# A Study of Stopping Rules in the Steepest Ascent Methodology for the Optimization of a Simulated Process

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Method

**a. Considerations of the case study**

**b. Fit the first order model**

**c. Determine the steepest path with the first order model**

**d. Application of MKSR, RPR and RPRE**

**e. Selection of the rule with best performance for each case.**

## 3. Results

^{®}, only factors $P,Q,S$ and U were significant for the response. The second case has factors $S,T,Y,Z,E,F$ and G. The low levels for each of these factors are 325, 650, −2, 1.4, 1, 28.5 and 9, respectively. The high levels for these factors are 350, 700, 0, 1.5, 3.0, 31 and 13, respectively. Only factors $Y,Z,E$ and G were significant. This information was used in both cases to build the steepest path using the procedure in [13] to determine the step size of the path for each of the significant factors. The sufficiency of diversity in the nature of these experiments to generalize findings and make more general recommendations will follow the conditions and properties of the experiment itself. This means that all conclusions will be highly valuable for oncoming situations with similar natures and characteristics to the ones here presented.

#### 3.1. Results for Case 1

- The selection of a step size for this path. The variable with the largest absolute regression coefficient is the one selected. In this case, factor Q is selected;
- The proposed natural step size for the factor Q is ${\Delta}_{Q}=1$. Through conversion from coded to natural units, the coded step size for Q is ${\Delta}_{{X}_{Q}}=0.1250$;
- The calculation of the coded step sizes for the rest of the variables is performed with (1).For example, the coded step size of P is:${\Delta}_{{X}_{P}}=\frac{-0.0367}{0.2123/0.1250}=-0.0216.$Thus, the coded step sizes for the significant factors are:${\Delta}_{{X}_{P}}=-0.0216$ for P,${\Delta}_{{X}_{Q}}=0.1250$ for Q,${\Delta}_{{X}_{S}}=-0.0224$ for S and;${\Delta}_{{X}_{U}}=0.0306$ for U.The natural step sizes for the same factors are:${\Delta}_{P}=-0.0270$ for P,${\Delta}_{Q}=1$ for Q,${\Delta}_{S}=-0.2804$ for S and;${\Delta}_{U}=5$ for U.The path for Case 1 is built next.

**Application of MKSR to Case 1**

**1. Assumption of behavior in Case 1**. The steepest direction is shown in Table 1, running 15 iterations. It starts with step 0, computing the center points of the experiment.

**2. Significance test in Case 1**. Iterations or individual experimentation through the steepest path stop when $y({n}_{i}+1)-y\left({n}_{i}\right)\le a$.

**3. Estimation for limits a and b in Case 1**. Limits $a=-0.744$ and $-b=0.744$ are calculated using (2) as shown in Table 2. Those limits are used to identify the moment where (3) is fulfilled. It is important to remember that the value of $\kappa $ is a guess of the number of individual experimentation runs to arrive at the improvement. In this case, the considered value of $\kappa =15$.

**4. Application of decision rule in Case 1**. The decision to stop is determined by (3). This means that the time t should stop when $y\left({n}_{i}\right)-y\left({n}_{i}-1\right)\le -0.744$. The behavior of the data is shown in Table 3.

**5. Selection of optimum response in Case 1**. If $y\left({n}_{i}\right)-y\left({n}_{i}-1\right)\le a$, the search stops and it returns to ${t}^{*}$ such that $Y\left({t}^{*}\right)={max}_{l=1,\dots ,t}\left\{Y\right(l\left)\right\}.$ As noted, the best performance for the MKSR us found in iteration 13, with a response of 6.37 units.

**Application of RPR to Case 1**

**1. Assumption of behavior in Case 1**. The steepest path applies the same way for this case. Now, Figure 4 shows the behavior of the response in the steepest path from $t=1$ to $t=14$. The curved line tries to illustrate the quadratic assumption of the response.

**2. Estimation of parameters ${\theta}_{0}$ and ${\theta}_{1}$ in Case 1**. The estimation starts with ${\theta}_{0}$, which is obtained by calculating the arithmetic mean of center points. In this case:

**3. Recursive estimation of parameters in Case 1.**Table 4 details the recursive estimation of parameters, which assists the stopping decision. For this case, ${P}_{0}={t}_{prior}=10.$

**4. Application of decision rule in Case 1.**The decision to stop is given when (9) is fulfilled. The "Status" column of Table 4 shows the moment that this occurs.

