Pareto plot

After data analysis, the statistically significant effects induced by certain factors are identified. This is typically achieved using standardized Pareto plots (Figure 14). The plot shows the standard values of the effects in descending order of magnitude. The length of each bar is proportional to the standardized effect, which is equal to the effect divided by the standard error. This is identical to the calculation of a lvalue for... 

FIGURE 14 Standardized Pareto plot. The standardized effects are plotted as a function of their extends.The dark line is the border of decision according to a t-test. Critical effects exceed this line. 

Figure 4.45. Pareto plot of interpolated catalysts predicted to be good compromises with respect to cost and activity for methanation. The positions of the interpolated catalysts are determined by the cost of their constituent elements vs. their distance from the optimal dissociative chemisorption energy for CO with respect to the experimentally observed optimum (see Figure 4.44 right-bottom). Adapted from Ref. [55].

Figure 7 Pareto plot demonstrating the relative importance of main effects and interaction effects for immunoaffinity chromatography step.

Figure 16.1 shows the graphical representation (Pareto plot) of the size effect of each of the parameters investigated upon resolution of the peaks... 

Figure 15 Pareto plot. The points connected by the line are selected as optima when maximizing both criteria C and S...

Plotting this data as a Pareto chart gives Figure 3. It shows that the load is the dominant variable in the problem and so the stress is very sensitive to changes in the load, but the dimensional variables have little impact on the problem. Under conditions where the standard deviation of the dimensional variables increased for whatever reason, their impact on the stress distribution would increase to the detriment of the contribution made by the load if its standard deviation remained the same. 

Palladium electrocatalysts, 183 Palladium-alloy electrocatalysts, 298-300 Pareto-optimal plot, 85 Platinum-alloy electrocatalysts, 6, 70-71, 284-288, 317-337 Platinum-bismuth, 86-87, 224 Platinum chromium, 361 362 Platinum-cobalt, 71, 257-260, 319, 321-330, 334-335 Platinum-iron, 319, 321, 334-335 Platinum-molybdenum, 253, 319-320... 

Figure 4.17 Plot of the feasible criteria space of the crushing strength and the disintegration time = Pareto-optimal point o = inferior point...

There were 87 compositions which showed no spot crossover. To select from these 87 compositions the Pareto Optimal points [22] were calculated (maximizing all four criteria). There were nine such points, these are given in Table 6.7. Plots of the minimum resolution for all these compositions were made, and finally the composition DEA=0.08, MeOH=0, CHCl3=0.16, EtAc=0.76 was selected as resulting in the best preferred separation. In Figure 6.7 the change of minimum resolution at this mixture composition at different temperatures and relative humidities is depicted. It is clear that the resolution is reasonably well for most temperatures and relative humidities, but at real humid situations the resolution declines. 

The resulting library consisted of 250 compounds representing different compromises between the two conflicting objectives supplied. Figure 3.5 presents a plot of the Pareto-approximation proposed by the software library (circles connected by line). Each of the remaining circles represents a solution from the initial population set after the hard filtering process. The x-axis represents similarity to ER-a ligands and the -axis dissimilarity (1-similarity)... 

The simplest way to analyse the effects is, perhaps, by means of proper graphics, among which the Pareto chart and the main effects or interactions effects plots are used widely. In a Pareto chart we represent the different effects ordered by magnitude (absolute value, on the vertical axis) and the magnitude... 

The answers need an analysis of the s/n ratio. Table 2.17 summarises the Pareto analysis of variance (see Figure 2.7 for the calculations involved) and Figure 2.8 displays the factor plots associated with the calculations. 

A useful plot for identifying factors that are important is a Pareto chart. The graph in Fig. 1 shows the t-test values in the horizontal axis and also includes a vertical line to indicate the p value (an effect that exceeds the vertical line maybe considered significant). As observed in the Pareto chart, enzyme concentration is the most significant variable influencing monolaurin molar fraction. 

Fig. 6.5. Time and the value of a hypothetical separation criterion for ten experiments in a Pareto optimality plot experiments I, 2 and 5 are Pareto optimal.

