrene mass percentage in styrene-butadiene rubber) follow this ideal blending rule exactly. It is also known that many polymer properties such as the weight average molecular weight approximately follow the ideal mixing rule (Grassman et al.3). We make no claim that all the properties must follow this ideal mixing rule. If there are known mixing rules with respect to the specific properties, they can be used to represent the mixing

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raw material property matrix Xmix. Furthermore, even if the mixing rule is unknown in advance and the characteristics are actually nonlinear, they can be approximately dealt with by applying the nonlinear PLS modeling methods (Wold et al.,5 Wold,6 and Fank7) to the modeling stage between Xmix () R‚ X) and Y (described in the next section). 2.3. Mixture PLS Modeling Using Raw Material Property Data Information. The traditional mixture models such as the Scheffe model (Cornell8) and mixture PLS models (KettanehWold9) represent only the relationship between mixture ratios matrix R and mixture property matrix Y. The model forms are generically expressed as

open the door to the simultaneous selection of raw materials and their ratios in the optimization, as shown in this paper. PLS regression modeling has extensively been described in the literature (Ho¨skuldsson,10 Burnham et al.,11 Martens and Naes,12 and Errikson et al.13), and only a brief description is given here. Prior to modeling, the matrices Xmix and Y are normally mean centered and scaled to unit variance. The mean centering is performed since we define a model only in the vicinity of the data, and scaling is performed to account for differences in the units of measurements of the different variables. PLS regression is performed by projecting the Xmix and Y onto lower dimensional subspaces:

Y ) f(R) + 

Xmix ) T‚PT + E

(3)

In this work, we use a new mixture modeling approach based on the mixing rules discussed above. The model is expressed as

Y ) f(Xmix) + 

(4)

where Xmix () R‚X) as defined in eq 2. The mixture PLS model of eq 4 represents the relationship between the blend rule raw material properties (Xmix) and the final product blend properties (Y). Combining the raw material properties (X) and their mixture ratios (R) through the use of mixing rules makes it possible to reduce the block matrices X, R, and Y with no common dimension to the matrices Xmix and Y having a common dimension M. A partial least-square (PLS) regression model will be used to obtain the relationship in eq 4. The PLS model will be briefly described in section 2.3.1. If the process operating conditions (Z) change between blending experiments, then the effect of these changes are easily accounted for by incorporating Z into the PLS model as follows:

Y ) f(Xmix,Z) + 

Y ) T‚QT + F

where the columns of T are values of latent variables (T ) XmixW*) that capture most of the variability in the data; W*, P, and Q are the loading matrices, and E and F are residual matrices. The PLS loading matrices are obtained by maximizing the covariance between Xmix and Y (Hoskuldsson10). Prediction of Y can be obtained from the PLS model as B

ˆ Y ˆ ) T‚QT ) Xmix(W*QT) ) XmixB

(7)

For any new (1× K) vector of mixture properties xTmixnew, one can compute a (1 × A) vector of new latent variable scores as τTnew ) xTmixnewW* and then predict the (1 × M) vector of blend properties as yˆTnew ) τTnewQT ) xTmixnewB ˆ . One can also compute two distance criteria to test the validity of the model for the new conditions. Hotelling’s T2 is expressed as

T ) 2

(5)

Z will be omitted in the modeling studies in this paper since in the industrial examples to be presented these conditions were constant. The new mixture PLS models need the following two assumptions. The first is that the ideal mixing rules are approximately valid. The second is that the raw material properties available correlate well with the final blend properties (Y). This will be naturally satisfied in most industrial applications because the raw material properties (X) are measured because of their expected importance in any polymer blending. Muteki et al.1 demonstrate that the new mixture PLS model works well in practical industrial examples, unless one of the above two conditions is severely violated. They also show several advantages of these new mixture PLS models: (1) they provide a direct interpretation of the effect of raw material properties on final blend properties, that is, what raw material properties affect what mixture properties, something that traditional mixture models using only the R + Y matrices cannot do; (2) the new mixture PLS models provide better estimates of the blend properties (Y) than the traditional mixture models in most industrial cases; (3) with new material grades which have never been used in the past, the new mixture PLS models can estimate final blend product properties by using only their raw material property data without implementing additional blending experiments; (4) introducing the Xmix () R‚X) matrix (i.e., ideal mixing rule) opens the door to new approaches to the design of mixture experiments (DOE) that can take into account both the effect of raw material properties and the effect of their blend ratios; and (5) the new mixture PLS models also

(6)