**5. Selection of optimum response in Case 1.**As seen in Table 8, the decision to stop occurred in iteration 5; nevertheless, the best response was at 4.62 units because it returns to ${t}^{*}$ such that $Y\left({t}^{*}\right)={max}_{l=1,\dots ,t}\left\{Y\right(l\left)\right\}.$

**Application of RPRE to Case 1**

**1. Assumption of behavior in Case 1.**Figure 5 shows the response behavior in the steepest path and the assumption of both quadratic and non-quadratic behavior. The straight and curved lines illustrate the capability of this procedure to assume both types of behavior.

**2. Estimation of parameters ${\theta}_{0}$, ${\theta}_{1}$ and ${\theta}_{2}$ when $t=0$ in Case 1.**

**3. Recursive estimation of ${\theta}_{2}$ and ${P}_{t}$ when $t<N-1$ in Case 1.**

**4. Application of decision rule in Case 1. If it is not fulfilled, modifications for $t\ge N-1$ are applied.**

**5. Selection of optimum response in Case 1.**Table 5 shows the decision to stop, which occurred in iteration 5. Nevertheless, the best response was at 4.62 units.

#### 3.2. Results for Case 2

- The selection of a step size for this path. The variable with the largest absolute regression coefficient is the one selected. In this case, factor E is the one selected to propose a natural step size;
- The proposed natural step size for factor E is the unit; thus, the step size for E is $\Delta E=1.0000$. Through conversion from coded to natural units, the coded step size for E is $\Delta {X}_{E}=1.0000$. By coincidence, it is equal for both coded and natural units;
- The calculation of coded step size in the other variables is performed with (1). For example, the coded step size of Z is:${\Delta}_{{X}_{Z}}=\frac{-0.2023}{31.1119/1.0000}=-0.0065.$Thus, the coded step sizes for the significant factors are:${\Delta}_{{X}_{Y}}=0.3056$ for Y,${\Delta}_{{X}_{Z}}=-0.0065$ for Z,${\Delta}_{{X}_{E}}=1.0000$ for E and;${\Delta}_{{X}_{G}}=0.9608$ for G.The natural step sizes for the same factors are:${\Delta}_{Y}=0.3056$ for Y,${\Delta}_{Z}=-0.0003$ for Z,${\Delta}_{E}=1.0000$ for E and;${\Delta}_{G}=1.9216$ for G.

**Application of MKSR to Case 2**

**1. Assumption of behavior in Case 2**. Table 6 shows the steepest path from iteration 0 to 14.

**2. Significance test in Case 2**. As in Case 1, individual experimentation over the steepest path stops when $y({n}_{i}+1)-y\left({n}_{i}\right)\le a$.

**3. Estimation for limits a and b in Case 2**. Limits $a=-3.668$ and $b=3.668$ are calculated using (2) with $\kappa =30$, as shown in Table 7. Those limits are used to identify the moment where (3) is fulfilled.

**4. Application of decision rule in Case 2**. The decision to stop occurs when

**5. Selection of optimum response in Case 2**. In this case, the best performance for the MKSR in Case 2 is the response of 232.75 units, because it stopped in ${t}^{*}$ such that $Y\left({t}^{*}\right)={max}_{l=1,\dots ,t}\left\{Y\right(l\left)\right\}.$

**Application of RPR to Case 2**

**1. Assumption of behavior in Case 2**. Figure 7 shows behavior of the response in the steepest path.

**2. Estimation of parameters ${\theta}_{0}$ and ${\theta}_{1}$ in Case 2**. These estimations are shown next:

**3. Recursive estimation of parameters in Case 2.**Table 9 shows the recursive estimation needed with ${P}_{0}={t}_{prior}=10.$

**4. Application of decision rule in Case 2.**The “Status” column in Table 9 shows the moment when (9) is fulfilled.

**5. Selection of optimum response in Case 2.**Table 11 illustrates the decision to stop, which occurred in iteration 4. However, the selection falls in ${t}^{*}$ such that $Y\left({t}^{*}\right)={max}_{l=1,\dots ,t}\left\{Y\right(l\left)\right\}$; therefore, the best response was at 232.75 units.

**Application of RPRE to Case 2**

**1. Assumption of behavior in Case 2.**Figure 8 shows the behavior of the response in the steepest path and the assumption of both quadratic and linear behavior.

