14.1 For each of the following equations determine the optimum response using the one-factor-at-a-time searching algorithm. Begin the search at (0,0) by first changing factor A, using a step-size of 1 for both factors. The boundary conditions for each response surface are 0 ≤ A ≤ 10 and 0 ≤ B ≤ 10. Continue the search through as many cycles as necessary until you find the optimum response. Compare your optimum response for each equation to the true optimum. \(R = 1.68 + 0.24A + 0.56B - 0.04A^2 - 0.04B^2 \hspace{30px} \mu_\ce{opt} = (3, 7)\) \(R = 4.0 - 0.4A + 0.08B \hspace{30px} \mu_\ce{opt} = (10, 10)\) \(R = 3.264 + 1.537A + 0.5664B - 0.1505A^2 - 0.02734B^2 - 0.05785AB \hspace{30px} \mu_\ce{opt} = (391, 6.22)\) Note: These equations are from Deming, S. N.; Morgan, S. L. Experimental Design: A Chemometric Approach , Elsevier: Amsterdam, 1987, and pseudo-three dimensional plots of the response surfaces can be found in their Figures 11.4, 11.5 and 11.14. 14.2. Determine the optimum response for the equation in Problem 14.1c, using the fixed-sized simplex searching algorithm. Compare your optimum response to the true optimum. 14.3 Show that equation 14.3 and equation 14.4 are correct. 14.4 A 2 k factorial design was used to determine the equation for the response surface in Problem 1b. The uncoded levels, coded levels, and the responses are shown in the following table. A B A* B* response 8 8 +1 +1 5.92 8 2 +1 –1 2.08 2 8 –1 +1 4.48 2 2 –1 –1 3.52 Determine the uncoded equation for the response surface. 14.5 Koscielniak and Parczewski investigated the influence of Al on the determination of Ca by atomic absorption spectrophotometry using the 2 k factorial design shown in the following table. 11 Ca2+ (ppm) Al3+ (ppm) Ca* Al* response 10 160 +1 +1 54.92 10 0 +1 –1 98.44 4 160 –1 +1 19.18 4 0 –1 –1 38.52 Determine the uncoded equation for the response surface. If you wish to analyze a sample that is 6.0 ppm Ca 2 + , what is the maximum concentration of Al 3 + that can be present if the error in the response must be less than 5.0%? 14.6 Strange reports the following information for a 2 3 factorial design used to investigate the yield of a chemical process. 12 factor high (+1) level low (–1) level X: temperature 140oC 120oC Y: catalyst type B type A Z: [reactant] 0.50 M 0.25 M run X* Y* Z* % yield 1 –1 –1 –1 28 2 +1 –1 –1 17 3 –1 +1 –1 41 4 +1 +1 –1 34 5 –1 –1 +1 56 6 +1 –1 +1 51 7 –1 +1 +1 42 8 +1 +1 +1 36 Determine the coded equation for this data. If β terms of less than ±1 are insignificant, what main effects and interaction terms in the coded equation are important? Write down this simpler form for the coded equation. Explain why the coded equation for this data can not be transformed into an uncoded form. Which is the better catalyst, A or B? What is the yield using this catalyst if the temperature is set to 125 o C and the concentration of the reactant is 0.45 M? 14.7 Pharmaceutical tablets coated with lactose often develop a brown discoloration. The primary factors affecting the discoloration are temperature, relative humidity, and the presence of a base acting as a catalyst. The following data have been reported for a 2 3 factorial design. 13 factor high (+1) level low (–1) level X: benzocaine present absent Y: temperature 40oC 25oC Z: relative humidity 75% 50% run X* Y* Z* color (arb. units) 1 –1 –1 –1 1.55 2 +1 –1 –1 5.40 3 –1 +1 –1 3.50 4 +1 +1 –1 6.75 5 –1 –1 +1 2.45 6 +1 –1 +1 3.60 7 –1 +1 +1 3.05 8 +1 +1 +1 7.10 Determine the coded equation for this data. If β terms of less than 0.5 are insignificant, what main effects and interaction terms in the coded equation are important? Write down this simpler form for the coded equation. 14.8 The following data for a 2 3 factorial design were collected during a study of the effect of temperature, pressure, and residence time on the % yield of a reaction. 14 factor high (+1) level low (–1) level X: temperature 200oC 100oC Y: pressure 0.6 MPa 0.2 MPa Z: residence time 20 min 10 min run X* Y* Z* percent yield 1 –1 –1 –1 2 2 +1 –1 –1 6 3 –1 +1 –1 4 4 +1 +1 –1 8 5 –1 –1 +1 10 6 +1 –1 +1 18 7 –1 +1 +1 8 8 +1 +1 +1 12 Determine the coded equation for this data. If β terms of less than 0.5 are insignificant, what main effects and interaction terms in the coded equation are important? Write down this simpler form for the coded equation. Three runs at the center of the factorial design—a temperature of 150 o C, a pressure of 0.4 MPa, and a residence time of 15 min—give percent yields of 12%, 8%, 9%, and 8.8%. Determine if a first-order empirical model is appropriate for this system at α = 0.05. 