A

τnew,a2

a)1

sa

∑

(8)

where sa is the variance of the score matrix T and A is the selected number of latent variables in the PLS model. It provides a measure of the distance from the center point in the latent space to the projection of the new observation onto the latent variable space. The SPE (square prediction error) in the X space is expressed as

SPEX )

∑(xmixnew - xˆ mixnew)2

(9)

where xˆ Tmixnew ) xTmixnewPTP is the predicted value of xTmixnew estimated from the PLS model. The SPE provides a measure of the orthogonal distance (residual) of the new point from the latent variable space. A large residual implies that the PLS model is not valid in the region of xTmixnew. 2.4. Optimization Based on the New Mixture PLS Models. In this section, we assume that the required data is available and a PLS model between Xmix and the final blend Y has been built. The objective is now to use this model to simultaneously optimize the selection of the best raw materials and their blend ratios in order to achieve the desired mixture property vector ydes with a minimum total raw material cost and a minimum number of raw materials. The formulation of the optimization is expressed as T T xmixnew)T‚W1‚(ydes - BPLS xmixnew) + min(ydes - BPLS rnew

NN

NN

rnew,jcj + w3∑δj ∑ j)1 j)1

w2

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s.t.

Ideal mixing rule {xTmixnew ) rTnew‚XDB PLS model constraint

{

k)1

SPEnew )

(xmixnew - xˆ mixnew)2 e  ∑ k)K

A

Tnew2 )

∑ a)1

τnew,a2 sa

e Tmax2

NN

Mixture constraints {

rnew,j ) 1, ∑ j)1

Binary variable constraint

{

{

0 e rnew,j e 1

1 rnew,j > 0 0 rnew,j ) 0 rnew,j e Mjδj

δj )

small amounts of a large number of raw materials. However, it was found that, due to different material costs, when this cost term was included, the simpler SQP approach almost always resulted in solutions with only a small number of raw materials. Therefore, in this study, solutions were found using both SQP and MINLP approaches. If process conditions also need to be considered, they should be included in the model (eq 5), the PLS model then built on the expanded X ) [Xmix,Z] matrix, and the optimization performed on both the ratios (rnew) and the independent dimensions of process conditions (znew). Previous literature18-21 has already discussed the optimization of only process conditions or of process conditions and blend ratios in the latent variable space of PLS models (linear and nonlinear). 3. Industrial Examples

(10)

where the (NN × 1) rnew vector contains the mixture ratios of all the raw materials available on the database XDB, the (L × 1) ydes vector refers to the desired final blend material properties, cj is the cost of raw material j, δj is the binary variable (0,1) that indicates if material j is used, Mj (e 1.0) is an upper limit on the allowable ratio for material j (the so-called “big-M”15), W1 is a diagonal weighting matrix providing the relative importance of each blend property, w2 is a penalty value on the total material costs, and w3 is a penalty value on the total number of raw materials used. The optimized variable is the mixture ratio vector rnew. The constraints on SPEnew and Hotelling’s Tnew2 force the solution to lie in the space of the PLS model. The Tmax2 value in the Tnew2 constraint may be taken as the 95 or 99% limit on T2 for the training data depending on how far from the training data one is willing to extrapolate. The  value in the SPE constraint can range from zero (perfect adherence to the model) up to some larger value such as the 95% limit on the SPE values from the training data. A larger  value may be used when willing to extrapolate or explore for ynew properties outside the training data region.1 In the industrial studies to follow, since the ydes values were within the bounds of the training data,  was taken as zero and Tmax2 was taken as the 95% limit. The first term of the objective function is a weighted measure of the estimation error between the desired blend properties and the estimated blend properties through the mixture PLS model. The new raw material properties χmixnew are estimated based on the ideal mixing rule between rnew and XDB. The second term in the objective function refers to the total raw material cost of the mixture. The third term in the objective function penalizes the total number of raw materials used to obtain the blend, since it is usually desirable to obtain the desired blend products using a minimum number of materials. Since the binary variable δj is involved, this is a mixed integer quadratic optimization.14-16 Solutions to these mixed integer nonlinear programming (MINLP) problems were obtained using the branch and bound algorithm14,16 in GAMS/SBB.15 This approach was found to work much better than that of GAMS/DICOPT15 on this problem. If the total number of raw materials is not penalized (last term in the objective function, eq 10), a solution can be obtained by a much easier sequential quadratic programming (SQP) approach17 (MATLAB and GAMS/MINOS were used). If the cost term in the objective function, eq 10, is also omitted, then this SQP approach will give unacceptable solutions having