**2. Estimation of parameters ${\theta}_{0}$, ${\theta}_{1}$ and ${\theta}_{2}$ when $t=0$ in Case 2.**

**3. Recursive estimation of ${\theta}_{2}$ and ${P}_{t}$ when $t<N-1$ in Case 2.**

**4. Application of decision rule in Case 2. If it is not fulfilled, modifications for $t\ge N-1$ are applied.**

**5. Selection of optimum response in Case 2.**Table 11 shows the decision to stop, which occurred in iteration 3. Nevertheless, the best response was at 232.75 units due to ${t}^{*}$ in $Y\left({t}^{*}\right)={max}_{l=1,\dots ,t}\left\{Y\right(l\left)\right\}.$

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Flow chart for the proposed method of the manuscript from the consideration of the case study to the selection of the rule with the best performance.

**Figure 3.**Graph with the steepest ascent path of the response Y with a straight line, which assumes normality in Case 1.

**Figure 4.**Graph with the steepest ascent path of the response y with a curved line, which assumes quadratic behavior in Case 1.

**Figure 5.**Graph with steepest ascent path of the response y with both straight and curved lines, which assumes both quadratic and non-quadratic behavior in Case 1.

**Figure 6.**Graph with the steepest descent path of the response Y with a straight line, which assumes normality in Case 2.

**Figure 7.**Graph with steepest descent path of y with a curved line, which assumes quadratic behavior in Case 2.

**Figure 8.**Graph with steepest descent path of y with straight and curved lines, which assumes both quadratic and non-quadratic behavior in Case 2.

t | P | Q | S | U | y |
---|---|---|---|---|---|

0 | 12.30 | 76.00 | 287.50 | 75.00 | 4.62 |

1 | 12.20 | 77.00 | 287.20 | 80.00 | 4.44 |

2 | 12.20 | 78.00 | 286.90 | 85.00 | 4.51 |

3 | 12.20 | 79.00 | 286.70 | 90.00 | 4.43 |

4 | 12.10 | 80.00 | 286.40 | 95.00 | 4.05 |

5 | 12.10 | 81.00 | 286.10 | 100.00 | 4.36 |

6 | 12.10 | 82.00 | 285.80 | 105.00 | 4.48 |

7 | 12.10 | 83.00 | 285.50 | 110.00 | 5.16 |

8 | 12.00 | 84.00 | 285.30 | 115.00 | 4.91 |

9 | 12.00 | 85.00 | 285.00 | 120.00 | 5.12 |

10 | 12.00 | 86.00 | 284.70 | 125.00 | 5.13 |

11 | 12.00 | 87.00 | 284.40 | 130.00 | 4.85 |

12 | 11.90 | 88.00 | 284.10 | 135.00 | 5.10 |

13 | 11.90 | 89.00 | 283.90 | 140.00 | 6.37 |

14 | 11.90 | 90.00 | 283.60 | 145.00 | 4.87 |

a | b | ${\mathsf{\Phi}}^{-1}\left(\right)open="("\; close=")">1/2\mathit{k}$ | ${\mathit{\sigma}}_{\mathit{\u03f5}}$ | $\sqrt{2}$ |
---|---|---|---|---|

−0.74 | 0.74 | −1.83 | 0.28 | 1.41 |

t | P | Q | S | U | y | $\mathit{y}({\mathit{n}}_{\mathit{i}}+1)-\mathit{y}\left({\mathit{n}}_{\mathit{i}}\right)$ | Status |
---|---|---|---|---|---|---|---|

