In a 4 x 3 factorial design, there are how many levels of the first grouping factor?

6.1 Introduction

It is widely accepted that the most commonly used experimental designs in manufacturing companies are full and fractional factorial designs at 2-levels and 3-levels. Factorial designs would enable an experimenter to study the joint effect of the factors (or process/design parameters) on a response. A factorial design can be either full or fractional factorial. This chapter is primarily focused on full factorial designs at 2-levels only. Factors at 3-levels are beyond the scope of this book. However, if readers wish to learn about experimental design for factors at 3-levels, the author would suggest them to refer to Montgomery (2001).

A full factorial designed experiment consists of all possible combinations of levels for all factors. The total number of experiments for studying k factors at 2-levels is 2k. The 2k full factorial design is particularly useful in the early stages of experimental work, especially when the number of process parameters or design parameters (or factors) is less than or equal to 4. One of the assumptions we make for factors at 2-levels is that the response is approximately linear over the range of the factor settings chosen. The first design in the 2k series is one with only two factors, say, A and B, each factor to be studied at 2-levels. This is called a 22 full factorial design.

Full factorial design

Full factorial design creates experimental points using all the possible combinations of the levels of the factors in each complete trial or replication of the experiments. The experimental design points in a full factorial design are the vertices of a hyper cube in the n-dimensional design space defined by the minimum and the maximum values of each of the factors. These experimental points are also called factorial points. For three factors having four levels of each factor, considering full factorial design, total 43 (64) numbers of experiments have to be carried out. If there are n replicates of complete experiments, then there will be n times of the single replication experiments to be conducted. In the experimentation, it must have at least two replicates to determine a sum of squares due to error if all possible interactions are included in the model.

4.2.5 Statistics

A full factorial design was made to obtain the drop of the interface pressure in 2 hours at all combination of the levels of the above factors. An N-way analysis of variance (ANOVA) for a fixed effect model was performed to check whether there were any significant différences in the mean values of the pressure drop at different levels of applied force, limb circumference and number of layers. A p-value less than 0.05 was considered as statistically significant. The number of repetitions of each individual test was kept fixed (i.e. 5) and this was ascertained after analyzing the mean values and variations of test results obtained for different number of replications.

Full Factorial Design (2k)

In a Full factorial design (FFD), the effect of all the factors and their interactions on the outcome (s) is investigated. A common experimental design is one, where all input factors are set at two levels each. These levels are termed high and low or + 1 and − 1, respectively. A design with all possible high/low groupings of all the input factors is termed as a full factorial design in two levels. If there are k factors, each at 2 levels, a full factorial design will be of 2k runs as mentioned earlier. As shown in Box 3.1, when the number of factors is more than five, a full factorial design requires a large number of experimental runs and is not effective. Therefore, a fractional factorial design or a Plackett-Burman design (PBD) is a better choice for five or more factors and is discussed in next section.

When a full factorial design for three input factors, each at two levels, is considered (23 design), it will have eight runs. Graphically, we can denote the 23 design by a cube shown in Fig. 3.4.

The arrows show the direction of increase of the factors. The numbers 1 through 8 at the corners of the design box represent the Standard Order of runs (Fig. 3.4).

In tabular form, this design can be represented as shown in Table 3.1.

Table 3.1. A 23 Two-Level, Full Factorial Design Table Showing Runs in ‘Standard Order’

The column on the left hand side of Table 3.1, which numbers up to 8, is called the Standard Run Order. These numbers are also depicted in Fig. 3.4. For example, run number 8th is made at the high setting of all three factors.

6.3.3 Factorial design

Factorial design is an useful technique to investigate main and interaction effects of the variables chosen in any design of experiment. This technique is helpful in investigating interaction effects of various independent variables on the dependent variables or process outputs. An example related to the current investigation is the effects of three variables on the response to fracture toughness, KIC. The variables and their levels are as follows: aspect ratio, AR (+1 and −1), interfacial strength, IS (+1 and −1), and volume fraction, VF (+1 and −1). According to factorial design, total runs to perform for these two level variables are, 23 = 8. The responses from the runs using main and interaction effects are shown in Table 6.1.

