**Appendix B Inducing Possibility Distributions (Fuzzy Numbers) from Numerical Data**

This appendix describes the mathematical procedures used to induce the possibility distributions (fuzzy numbers) from the time series of machining forces. The same procedure can be found in [25,26]. Let *x*(*t*) ∈ , *t* = 0, ... , *n* − 1 be *n* data points in the form of a time series, as shown in Figure A4.

**Figure A4.** A given set of numerical data.

Let (*x*(*t*), *x*(*t* + 1)), *t* = 0, ... , *n* − 1 be a point-cloud in the universe of discourse *X* = [*x*min, *x*max] so that *x*min < min(*x*(*t*)| ∀*t* ∈ {0, ... , *n*}) and *x*max > max(*x*(*t*)| ∀*t* ∈ {0, ... , *n*}). Let *A* and *B* two square boundaries so that the vectors of the vertices of *A* and *B* (in the anti-clockwise direction) are ((*x*min, *x*min), (*x*, *x*min), (*x*, *x*), (*x*min, *x*)) and ((*x*max, *x*max), (*x*, *x*max), (*x*, *x*), (*x*max, *x*)), respectively, ∀*x* ∈ *X*. As such, (*x*, *x*) is the common vertex of A and B. For example, consider the arbitrary point-cloud shown in Figure A5. According to Figure A5, the universe of discourse is as follows, *X* = [20, 80]. Notice the relative positions of the boxes denoted as *A* and *B* in Figure A5. The boxes are connected at their common vertices.

**Figure A5.** Relative position of *A* and *B* in the point-cloud (*x*(*t*), *x*(*t* + 1)).

Let Pr*A*(*x*) and Pr*B*(*x*) be two subjective probabilities, wherein Pr*A*(*x*) and Pr*B*(*x*) represent the degrees of chance that the points in the point-cloud are in *A* and *B*, respectively. As such, these functions are defined by the following mappings:

$$\begin{aligned} &X \to [0,1] \\ &\mathbf{x} \mapsto \text{Pr}\_A(\mathbf{x}) = \frac{\sum\_{i=0}^{n-1} \Theta(t)}{n-1} \\ &\Theta(t) = \begin{cases} 1 & \left( (\mathbf{x}(t) \le \mathbf{x}) \land (\mathbf{x}(t+1) \le \mathbf{x}) \right) \\ 0 & \text{otherwise} \end{cases} \end{aligned} \tag{A1}$$

$$\begin{aligned} &X \to [0,1] \\ &\mathbf{x} \mapsto \text{Pr}\_{B}(\mathbf{x}) = \frac{\sum\_{i=0}^{n-1} \Omega(t)}{\frac{n-1}{n-1}} \\ &\Omega(t) = \begin{cases} 1 & \left( (\mathbf{x}(t) \ge \mathbf{x}) \wedge (\mathbf{x}(t+1) \ge \mathbf{x}) \right) \\ 0 & \text{otherwise} \end{cases} \end{aligned} \tag{A2}$$

The typical natures of the functions defined in Equations (A1) and (A2) are illustrated in Figure A6, using the information of the point-cloud shown in Figure A5. Note that Pr*A*(*x*) increases with the increase in *x*, and the opposite is true for Pr*B*(*x*). It is worth mentioning that Pr*A*(*x*) + Pr*B*(*x*) ≤ 1 for the point-cloud, though for some cases, Pr*A*(*x*) + Pr*B*(*x*) = 1 (see Figure A7). This means that the expression Pr*A*(*x*) + Pr*B*(*x*) does not serve the role of "cumulative probability distribution". A cumulative probability distribution can, however, be formulated by using the information of Pr*A*(*x*) and Pr*B*(*x*), as shown in Figure A7.

**Figure A6.** The typical nature of Pr*A*(*x*) and Pr*B*(*x*) for unimodal quantity.

**Figure A7.** Nature of Pr*A*(*x*) + Pr*B*(*x*) and min(Pr*A*(*x*), Pr*B*(*x*)) for unimodal data.

Consider a mapping that maps *x* into the minimum of Pr*A*(*x*) and Pr*B*(*x*), as follows:

$$\begin{aligned} \mathbf{x} &\to [0, a] \\ \mathbf{x} &\mapsto \mathbf{g}(\mathbf{x}) = \min(\text{Pr}\_A(\mathbf{x}), \text{Pr}\_B(\mathbf{x})) \end{aligned} \tag{A3}$$

In Equation (A3), *a* = 1 if the point-cloud is a point; otherwise, *a* < 1. Figure A7 shows the nature of *g*(*x*) with respect to Pr*A*(*x*) + Pr*B*(*x*). The area under g(*x*) is given by:

$$Q = \int\_X \mathbf{g}(\mathbf{x})d\mathbf{x} \tag{A4}$$

There is no guarantee that *Q* = 1. Otherwise, *g*(*x*) could have been considered a probability distribution of the underlying point-cloud. However, a function *F*(*x*) can be defined as follows:

$$\begin{array}{c} \left[0, a\right] \to \left[0, 1\right] \\\\ \mathbf{x} \mapsto F(\mathbf{x}) = \frac{\int\_{x\_{\min}}^{x} g(\mathbf{x}) dx}{Q} \end{array} \tag{A5}$$

