*2.5. Design of Experiments (DOE)*

For each honing step, a central composite design was conducted in order to be able to obtain second order models for the responses. Minitab statistical software version 19, (Minitab LLC, State College, PA, USA) was used. The cube experimental runs were defined as a fractional factorial design 25−<sup>1</sup> with 16 runs. The axial runs were defined with 10 facecentered points plus three central points. Table 1 shows the variables and levels employed for rough, semi-finish and finish operations.

**Table 1.** Low and high levels for the different variables employed in the rough, semi-finish and finish experiments.


As can be seen in Table 1, the same levels were used for pressure, tangential speed and linear speed for the three honing steps. The values for the different parameters were selected according to the manufacturers' recommendation and to the literature. For instance, Vrac et al. [7] recommended grain size 181 and 151 in normal honing. These values lie within the range that was selected in the present work for rough honing. Grain size and abrasive density usually decrease as the honing process advances in order to achieve finer and finer surfaces.

## *2.6. Multiobjective Optimization*

In the present paper, the desirability function method was used to carry out multiobjective optimization [14].

The process searches for a combination of the factors that gives the best possible compromise for all the factors. This is achieved following these steps:


The individual desirability functions map each one of the responses onto a value ranging from 0 to 1 (0 meaning that the level of the response is not what was wanted; 1 meaning that the level of the response is most preferred, the target). The formula depends on whether one wants to minimize the response, maximize the response or set the response to a target. In our study, we want to minimize roughness, Ra, and tool wear, Qp, and maximize the material removal rate, Qm.

Figure 4 shows the shape of the function when minimizing (left) or maximizing (right) the response. In our study, we use the target and upper and lower bounds as the maximum and minimum response values obtained, depending on the situation. A weight of 1 was used in all cases, corresponding to the use of a linear function.

**Figure 4.** Desirability functions when minimizing (left) and maximizing (right) the response.

The composite desirability function D is computed using the formula shown in Equation (4).

$$\mathbf{D} = \left(\prod \mathbf{d}\_i^{\mathrm{Imp}\_i}\right)^{\mathrm{MW}} \tag{4}$$

Impi is the importance given to response i. IMP is the sum of all importance values, ∑ Impi . One can set the importance for each response so that the sum is one, thus simplifying the formula and giving the idea that the importance for each response is a percentage of importance.

The importance values for each of the three responses in this study are shown in Table 2. They were selected from previous honing experiments. The following criteria were employed: in rough operations it is important to remove as much material as possible, while in finish operations surface finish is crucial. Thus, the importance values increase for roughness in subsequent honing operations, while they decrease for material removal rate and tool wear. In other words, in rough honing high importance values of Qm and Qp, as well as low values for Ra, are recommended. On the contrary, in finish honing high importance values are required for Ra and low values for Qm and Qp.

**Table 2.** Importance values used for each response and honing phase in the optimization.


One of the main objectives of this study is assessing to what extend the results are dependent on the importance given to each of the responses. To achieve this objective, the importance of each response was later varied, in order to perform a sensitivity analysis of the optimization process (Section 2.7).

### *2.7. Sensitivity Analysis*

The purpose of the sensitivity analysis is to determine the effect of a certain change in the importance values of the responses on the optimal values of the variables that are obtained from the multi-objective optimization. In order to achieve this, the values of the importance for the different responses were varied from the initially defined values in Table 2 with the help of a mixture design. Values of importance were varied from a slight degree (1%) to a considerable degree (15%) (the higher the variation in the initial importance values, the higher the expected impact on the optimization results).

Mixture designs are special experiments in which the product being studied is composed of different ingredients. These ingredients cannot be modified independently: if the percentage of one ingredient in the formula increases, the percentages of others must decrease, as the total always sums to 1 [29]. These experiments are commonly used in pharma or food investigations. We have used a mixture design to change in an organized and balanced way the importance of each response in our optimization problem.

For instance, Figure 5 shows the experiments performed for the finish step. The central point corresponds to the initial importance values shown in Table 2 (Ra = 0.8, Qm = 0.1, Qp = 0.1). The other points are slight variations of these importance values, always summing to 1.

**Figure 5.** Optimization runs performed for the sensibility analysis in the finish step, with a 1% variation of the importance.

For each one of the runs coming from the mixture design we have a combination of values of the variables Gs, De, Pr, Vt and Vl that globally optimize the three responses. In order to see the extent to which these values vary depending on the run, the coefficient of variation CV was calculated for each variable, Gs, De, Pr, Vt and Vl, and for each percentage of variation of importance (1%, 3%, 5%, 10%, 15%).