Two real industrial examples using this product development approach are presented in this section. Both examples feature the manufacture of thermoplastic materials which are manufactured from mixtures of rubbers, polypropylenes, and oils. It can be assumed that the mixture property matrix Y is influenced by only the blend ratios and the properties of the raw materials, since the same manufacturing equipment was used and all processing conditions such as shear and extruder and mold temperatures were kept constant (Z ) constant in Figure 2). 3.1. Structure of the Industrial Data. The structure of the industrial data when three classes of materials such as polypropylene, oil, and rubber are blended is shown in Figure 3. This is simply a more complex extension of the structure shown in Figure 2 for a single class of materials. The raw materials consist of various types of rubbers, oils, and polypropylenes, and the blend ratios matrix R can be partitioned into three blend ratio matrices R ) [RrubberRoilRPP]. The corresponding raw material property data matrices on each T , respectively. class of raw materials is XTrubber, XToil, and XPP These raw material property data matrices can be obtained from the much larger raw material property data matrices, Xrubber_DB, Xoil_DB, and XPP_DB. As can be seen in Figure 3, the raw material property data matrices result in a staircase type of data structure and are not overlapped with each other because their raw material properties contain measurements of different variables due to the different classes of materials. In the current industrial application, the (111 × 13) rubber mixture ratio matrix Rrubber consists of 111 blends and the ratios of 13 rubber materials used in each blend, the (111 × 1) oil mixture ratio matrix Roil consists of 111 blends and the ratio of a single oil material used in each blend, and the (111 × 4) polypropylene mixture ratio matrix RPP consists of 111 blends and the ratios of 4 polypropylene materials used in each blend. A total of 18 raw materials were used in the manufacture of previous blends including the following: 13 rubbers, 1 oil, and 4 polypropylenes. For these blend ratio matrices, a D-optimal design was used to select the blend ratios and, therefore, the conditioning is relatively good. The (111 × 7) mixture property matrix Y consists of 7 polymer blend properties measured on each of the 111 blends. The (11 × 13) rubber material property matrix XTrubber consists of measurements of 11 properties on the 13 rubbers. The larger (11 × 30) rubber material property T database matrix Xrubber_DB consists of measurements of 11 properties on 30 rubbers. The mixture properties are mainly affected by the rubber properties, and the rubber property data is critical for the design of any blended product. Although an oil material property database matrix XToil_DB (2 × 5) is

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Figure 3. Industrial polymer blending data structure (raw materials: rubber, polypropylene, and oil). Table 1. R2 and Q2 Values of the PLS Model PLS comp

R2

Q2

1 2 3 4 5 6 7

0.513 0.644 0.797 0.856 0.887 0.907 0.913

0.484 0.590 0.748 0.797 0.819 0.839 0.837

available, since only one specific oil was used in every blend (i.e., Roil (111 × 1)), the oil material property data information is not used for the analysis. A polypropylene material property T data matrix XPP_DB is not available in this case. Therefore, the optimization problem will not consider the selection of any new oils or polypropylenes. The optimal blend design problem will therefore consist of selecting which of the 30 rubbers T to select and what ratios to use for the selected from Xrubber_DB rubbers and for the oil and polypropylene. Therefore, the Xmix matrix of the mixture PLS model consists of [Rrubber‚Xrubber Roil RPP]. A summary of the fit of the PLS model is shown in Table 1. R2 is the fraction of the cumulative sum of squares of Y explained by the model fit, and Q2 is the fraction of the cumulative sum of squares of predictions obtained from cross validation. As can be seen in Table 1, the new PLS model provides a good estimate and good prediction of the final blend properties. The six PLS components that correspond to the maximum of the Q2 value are selected and used for the modeling. Muteki et al.1 show that this PLS mixture model which uses the raw material data is much more predictive than a traditional PLS mixture model using only the ratios matrix. 3.2. Optimization (Replacement of Raw Materials). We show two industrial examples where the above PLS modeling and the optimization algorithm were used to design new materials. They both involve the problem of having to replace materials (rubbers) for business reasons, while maintaining the same final product blend properties, and doing so with minimum total raw material cost. In this case, the total number of raw

