Engineering Statistics Handbook Episode 7 Part 8 doc

Optimization performed by Design-Expert software Optimization of D with respect to x was carried out using the Design-Expert software. Figure 5.7 shows the individual desirability functions d i ( i ) for each of the four responses. The functions are linear since the values of s and t were set equal to one. A dot indicates the best solution found by the Design-Expert solver. Diagram of desirability functions and optimal solutions FIGURE 5.7 Desirability Functions and Optimal Solution for Example Problem Best Solution The best solution is (x * )' = (-0.10, 0.15, -1.0) and results in: d 1 ( 1 ) = 0.34 ( 1 (x * ) = 136.4) d 2 ( 2 ) = 1.0 ( 2 (x * ) = 157.1) d 3 ( 3 ) = 0.49 ( 3 (x * ) = 450.56) d 4 ( 4 ) = 0.76 ( 4 (x * ) = 69.26) The overall desirability for this solution is 0.596. All responses are predicted to be within the desired limits. 5.5.3.2.2. Multiple responses: The desirability approach http://www.itl.nist.gov/div898/handbook/pri/section5/pri5322.htm (4 of 5) [5/1/2006 10:31:15 AM] 3D plot of the overall desirability function Figure 5.8 shows a 3D plot of the overall desirability function D(x) for the (x 2 , x 3 ) plane when x 1 is fixed at -0.10. The function D(x) is quite "flat" in the vicinity of the optimal solution, indicating that small variations around x * are predicted to not change the overall desirability drastically. However, the importance of performing confirmatory runs at the estimated optimal operating conditions should be emphasized. This is particularly true in this example given the poor fit of the response models (e.g., 2 ). FIGURE 5.8 Overall Desirability Function for Example Problem 5.5.3.2.2. Multiple responses: The desirability approach http://www.itl.nist.gov/div898/handbook/pri/section5/pri5322.htm (5 of 5) [5/1/2006 10:31:15 AM] experimental data were obtained. For example, if the experimental design is a central composite design, choosing (axial distance) is a logical choice. Bounds of the form L x i U can be used instead if a cubical experimental region were used (e.g., when using a factorial experiment). Note that a Ridge Analysis problem is related to a DRS problem when the secondary constraint is absent. Thus, any algorithm or solver for DRS's will also work for the Ridge Analysis of single response systems. Nonlinear programming software required for DRS In a DRS, the response models and can be linear, quadratic or even cubic polynomials. A nonlinear programming algorithm has to be used for the optimization of a DRS. For the particular case of quadratic responses, an equality constraint for the secondary response, and a spherical region of experimentation, specialized optimization algorithms exist that guarantee global optimal solutions. In such a case, the algorithm DRSALG can be used (download from http://www.nist.gov/cgi-bin/exit_nist.cgi?url=http://www.stat.cmu.edu/jqt/29-3), but a Fortran compiler is necessary. More general case In the more general case of inequality constraints or a cubical region of experimentation, a general purpose nonlinear solver must be used and several starting points should be tried to avoid local optima. This is illustrated in the next section. Example for more than 2 responses Example: problem setup The values of three components (x 1 , x 2 , x 3 ) of a propellant need to be selected to maximize a primary response, burning rate (Y 1 ), subject to satisfactory levels of two secondary reponses; namely, the variance of the burning rate (Y 2 ) and the cost (Y 3 ). The three components must add to 100% of the mixture. The fitted models are: 5.5.3.2.3. Multiple responses: The mathematical programming approach http://www.itl.nist.gov/div898/handbook/pri/section5/pri5323.htm (2 of 3) [5/1/2006 10:31:16 AM] The optimization problem The optimization problem is therefore: maximize 1 (x) subject to: 2 (x) -4.5 3 (x) 20 x 1 + x 2 + x 3 = 1.0 0 x 1 1 0 x 2 1 0 x 3 1 Solve using Excel solver function We can use Microsoft Excel's "solver" to solve this problem. The table below shows an Excel spreadsheet that has been set up with the problem above. Cells B2:B4 contain the decision variables (cells to be changed), cell E2 is to be maximized, and all the constraints need to be entered appropriately. The figure shows the spreadsheet after the solver completes the optimization. The