Table 6.1. Responses from different runs using main and interaction effects

Table 6.2. Variables with two levels for investigations of fractional factorial design

From the eight runs, the main effect of aspect ratio (AR) is found from the average difference in fracture toughness when the aspect ratio is set high (+1) and low (−1). In other words, it can be found using the following equation:

(6.2)(ΔKIC)AR=Y2+Y4+Y6+Y84− Y1+Y3+Y5+Y74

Similarly, the main effect of interfacial strength (IS) can be found using the following equation:

(6.3)(ΔKIC)IS=Y3+Y4+ Y7+Y84−Y1+Y2+Y5+Y64

The main effect of volume fraction (VF) can be found using the following equation:

(6.4)(ΔKIC)VF=Y5+Y6+Y7+Y84−Y1+Y2+Y 3+Y44

Similarly, interaction effects of aspect ratio and interfacial strength (AR∗IS) can be found using the following equation:

(6.5)(ΔK IC)AR∗IS=Y1+Y4+Y5+Y84−Y2+Y3+Y6+Y74

Interaction effects of other two combinations can be found using the following equations:

(6.6) (ΔKIC)AR∗VF=Y1+Y3+Y6+Y84−Y2+Y4+Y5+Y7 4

(6.7)(ΔKIC)IS∗VF=Y1+Y2+Y7+Y84−Y3+Y4+Y 5+Y64

There are several statistical tools available for analyzing and presenting the effects of the variables obtained from the design of experiments. Among those methods the current study involves the analysis of variations (ANOVA) and Pareto analysis.

Pareto chart presents the cumulative effect of individual variables and their interactions on the fracture toughness of MEYEB. The Pareto effects can be calculated using the following equation:

(6.8)Ei%=Ei−1+ Ei∑i=1nEi×100%

where Ei−1 and Ei stand for the effect of any variable and the effect of the next one, where all the effects are ordered from high to low.

A full factorial design is developed to investigate the potential effects of several independent variables on the fracture toughness (KIC) of MEYEB. These variables are aspect ratio (AR), interfacial strength (IS), volume fraction (VF), temperature (T), and environment (E). These variables are expected to have combinations of linear and nonlinear effects on the dependent variable, fracture toughness. For the simplicity in the design, the linear effects of aspect ratio and environment are investigated with nonlinear effects of interfacial strength, volume fraction, and temperature. Linear effects can be measured with the two level values, low and high. However nonlinear effects can be estimated with several intermediate levels along with low and high. By combining all these linear and nonlinear effects, a full factorial design is performed with the combinations of two, three, and four- level variables. A full factorial design using five variables above allows the investigation of main effects and five way interactions between the independent variables.

4.1 Simplified models for the molecular weight and the conversion of the monomer

A way to obtain simplified models is using full factorial designs, which are important means to evaluate the influence of the factors on response. However, they have the inconvenience of requiring too many runs when working with a great number of factors. As the amount of runs increases exponentially with the number of involved variables,it is interesting to pick up the most significant factors before run the full factorial design when there are too many options to be studied. For screening purposes, the influence of all the 30 kinetic parameters on number average molecular weight (MWN) and monomer conversion (XCL) were evaluated by using of a2-level fractional experimental design and a central composite design. These additional runs are used in order to generate a quadratic model for the responses. Because of the parameters E1c,E30,E3c for XCL and E2c,E20,E3c for MWN were statistically most significant, these were chosen to develop the mathematical models. Even though experimental data of two process variables were considered to estimate the five parameters of the models, alternative computational tools were used to achieve satisfactory results and to validate these complex models. It is important to emphasize these statistic models were formulated in a specific behavior range of the experimental variable, thus calculations out of this range could result in less reliable results.

To obtain the simplified models, composite factorial designs were performed with the selected parameters. Level zero values were defined as the same as those estimated by Arai et al. (1981) and the variations of levels −1 and +1 were 15 % from level 0. The simplified models achieved for MWN and XCL (variables in coded form) can be found at Eqs. (3) and (4), respectively. Analyses were carried out by using of the software Statistica. Furthermore, the good fit Eqs. (3) and (4) provides can be confirmed by the F-test: it is almost 3 times greater than the F-tabled value for 95 % confidence level.

(3) MWN=6459.004−446.833E20−93.710(E20 )2−578.825E2c−3636.732E3c−429.071(E2c)2−258.096E20E2c+546.275E20 E3c+299.598E2cE3c

(4)XCL=0.889−0.088E1c−0.101(E1c)2 −0.132E30+0.015(E30)2−0.242E3c−0.126(E 3c)2+0.006E1cE30−0.011E1cE 3c−0.219E30E3c

3.5.3.2 Two-level fractional factorial designs

It is obvious as the number of factors in full factorial design rises, the number of realizations increases. For example, a complete 26 design requires 64 runs. In this design only 6 of the 64 runs correspond to the main effects, and 15 runs correspond to two-factor interactions. The remaining runs are associated with three-factor and higher interactions. The more runs that are done, the more budget and time required. If the engineers can reasonably assume that some higher-order interactions (e.g. third-order and higher) are not important, then information on the main effects and two-order interactions can be obtained by running only a fraction of the full factorial experiment. This design is called fractional factorial design. Fractional factorial designs are the most widely and commonly used types of design in industry. These designs are generally represented in the form 2(k−p), where k is the number of factors and 1/2p represents the fraction of the full factorial of 2k. For example, 2(6−2) is a {1/4} fraction of a 64 full factorial experiment. This means that one may be able to study 6 factors at two levels in just 16 runs rather than 64 runs.