14.9 Duarte and colleagues used a factorial design to optimize a flow-injection analysis method for determining penicillin. 15 Three factors were studied: reactor length, carrier flow rate, and sample volume, with the high and low values summarized in the following table. factor high (+1) level low (–1) level X: reactor length 1.5 cm 2.0 cm Y: carrier flow rate 1.6 mL/min 2.2 mL/min Z: sample volume 100 μL 150 μL The authors determined the optimum response using two criteria: the greatest sensitivity, as determined by the change in potential for the potentiometric detector, and the largest sampling rate. The following table summarizes their optimization results. run X* Y* Z* ∆E(mV) samples/h 1 –1 –1 –1 37.45 21.5 2 +1 –1 –1 31.70 26.0 3 –1 +1 –1 32.10 30.0 4 +1 +1 –1 27.20 33.0 5 –1 –1 +1 39.85 21.0 6 +1 –1 +1 32.85 19.5 7 –1 +1 +1 35.00 30.0 8 +1 +1 +1 32.15 34.0 Determine the coded equation for the response surface where ∆ E is the response. Determine the coded equation for the response surface where sample/h is the response. Based on the coded equations, do conditions favoring sensitivity also improve the sampling rate? What conditions would you choose if your goal is to optimize both sensitivity and sampling rate? 14.10 Here is a challenge! McMinn, Eatherton, and Hill investigated the effect of five factors for optimizing an H 2 -atmosphere flame ionization detector using a 2 5 factorial design. 16 The factors and their levels were factor high (+1) level low (–1) level A: H2 flow rate 1460 mL/min 1382 mL/min B: SiH4 20.0 ppm 12.2 ppm C: O2 + N2 flow rate 255 mL/min 210 mL/min D: O2/N2 1.36 1.19 E: electrode height 75 (arb. unit) 55 (arb. unit) The coded (“+” = +1, “–” = –1) factor levels and responses, R , for the 32 experiments are shown in the following table run A* B* C* D* E* run.1 A*.1 B*.1 C*.1 D*.1 E*.1 1 – – – – – 17 – – – – + 2 + – – – – 18 + – – – + 3 – + – – – 19 – + – – + 4 + + – – – 20 + + – – + 5 – – + – – 21 – – + – + 6 + – + – – 22 + – + – + 7 – + + – – 23 – + + – + 8 + + + – – 24 + + + – + 9 – – – + – 25 – – – + + 10 + – – + – 26 + – – + + 11 – + – + – 27 – + – + + 12 + + – + – 28 + + – + + 13 – + + + – 29 – – + + + 14 + – + + – 30 + – + + + 15 – + + + – 31 – + + + + 16 + + + + – 32 + + + + + Determine the coded equation for this response surface, ignoring β terms less than ±0.03. A simplex optimization of this system finds optimal values for the factors of A = 2278 mL/min, B = 9.90 ppm, C = 260.6 mL/min, and D = 1.71. The value of E was maintained at its high level. Are these values consistent with your analysis of the factorial design? 14.11 A good empirical model provides an accurate picture of the response surface over the range of factor levels within the experimental design. The same model, however, may yield an inaccurate prediction for the response at other factor levels. For this reason, an empirical model, is tested before extrapolating to conditions other than those used in determining the model. For example, Palasota and Deming studied the effect of the relative amounts of H 2 SO 4 and H 2 O 2 on the absorbance of solutions of vanadium using the following central composite design. 17 run drops 1% H2SO4 drops 20% H2O2 1 15 22 2 10 20 3 20 20 4 8 15 5 15 15 6 15 15 7 15 15 8 15 15 9 22 15 10 10 10 11 20 10 12 15 8 The reaction of H 2 SO 4 and H 2 O 2 generates a red-brown solution whose absorbance is measured at a wavelength of 450 nm. A regression analysis on their data yielded the following uncoded equation for the response (absorbance × 1000). \[R = 835.90 - 36.82X_1 - 21.34X_2 + 0.52(X_1)^2 + 0.15(X_2)^2 + 0.98X_1X_2\] where X 1 is the drops of H 2 O 2 , and X 2 is the drops of H 2 SO 4 . Calculate the predicted absorbances for 10 drops of H 2 O 2 and 0 drops of H 2 SO 4 , 0 drops of H 2 O 2 and 10 drops of H 2 SO 4 , and for 0 drops of each reagent. Are these results reasonable? Explain. What does your answer tell you about this empirical model? 14.12 A newly proposed method is to be tested for its single-operator characteristics. To be competitive with the standard method, the new method must have a relative standard deviation of less than 10%, with a bias of less than 10%. To test the method, an analyst performs 10 replicate analyses on a standard sample known to contain 1.30 ppm of analyte. The results for the 10 trials are 1.25 1.26 1.29 1.56 1.46 1.23 1.49 1.27 1.31 1.43 Are the single-operator characteristics for this method acceptable? 