solution is (x * )' = (0.212, 0.343, 0.443) which provides 1 = 106.62, 2 = 4.17, and 3 = 18.23. Therefore, both secondary responses are below the specified upper bounds. The solver should be run from a variety of starting points (i.e., try different initial values in cells B1:B3 prior to starting the solver) to avoid local optima. Once again, confirmatory experiments should be conducted at the estimated optimal operating conditions. Excel spreadsheet A B C D E 1 Factors Responses 2 x1 0.21233 Y1(x) 106.6217 3 x2 0.343725 Y2(x) 4.176743 4 x3 0.443946 Y3(x) 18.23221 5 Additional constraint 6 x1 + x2 + x3 1.000001 5.5.3.2.3. Multiple responses: The mathematical programming approach http://www.itl.nist.gov/div898/handbook/pri/section5/pri5323.htm (3 of 3) [5/1/2006 10:31:16 AM] Purpose of a mixture design In mixture problems, the purpose of the experiment is to model the blending surface with some form of mathematical equation so that: Predictions of the response for any mixture or combination of the ingredients can be made empirically, or 1. Some measure of the influence on the response of each component singly and in combination with other components can be obtained. 2. Assumptions for mixture experiments The usual assumptions made for factorial experiments are also made for mixture experiments. In particular, it is assumed that the errors are independent and identically distributed with zero mean and common variance. Another assumption that is made, as with factorial designs, is that the true underlying response surface is continuous over the region being studied. Steps in planning a mixture experiment Planning a mixture experiment typically involves the following steps (Cornell and Piepel, 1994): Define the objectives of the experiment.1. Select the mixture components and any other factors to be studied. Other factors may include process variables or the total amount of the mixture. 2. Identify any constraints on the mixture components or other factors in order to specify the experimental region. 3. Identify the response variable(s) to be measured.4. Propose an appropriate model for modeling the response data as functions of the mixture components and other factors selected for the experiment. 5. Select an experimental design that is sufficient not only to fit the proposed model, but which allows a test of model adequacy as well. 6. 5.5.4. What is a mixture design? http://www.itl.nist.gov/div898/handbook/pri/section5/pri54.htm (2 of 2) [5/1/2006 10:31:16 AM] 5. Process Improvement 5.5. Advanced topics 5.5.4. What is a mixture design? 5.5.4.2.Simplex-lattice designs Definition of simplex- lattice points A {q, m} simplex-lattice design for q components consists of points defined by the following coordinate settings: the proportions assumed by each component take the m+1 equally spaced values from 0 to 1, x i = 0, 1/m, 2/m, , 1 for i = 1, 2, , q and all possible combinations (mixtures) of the proportions from this equation are used. Except for the center, all design points are on the simplex boundaries Note that the standard Simplex-Lattice and the Simplex-Centroid designs (described later) are boundary-point designs; that is, with the exception of the overall centroid, all the design points are on the boundaries of the simplex. When one is interested in prediction in the interior, it is highly desirable to augment the simplex-type designs with interior design points. Example of a three- component simplex lattice design Consider a three-component mixture for which the number of equally spaced levels for each component is four (i.e., x i = 0, 0.333, 0.667, 1). In this example q = 3 and m = 3. If one uses all possible blends of the three components with these proportions, the {3, 3} simplex-lattice then contains the 10 blending coordinates listed in the table below. The experimental region and the distribution of design runs over the simplex region are shown in the figure below. There are 10 design runs for the {3, 3} simplex-lattice design. 5.5.4.2. Simplex-lattice designs http://www.itl.nist.gov/div898/handbook/pri/section5/pri542.htm (1 of 7) [5/1/2006 10:31:17 AM] Design table TABLE 5.3 Simplex Lattice Design X1 X2 X3 0 0 1 0 0.667 0.333 0 1 0 0.333 0 0.667 0.333 0.333 0.333 0.333 0.6667 0 0.667 0 0.333 0.667 0.333 0 1 0 0 Diagram showing configuration of design runs FIGURE 5.9 Configuration of Design Runs for a {3,3} Simplex-Lattice Design The number of design points in the simplex-lattice is (q+m-1)!