The 23−1 design is called a resolution III design, where a main effect is aliased with two-factor interactions (C=AB). Table 3.6 shows this type of design. In a resolution IV design, two-factor interactions are aliased with each other (D=ABC); see Table 3.7. Table 3.8 shows a resolution V for five factors where two-factor interactions are aliased with three-factor interactions [Montgomery, 2001].

Table 3.6. Fractional factorial design, Resolution III

Table 3.7. Fractional factorial design, Resolution IV

Table 3.8. Fractional factorial design, Resolution V

7.4.3 Two-Level Full Factorial Design

We performed 16 simulation cases based on the two-level full factorial design. The half normal plots from the design are shown in Fig. 7.10. The terms with highest effect are on the right side of the plot. It can be seen that the large effects came from the two-way and three-way interaction terms between matrix permeability, fracture conductivity, and fracture height, which were selected in Fig. 7.10A (highlighted points). The effect of fracture half-length is not as important as others, which can be approximated by normal distribution line (orange line in the plot) along with others with small effect. The Shapiro-Wilk test (Shapiro and Wilk, 1965) is used to evaluate if the selected parameters are relatively important through rejection of null hypothesis, which means that the effects of unselected parameters are normally distributed. We also need to compare the Shapiro-Wilk test when no selection is made and make sure that the effects of selected parameters are indeed significant and cannot be approximated by normal distribution.

The Shapiro-Wilk P-value is determined as 0.107 when significant effects listed earlier are selected. The null hypothesis cannot be rejected since the value was slightly higher than an assumed alpha level of 0.10. However, when no terms were selected in Fig. 7.10B, the Shapiro-Wilk P-value is 0.108, which is still larger than the assumed alpha level of 0.10, indicating that no individual effect in the design is statistically insignificant (Stat-Ease, 2015); in other words, the null hypothesis cannot be rejected. Hence, we cannot drop any parameters to reduce uncertainties. Table 7.2 presents the summary of every uncertain parameter with the minimum and maximum value and response parameters. The comparison of gas flow rate and cumulative gas production between 16 simulation cases and actual production data is shown in Fig. 7.11. It can be observed that the simulation results under the given range of uncertain parameters have expanded throughout the actual data. Hence, it is guaranteed that the history-matching solutions will be within the design space of uncertain parameters.

Table 7.2. Uncertain and Response Parameters Considered in This Study(Wantawin et al., 2017)

6.4.2 Box–Behnken designs

The Box–Behnken designs of experiments provide modeling of the response surface. These designs are not based on full or fractional factorial designs. The design points are positioned at the middle of the subareas of the dimension k-1. In the case of three factors, for instance, the points are located in the middle of the edges of the experimental domain (see Figure 6.4).

These designs require three levels per factor. The Box–Behnken design for three factors does not comply with the criteria of iso-variance per rotation. However, the designs above, having more than three factors, can meet the iso-variance criteria if center points are added. These designs can also respect the orthogonality criteria.

The Box–Behnken designs make it possible to study sequentially the effect of the various factors of the design if, during the study of the first factors, the other factors are maintained at a constant level. Table 6.4 shows for a Box–Behnken DOE the number of trials and number of coefficients to estimate in case of 3, 4, 5 and 6 factors.

Table 6.4. Number of trials in a Box–Behnken design (one trial in the center point)

5 Conclusions, discussion, and way forward

We have adopted a complex systems perspective and built an agent based simulation model accordingly. This simulation model allows for full-factorial design experiments, without the need to construct multi-dimensional, highly complicated and interdependent superstructures. First simulation results provide insights in some of the different parameters that may drive technology choices for central and decentral networks. Agent based models like this may in future thus support early stage field developers in exploring vast design spaces, and through such exploratory modeling the main cost drivers can be identified.

The current model has some important shortcomings Geographical distances and construction time are not taken into account in the current model. Including the costs of the pipelines in the costs calculation could lead to more realistic visualization of different network configurations in the model and could positively influence the profitability of decentral line ups as pipelines for sour gas are more expensive than carbon steel pipelines. Finally, compared to agent based models that are used for social simulation, the model is limited with respect to human decision making.