14.13 A proposed gravimetric method was evaluated for its ruggedness by varying the following factors. Factor A: sample size A = 1 g a = 1.1 g Factr B: pH B = 6.5 b = 6.0 Factor C: digestion time C = 3 h c = 1 h Factor D: number rinses D = 3 d = 5 Factor E: precipitant E = reagent 1 e = reagent 2 Factor F: digestion temperature F = 50oC f = 60oC Factor G: drying temperature G = 110oC g = 140oC A standard sample containing a known amount of analyte was carried through the procedure using the experimental design in Table 14.5. The percentage of analyte actually found in the eight trials were found to be 0 1 2 3 R1 = 98.9 R5 = 98.8 R2 = 98.5 R6 = 98.5 R3 = 97.7 R7 = 97.7 R4 = 97.0 R8 = 97.3 Determine which factors, if any, appear to have a significant affect on the response, and estimate the expected standard deviation for the method. 14.14 The two-sample plot for the data in Example 14.6 is shown in Figure 14.21. Identify the analyst whose work is (a) the most accurate, (b) the most precise, (c) the least accurate, and (d) the least precise. 14.15 Chichilo reports the following data for the determination of the %w/w Al in two samples of limestone. 18 analyst sample 1 sample 2 1 1.35 1.57 2 1.35 1.33 3 1.34 1.47 4 1.50 1.60 5 1.52 1.62 6 1.39 1.52 7 1.30 1.36 8 1.32 1.53 Construct a two-sample plot for this data and estimate values for σ rand and σ syst . 14.16 The importance of between-laboratory variability on the results of an analytical method can be determined by having several laboratories analyze the same sample. In one such study, seven laboratories analyzed a sample of homogenized milk for a selected alfatoxin. 19 The results, in ppb, are summarized below. lab A lab B lab C lab D lab E lab F lab G 1.6 4.6 1.2 1/5 6.0 6.2 3.3 2.9 2.8 1.9 2.17 3.9 3.8 3.8 3.5 3.0 2.9 3.4 4.3 5.5 5.5 1.8 4.5 1.1 2.0 5.8 4.2 4.9 2.2 3.1 2.9 3.4 4.0 5.3 4.5 Determine if the between-laboratory variability is significantly greater than the within-laboratory variability at α = 0.05. If the between-laboratory variability is significant, then determine the source(s) of that variability. Estimate values for σ rand 2 and σ syst 2 . 14.17 Show that the total sum-of-squares ( SS t ) is the sum of the within-sample sum-of-squares ( SS w ) and the between-sample sum-of-squares ( SS b ). See Table 14.7 for the relevant equations. 14.18 Eighteen analytical students are asked to determine the %w/w Mn in a sample of steel, with the results shown here. 0.26% 0.28% 0.27% 0.24% 0.26%.1 0.25% 0.26% 0.28% 0.25% 0.24% 0.26% 0.25% 0.29% 0.24% 0.27% 0.23% 0.26% 0.24% (a) Given that the steel sample is 0.26% w/w Mn, estimate the expected relative standard deviation for the class’ results. (b) The actual results obtained by the students are shown here. Are these results consistent with the estimated relative standard deviation? 14.5.3 Solutions to Practice Exercises Practice Exercise 14.1 If we hold factor A at level A 1 , changing factor B from level B 1 to level B 2 increases the response from 40 to 60, or a change ∆ R , of \[R = 60 - 40 = 20\] If we hold factor A at level A 2 , we find that we have the same change in response when the level of factor B changes from B 1 to B 2 . \[R = 100 - 80 = 20\] Click here to return to the chapter. Practice Exercise 14.2 If we hold factor B at level B 1 , changing factor A from level A 1 to level A 2 increases the response from 20 to 80, or a change ∆ R , of \[R = 80 - 20 = 60\] If we hold factor B at level B 2 , we find that the change in response when the level of factor A changes from A 1 to A 2 is now 20. \[R = 80 - 20 = 60\] Click here to return to the chapter. Practice Exercise 14.3 Answers will vary here depending on the options you decided to explore. The last response surface is an interesting one to explore. Figure 14.26 shows the response surface as a level plot and a contour plot. The interesting feature of this surface is the saddle point on a ridge connecting a local optimum (maximum response of 4.45) and the global optimum (maximum response of 10.0). All three optimization strategies are very sensitive to the initial position and the step-size. Click here to return to the chapter. Contributors David Harvey (DePauw University)