*F*(*x*) can be considered a cumulative probability distribution because max(*F*(*x*)) = 1, *F*(*x*) ≥ *F*(*z*) for *x* ≥ *z*, *F*(*x*) ∈ [0, 1], ∀*x*, *z* ∈ *X*. Figure A8 shows the nature of *F*(*x*) derived from *g*(*x*) shown in Figure A7. The cumulative probability distribution defined in Equation (A5) produces a probability distribution Pr(*x*). Thus, the following formulation holds:

$$\Pr(\mathbf{x}) = \frac{dF(\mathbf{x})}{d\mathbf{x}} \tag{A6}$$

Figure A9 shows the probability distribution Pr(*x*) that corresponds to *F*(*x*) as shown in Figure A8. The area under the probability distribution Pr(*x*) is unit and Pr(*x*) remains in the bound of [0, 1].

From the induced probability distribution Pr(*x*), a possibility distribution given by the membership function *μI*(*x*)) can be defined based on the heuristic rule of probability-possibility transformation—the degree of possibility is greater than or equal to the degree of probability. The easiest formulation is to normalize Pr(*x*) by its maximum value, max(Pr(*x*) | ∀*x* ∈ *X*), yielding the following formulation:

$$\begin{aligned} [0,1] &\rightarrow [0,1] \\ \Pr(\mathbf{x}) &\rightarrow \mu\_I(\mathbf{x}) = \frac{\Pr(x)}{\max(\Pr(x)|\forall x \in X)} \end{aligned} \tag{A7}$$

Figure A10 shows the possibility distribution *μI*(*x*) derived from the probability distribution Pr(*x*) shown in Figure A9. The shape of the induced probability and possibility distributions are identical, as evident from Figures A9 and A10, respectively. Other formulations can be used instead of the formulation (A7), if needed.

**Figure A8.** Nature of cumulative probability distribution of a point-cloud.

**Figure A9.** The nature of the probability distribution of a unimodal point-cloud.

**Figure A10.** The nature of the possibility distribution of a unimodal point-cloud.

However, it is observed that when the point-cloud resembles the point-cloud of a bimodal quantity, the induced possibility distribution resembles a trapezoidal fuzzy number. In addition, when the point-cloud is a point, the induced possibility distribution becomes a fuzzy singleton. Moreover, when the point-cloud resembles the point-cloud of unimodal data, the induced probability/possibility distribution resembles a triangular fuzzy number. To define the membership function of an induced fuzzy number in the form of a triangular fuzzy number, the following formulation can be used.

Let *u*, *v*, and *w* be three points in ascending order in the universe of discourse *X*, *u* ≤ *v* ≤ *w* ∈ *X*. Let the interval [*u*, *w*] be the *support* of a triangular fuzzy number and the point *v* be the *core*. The following procedure can be used to determine the values of *u*, *v*, and *w* from the induced fuzzy number *μI*(*x*) (Equation (A7)):

$$\begin{array}{l} u \le v \le w \in X \\ u = \mathbf{x} \quad \left(\mu\_I(\mathbf{x}) = 0 \land \mu\_I(\mathbf{x} + d\mathbf{x}) > 0\right) \\ v = \mathbf{x} \quad \left(\mu\_I(\mathbf{x} - d\mathbf{x}) < 1 \land \mu\_I(\mathbf{x}) = 1\right) \\ w = \mathbf{x} \quad \left(\mu\_I(\mathbf{x} - d\mathbf{x}) > 0 \land \mu\_I(\mathbf{x}) = 0\right) \end{array} \tag{A8}$$

As defined in (A8), *u* is the point after which the membership value *μI*(*x*) is greater than zero, *v* is the point corresponding to the maximum membership value max(*μI*(*x*)), and *w* is the point from/beyond which the membership value *μI*(*x*) again becomes/remains zero. Thus, the membership function of the induced triangular fuzzy number denoted as *μT*(*x*) is as follows:

$$\begin{array}{l} X \to [0, 1] \\ \mathbf{x} \mapsto \mu\_T(\mathbf{x}) = \max\left(0, \min\left(\frac{\mathbf{x} - \mathbf{u}}{\mathbf{v} - \mathbf{u}}, \frac{\mathbf{w} - \mathbf{x}}{\mathbf{w} - \mathbf{v}}\right)\right) \end{array} \tag{A9}$$

In this article, the formulations up to (A7) were used, i.e., the regular fuzzy number was not constructed. The triangular fuzzy numbers are particularly important when the optimization of cutting conditions is carried out using the experimental data obtained by using a statistical procedure (e.g., design of experiment), as shown in [26].