0 | 12.30 | 76.00 | 287.50 | 75.00 | 4.62 | - | Starts |

1 | 12.20 | 77.00 | 287.20 | 80.00 | 4.44 | −0.18 | Continues |

2 | 12.20 | 78.00 | 286.90 | 85.00 | 4.51 | 0.07 | Continues |

3 | 12.20 | 79.00 | 286.70 | 90.00 | 4.43 | −0.08 | Continues |

4 | 12.10 | 80.00 | 286.40 | 95.00 | 4.05 | −0.38 | Continues |

5 | 12.10 | 81.00 | 286.10 | 100.00 | 4.36 | 0.31 | Continues |

6 | 12.10 | 82.00 | 285.80 | 105.00 | 4.48 | 0.12 | Continues |

7 | 12.10 | 83.00 | 285.50 | 110.00 | 5.16 | 0.68 | Continues |

8 | 12.00 | 84.00 | 285.30 | 115.00 | 4.91 | −0.25 | Continues |

9 | 12.00 | 85.00 | 285.00 | 120.00 | 5.12 | 0.21 | Continues |

10 | 12.00 | 86.00 | 284.70 | 125.00 | 5.13 | 0.01 | Continues |

11 | 12.00 | 87.00 | 284.40 | 130.00 | 4.85 | −0.28 | Continues |

12 | 11.90 | 88.00 | 284.10 | 135.00 | 5.10 | 0.25 | Continues |

13 | 11.90 | 89.00 | 283.90 | 140.00 | 6.37 | 1.27 | Continues |

14 | 11.90 | 90.00 | 283.60 | 145.00 | 4.87 | −1.50 | Stops |

t | $\mathit{y}\left(\mathit{t}\right)$ | ${\mathit{\theta}}_{2}^{\left(\mathit{t}\right)}$ | $\mathit{P}\mathit{t}$ | ${\mathit{\theta}}_{1}+2{\mathit{\theta}}_{2}^{\left(\mathit{t}\right)}\mathit{t}$ | ${\mathit{\sigma}}_{{\mathit{\theta}}_{1}+2{\mathit{\theta}}_{2}^{\left(\mathit{t}\right)}\mathit{t}}^{2}$ | $-3\sqrt{{\mathit{\sigma}}_{{\mathit{\theta}}_{1}+2{\mathit{\theta}}_{2}^{\left(\mathit{t}\right)}\mathit{t}}^{2}}$ | Status |
---|---|---|---|---|---|---|---|

0 | 4.62 | −0.01 | 10 | 0.22 | 0.00 | 0.00 | Starts |

1 | 4.44 | −0.05 | 0.91 | 0.12 | 0.30 | −1.64 | Continues |

2 | 4.51 | −0.05 | 0.06 | 0.01 | 0.08 | −0.83 | Continues |

3 | 4.43 | −0.06 | 0.01 | −0.11 | 0.03 | −0.52 | Continues |

4 | 4.05 | −0.07 | 0.00 | −0.31 | 0.01 | −0.37 | Continues |

5 | 4.36 | −0.05 | 0.00 | −0.28 | 0.00 | −0.27 | Stops |

t | $\mathit{Y}\left(\mathit{t}\right)$ | ${\mathit{\theta}}_{0}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{P}}_{\mathit{t}}$ | ${\mathit{d}}_{\mathit{t}}^{\prime}{\mathit{\theta}}^{\left(\mathit{t}\right)}$ | $-1.645{\mathit{\sigma}}_{\mathit{\u03f5}}\sqrt{{\mathit{d}}_{\mathit{t}}^{\prime}{\mathit{P}}_{\mathit{t}}{\mathit{d}}_{\mathit{t}}}$ | ||
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0 | |||||||