/(m!(q-1)!). 5.5.4.2. Simplex-lattice designs http://www.itl.nist.gov/div898/handbook/pri/section5/pri542.htm (2 of 7) [5/1/2006 10:31:17 AM] Definition of canonical polynomial model used in mixture experiments Now consider the form of the polynomial model that one might fit to the data from a mixture experiment. Due to the restriction x 1 + x 2 + + x q = 1, the form of the regression function that is fit to the data from a mixture experiment is somewhat different from the traditional polynomial fit and is often referred to as the canonical polynomial. Its form is derived using the general form of the regression function that can be fit to data collected at the points of a {q, m} simplex-lattice design and substituting into this function the dependence relationship among the x i terms. The number of terms in the {q, m} polynomial is (q+m-1)!/(m!(q-1)!), as stated previously. This is equal to the number of points that make up the associated {q, m} simplex-lattice design. Example for a {q, m=1} simplex- lattice design For example, the equation that can be fit to the points from a {q, m=1} simplex-lattice design is Multiplying 0 by (x 1 + x 2 + + x q = 1), the resulting equation is with = 0 + i for all i = 1, , q. First- order canonical form This is called the canonical form of the first-order mixture model. In general, the canonical forms of the mixture models (with the asterisks removed from the parameters) are as follows: Summary of canonical mixture models Linear Quadratic Cubic Special Cubic 5.5.4.2. Simplex-lattice designs http://www.itl.nist.gov/div898/handbook/pri/section5/pri542.htm (3 of 7) [5/1/2006 10:31:17 AM] Linear blending portion The terms in the canonical mixture polynomials have simple interpretations. Geometrically, the parameter i in the above equations represents the expected response to the pure mixture x i =1, x j =0, i j, and is the height of the mixture surface at the vertex x i =1. The portion of each of the above polynomials given by is called the linear blending portion. When blending is strictly additive, then the linear model form above is an appropriate model. Three- component mixture example The following example is from Cornell (1990) and consists of a three-component mixture problem. The three components are Polyethylene (X1), polystyrene (X2), and polypropylene (X3), which are blended together to form fiber that will be spun into yarn. The product developers are only interested in the pure and binary blends of these three materials. The response variable of interest is yarn elongation in kilograms of force applied. A {3,2} simplex-lattice design is used to study the blending process. The simplex region and the six design runs are shown in the figure below. The figure was generated in JMP version 3.2. The design and the observed responses are listed in the table below. There were two replicate observations run at each of the pure blends. There were three replicate observations run at the binary blends. There are o15 observations with six unique design runs. Diagram showing the designs runs for this example 5.5.4.2. Simplex-lattice designs http://www.itl.nist.gov/div898/handbook/pri/section5/pri542.htm (4 of 7) [5/1/2006 10:31:17 AM] [...]... 9 14 Sum of Squares 1 28. 29600 6.56000 134 .85 600 Mean Square 25.6592 0 .72 89 F Ratio 35.2032 Prob > F < 0001 Tested against reduced model: Y=mean Parameter Estimates Term X1 X2 X3 X2*X1 X3*X1 X3*X2 Estimate 11 .7 9.4 16.4 19 11.4 -9.6 Std Error 0.603692 0.603692 0.603692 2.6 082 49 2.6 082 49 2.6 082 49 t Ratio 19. 38 15. 57 27. 17 7. 28 4. 37 -3. 68 Prob>|t| . 0.603692 15. 57 <.0001 X3 16.4 0.603692 27. 17 <.0001 X2*X1 19 2.6 082 49 7. 28 <.0001 X3*X1 11.4 2.6 082 49 4. 37 0.00 18 X3*X2 -9.6 2.6 082 49 -3. 68 0.0051 Interpretation of the JMP output Under. designs http://www.itl.nist.gov/div8 98 /handbook/ pri/section5/pri542.htm (7 of 7) [5/1/2006 10:31: 17 AM] 5.5.4.3. Simplex-centroid designs http://www.itl.nist.gov/div8 98 /handbook/ pri/section5/pri543.htm. X3 Observed Elongation Values 0.0 0.0 1.0 16 .8, 16.0 0.0 0.5 0.5 10.0, 9 .7, 11 .8 0.0 1.0 0.0 8. 8, 10.0 0.5 0.0 0.5 17. 7, 16.4, 16.6 0.5 0.5 0.0 15.0, 14 .8, 16.1 1.0 0.0 0.0 11.0, 12.4 Fit a quadratic mixture model