materials (rubbers) was not penalized and the objective function in the optimization (eq 10) consisted of only the first two terms’ error in the final product properties and the total raw material costs. The number of variables in the optimization was 30, and the number of constraints was 93. The optimization was therefore computed by an SQP algorithm17 (GAMS/MINOS and MATLAB were used). Feasibility Study on Target Products. The first step in the product development is to specify a set of desired polymer blend properties, ydes. This cannot be specified arbitrarily because only certain combinations of blend properties (y) can be achieved in any given process with certain materials. To verify the feasibility of any specified vector, ydes, one can project this vector onto a multivariate PCA model built from past product data (Y) made on this process. If the residual magnitude, as measured by the squared projection error (SPEY ) ∑(ydes - yˆdes)2), is large (say greater than the 99% confidence limit), then it is questionable whether this product can be made in the current process.18-21 In such a situation, one should consider a modified ydes that appears achievable. The SPEY plot from a PCA on the existing blend properties (Y) and on the two target product samples is shown at Figure 4. Since this study concerns simply material replacement of existing products, the SPEY values for the two target properties (ydes) are very small indicating what we clearly already know, namely, that they are feasible. Results of Optimization. Optimal product formulations for the two target products (replacement of the rubber materials) were obtained by the optimization of eq 9. These products were then manufactured by the company involved using the blend formulations provided by the optimization. The results are shown in Tables 2 and 3 for the two products, respectively. In both tables, part a shows the materials and ratios of rubbers, oil, and polypropylene used to make the existing product and the total cost of this formulation; part b shows the measured blend properties of the current product; part c shows the new materials and their calculated ratios, obtained from our blend optimization approach, and the total cost of materials for the new product;

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Figure 4. SPEY plot on the training blending products and the two target products: (line) 111 past (training) samples; (o) two target products.

part d shows the property predictions (from the PLS model), the experimentally observed values of the final properties of the manufactured blend, and the quadratic distances of these properties from the target values (ydes - y)TW1(ydes - y). Since the features of both examples are similar, they are explained together. The raw materials that are to be replaced are Rub5 in the first example and Rub6 in the second example (see part a in Tables 2 and 3). In the optimization solutions (part c in Table 2 and Table 3), they are replaced by a selected ratio of three rubbers (Rub1, Rub2, and Rub_new1) in the first example and by a selected ratio of another three rubbers (Rub1, Rub2, and Rub_new2) in the second example. Note that

Rub_new1 and Rub_new2 are rubber materials that had never been blended by the company in the past, but they were available in the database (XDB_rubber). The other rubber materials such as Rub1 and Rub2 had been often blended by the company in the past. This shows that the selection of raw materials has been truly implemented in the optimization. The effectiveness of this optimal material replacement solution can be seen by comparing the final blend properties of the existing material (targets for the new replacement blend) in part b of Tables 2 and 3 with the estimated (from the PLS model) and measured values (from actually manufactured products) in part d of Tables 2 and 3. In both cases, the predicted and actually achieved products had 7 properties very close to those desired. To test whether the actually achieved final blend properties are statistically consistent with the target properties, one should use a Hotelling T2 test with a covariance matrix estimated from replicate production runs for various products. However, such replicated experiments were not available to us. Therefore, the success of the optimization in providing a new formulation was judged by the company chemists. They considered the results to be within these replication errors on essentially all properties and accepted both product development results as unqualified success. Furthermore, as shown in Tables 2 and 3, the total material costs per unit weight of the newly designed replacement are significantly lower in both cases. In addition to the above results, we show the calculated raw material properties of the mixed rubbers for the existing products and the new products (Tables 4 and 5, respectively). In both tables, part a shows the raw material properties of the existing rubbers from the blending rules (i.e., rrubberxrubber)and part b shows the predicted raw material properties of the blended rubbers from the optimization (i.e., rnewrubberxrubber).

Table 2. Result of Optimization for Target Product 1a

a Rub1, Rub2, and Rub5 are grades from rubber materials that have been blended in the past. Rub_new1 is a rubber grade that has not been blended in the past. Oil1 is the grade of oil material. PP4 is the grade of polypropylene material. Y1-Y7 are properties of the final blends.

Table 3. Result of Optimization for Target Product 2a

a Rub1, Rub2, and Rub6 are grades from rubber materials that have been blended in the past. Rub_new2 is a rubber grade that has not been blended in the past. Oil1 is the grade of oil material. PP3 and PP4 are the grades of polypropylene materials. Y1-Y7 are properties of the final blends.

Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4659 Table 4. Result of Raw Material Properties of Rubbers for Target Product 1a

a

Raw1-Raw11 are the rubber material properties (predicted for the new mixture).