Figure 18.43 Pareto optimal solution for the multiobjective optimization (maximum purity and production rate) of the SMB unit for various particle sizes. The (a) purity of the extract, (b) flow rate, (c) number of theoretical stages per column, and (d) pressure drop are plotted as the function of production rate. Reproduced with permission from Z. Zhang et ah,. Chromatogr., 989 (2003) 95 (Fig. 4).

The resulting print-out from MINITAB is given in Figure 2. However, the easiest way to interpret the data is to look at the pareto chart of calculated effects (Fig. 3) and related normal probability plot (Fig. 4). The vertical line on the pareto chart in Figure 3 corresponds to a P-value of 0.10 for each calculated effect. In other words. 

The MINITAB analysis Printout is given in Figure 8. The pareto chart (Fig. 9) and corresponding normal probability plot (Fig. 10) show that there are no statistically significant main effects or interactions present. Note that the bars are all to the left of the verticle line on the pareto chart and none of the points on the probability plot are labeled. In addition, looking at the data... 

In Figure 1.5, the first plot shows the two objectives (P on the x-axis and X7 on the y-axis) and the remaining plots show the optimal values of all other decision variables versus P. Increase in P from 1,000 to 1,106 /day is accompanied by x decreasing from 95.3 to 94.3 thus, the two objectives, P and x-j are contradictory leading to the optimal Pareto front in the first plot in Figure 1.5. All other decision variables with the exception of X2 (isobutane recycle) which remains at its upper bound, contribute to the optimal Pareto front. Interestingly, each of them varies with P at certain rate until P is about l,110/day and then follows a different trend - xi, X4, xs and xg become constant, xs and xe start to decrease at P >, 1110/day. Of these, the trend of xs is striking - it increases with P initially and then decreases for P > 1,110/day. 

Fig. 1.5 Pareto-optimal solutions for maximizing profit and octane number (xq) by the -constraint method profit is shown on the x-axis in all plots.

NAGA was used by Mu et al. (2004) to optimize the operahon of a paraxylene oxidahon process to give terephthalic acid. They consider two objectives minimization of the concentration of the undesirable 4-carboxy-benzaldehyde (4-CBA) in the product stream, and maximization of the feed flow rate of the paraxylene. They consider four ophmization problems using a different number of decision variables (1, 2, 4 and 6 variables). The problem has two constraints. The plot of the Pareto front obtained presented a convex and conhnuous curve. 

Fig. 4.12 Plots of non-dominated solutions obtained with NSGA-II-JG after 240,000 -300,000 function evaluations (fn. evals.) and for NSGA-II for 320,000 - 400,000 fn. evals. for the ZDT4 problem. Note that /j and I2 extend over [0, 1] (global Pareto set) for NSGA-II-JG only after about 300,000 function evaluations, and do not show this characteristic for NSGA-II...

Since most of this book is devoted to evolutionary methods for multiobjective optimization, we here only wish to discuss some differences between EMO approaches and scalarization based approaches. As mentioned before, EMO approaches are a posteriori type of methods and they try to generate an approximation of the Pareto optimal set. In bi-objective optimization problems, it is easy to plot the objective vectors produced on a plane and ask the DM to select the most preferred one. While looking at the... 

Fig. 7.4 shows the ranked Pareto domain for the two-objective optimization of gluconic acid. This plot of Pf Its versus Pf is a typical Pareto domain for a two-objective optimization where both objective functions need to be maximized. It may be tempted to believe that maximizing Pf and minimizing tg would be equivalent to this two-objective problem. However, this is not the case. Indeed, a very different Pareto domain, which would include very low and very high values of the batch time, would be obtained. Using the productivity Pfltg) is truly the best way to define the desired objective. 

Fig. 7.5 Plot of the input space for the two-objective optimization problem associated with the Pareto domain Kia versus...

Fig. 7.11 Plot of the input space for the three-objective Pareto domain using RSM A a versus ts-...

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