Table 5. Result of Raw Material Properties of Rubbers for Target Product 2a

a

Raw1-Raw11 are the rubber material properties (predicted for the new mixture).

Table 6. Alternative Solutions for Target Product 2: Mixture Results, Costs, Final Blend Properties, and Quadratic Errors in the Predicted Properties

Comparing parts a and b in both Tables 4 and 5 shows that the properties of the replacement mixture of rubber materials as estimated by the mixture rules are able to provide a reasonable approximation to the properties of current rubber materials (Rub5 in the first example and Rub2 plus Rub6 in the second example). This rough confirmation is important for practical industrial applications because it allows one to check that the new mixture of rubbers has similar properties to the rubber mixture it is to replace prior to implementing the blending experiment. It was important in these cases for convincing the industrial personnel to proceed with the new formulations, especially since they both contained rubbers never before used in these applications. 3.3. Sensitivity Analysis: Alternative Blends with Similar Cost and Similar Properties. Although only one optimal new blend product has been shown on each case in the previous section, there exist alternative blends with similar properties, similar costs, and similar total number of raw materials. It is important to check the sensitivity of the blend optimization to small changes in the objective function and to look at alternative solutions to the blending problem provided by these sensitivity studies. Alternative solutions were obtained by slightly changing the weight values in the objective function (W1, w2, and w3 in eq 10). As mentioned earlier, when including the total number of raw materials term in the objective function, the selection can be computed as a MINLP problem using GAMS/SBB.15

As an example, four alternative blends (Cases 1 and 2 NLP; Cases 3 and 4 MINLP) obtained in this way for the target product 2 are shown in Table 6. These should be compared with the target properties in Table 3b and with the optimization solution actually used in Table 3c and d. Case 1 provides a more cost-effective blend than the optimal blend shown in Table 3, but the quadratic error in the predicted properties is much larger and is unacceptable for the researchers. Cases 2 and 3 provide more expensive blends than the optimal blend shown in Table 3, but they achieve the target mixture properties more closely. Case 4 provides an almost equivalent solution in terms of cost and quadratic error to the one actually implemented for product 2 in Table 3. The most significant feature is that all the blend conditions in Cases 1-4 select very different combinations of raw materials. With such sensitivity results, the researcher can consider tradeoffs between the closeness to the target properties, the total material cost, the total number of raw materials, and other intangibles such as availability and reliability of suppliers, etc. 4. Conclusion An optimization approach to the development of new products with desired final properties is presented. The approach simultaneously considers the selection of raw materials and their required blending ratios. The approach is based on mixture PLS

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models built from readily available databases on the raw material properties and on past blending runs performed within the company. If these databases are improved and expanded through collecting data on additional raw materials and designed blending experiments in new regions, the companies can rapidly develop new products, improve existing products, and reduce product costs with a minimum of effort. The methodology was applied very successfully to two industrial polymer blending problems involving the replacement of materials. However, the methodology is much more general and should be applicable to a wide variety of product development problems, such as the development of new catalysts, food products, pharmaceuticals, etc. The methodology is also capable of optimizing over the process conditions (znew) to be used for processing the new materials. The simultaneous selection of all these degrees of freedom (X, R, and Z) will be illustrated in a future paper on the development of other products. Acknowledgment The authors wish to thank Mitsubishi Chemical Corporation, for providing the data and funding for this study. Literature Cited (1) Muteki, K.; MacGregor, J. F.; Ueda, T. Mixture designs and models for the simultaneous selection of ingredients and their ratios. Chemom. Intell. Lab. Syst., to be submitted. (2) Muteki, K.; MacGregor, J. F.; Ueda, T. Estimation of missing data using latent variable methods with auxiliary information. Chemom. Intell. Lab. Syst. 2005, 78, 41-50. (3) Grassmann, P.; Sawistowski, H.; Hardbottle, R. Physical principles of chemical engineering; Pergamon Press: New York, 1971. (4) Muteki, K.; MacGregor, J. F. Multiblock PLS for L-shape data structures in mixture design. Chemom. Intell. Lab. Syst., to be submitted. (5) Wold, S.; Kettaneh-Wold, N.; Skagerberg, B. Nonlinear PLS modeling. Chemom. Intell. Lab. Syst. 1989, 7, 53-65. (6) Wold, S. Nonlinear estimation by iterative least squares procedure. In Research Papers in Statistics; Wiley: New York, 1966, pp 411-444.

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ReceiVed for reView August 19, 2005 ReVised manuscript receiVed January 24, 2006 Accepted February 1, 2006 IE050953B