0 | 4.62 | 4.62 | 0.23 | −0.01 | 0 | 1 | 0 | 0.23 | NA |

0 | 0 | 10 | |||||||

0.90 | −0.10 | −0.80 | |||||||

1 | 4.44 | 4.59 | 0.19 | −0.31 | −0.10 | 0.90 | −0.80 | −0.43 | −0.91 |

−0.80 | −0.80 | 2.30 | |||||||

0.70 | −0.20 | −0.10 | |||||||

2 | 4.51 | 4.51 | 0.15 | −0.08 | −0.20 | 0.90 | −0.40 | −0.19 | −0.53 |

−0.10 | −0.40 | 0.30 | |||||||

0.60 | −0.30 | 0.00 | |||||||

3 | 4.43 | 4.49 | 0.12 | −0.05 | −0.30 | 0.70 | −0.20 | −0.18 | −0.38 |

0.00 | −0.20 | 0.10 | |||||||

0.60 | −0.30 | 0.00 | |||||||

4 | 4.05 | 4.50 | 0.14 | −0.06 | −0.30 | 0.60 | −0.10 | −0.28 | −0.30 |

0.00 | −0.10 | 0.00 | |||||||

0.60 | −0.30 | 0.00 | |||||||

5 | 4.36 | 4.50 | 0.02 | −0.02 | −0.30 | 0.40 | −0.10 | −0.48 | −0.25 |

0.00 | −0.10 | 0.00 |

t | Y | Z | E | G | y |
---|---|---|---|---|---|

0 | −1.00 | 1.50 | 2.00 | 11.00 | 163.41 |

1 | −0.70 | 1.40 | 3.00 | 12.90 | 212.22 |

2 | −0.40 | 1.40 | 4.00 | 14.80 | 232.75 |

3 | −0.10 | 1.40 | 5.00 | 16.80 | 226.16 |

4 | 0.20 | 1.40 | 6.00 | 18.70 | 191.12 |

5 | 0.50 | 1.40 | 7.00 | 20.60 | 130.32 |

6 | 0.80 | 1.40 | 8.00 | 22.50 | 42.51 |

7 | 1.10 | 1.40 | 9.00 | 24.50 | −82.49 |

8 | 1.40 | 1.40 | 10.00 | 26.40 | −233.67 |

9 | 1.80 | 1.40 | 11.00 | 28.30 | −405.47 |

10 | 2.10 | 1.40 | 12.00 | 30.20 | −573.87 |

11 | 2.40 | 1.40 | 13.00 | 32.10 | −849.67 |

12 | 2.70 | 1.40 | 14.00 | 34.10 | −1094.76 |

13 | 3.00 | 1.40 | 15.00 | 36.00 | −1356.20 |

14 | 3.30 | 1.40 | 16.00 | 37.90 | −1605.82 |

a | b | ${\mathsf{\Phi}}^{-1}\left(\right)open="("\; close=")">1/2\mathit{k}$ | ${\mathit{\sigma}}_{\mathit{\u03f5}}$ | $\sqrt{2}$ |
---|---|---|---|---|

−3.67 | 3.67 | −1.83 | 1.41 | 1.41 |

t | Y | Z | E | G | y | $\mathit{y}(\mathit{ni}+1)-\mathit{y}\left(\mathit{ni}\right)$ | Status |
---|---|---|---|---|---|---|---|

0 | −1.00 | 1.50 | 2.00 | 11.00 | 163.41 | - | Starts |

1 | −0.70 | 1.40 | 3.00 | 12.90 | 212.22 | 48.81 | Continues |

2 | −0.40 | 1.40 | 4.00 | 14.80 | 232.75 | 20.54 | Continues |

3 | −0.10 | 1.40 | 5.00 | 16.80 | 226.16 | −6.60 | Stops |

t | $\mathit{y}\left(\mathit{t}\right)$ | ${\mathit{\theta}}_{2}^{\left(\mathit{t}\right)}$ | $\mathit{Pt}$ | ${\mathit{\theta}}_{1}+2{\mathit{\theta}}_{2}^{\left(\mathit{t}\right)}\mathit{t}$ | ${\mathit{\sigma}}_{{\mathit{\theta}}_{1}+2{\mathit{\theta}}_{2}^{\left(\mathit{t}\right)}\mathit{t}}^{2}$ | $-3\sqrt{{\mathit{\sigma}}_{{\mathit{\theta}}_{1}+2{\mathit{\theta}}_{2}^{\left(\mathit{t}\right)}\mathit{t}}^{2}}$ | Status |
---|---|---|---|---|---|---|---|

0 | 163.41 | −2.21 | 10.00 | 44.18 | 0.00 | 0.00 | Starts |

1 | 212.22 | 4.64 | 0.91 | 53.46 | 7.27 | −8.09 | Continues |

2 | 232.75 | −3.99 | 0.06 | 28.23 | 1.87 | −4.10 | Continues |

3 | 226.16 | −7.03 | 0.01 | 1.98 | 0.73 | −2.57 | Continues |

4 | 191.12 | −8.65 | 0.00 | −25.02 | 0.36 | −1.80 | Stops |

t | $\mathit{y}\left(\mathit{t}\right)$ | ${\mathit{\theta}}_{0}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{P}}_{\mathit{t}}$ | ${\mathit{d}}_{\mathit{t}}^{\prime}{\mathit{\theta}}^{\left(\mathit{t}\right)}$ | $-1.645{\mathit{\sigma}}_{\mathit{\u03f5}}\sqrt{{\mathit{d}}_{\mathit{t}}^{\prime}{\mathit{P}}_{\mathit{t}}{\mathit{d}}_{\mathit{t}}}$ | ||
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0 | |||||||

0 | 163.41 | 163.41 | 44.18 | −2.21 | 0 | 1 | 0 | 44.18 | −0.70 |

0 | 0 | 10 | |||||||

0.90 | −0.10 | −0.80 | |||||||

1 | 212.22 | 163.94 | 44.71 | 3.05 | −0.10 | 0.90 | −0.80 | 50.80 | −1.86 |

−0.80 | −0.80 | 2.30 |

t | $\mathit{Y}\left(\mathit{t}\right)$ | $\mathit{YN}\left(\mathit{t}\right)$ | ${\mathit{b}}^{\prime}\mathit{NYN}\left(\mathit{t}\right)$ | $-1.645{\mathit{\sigma}}_{\mathit{\u03f5}}\sqrt{\mathit{vN}}$ | ||
---|---|---|---|---|---|---|

2 | 232.75 | 163.41 | 212.22 | 232.75 | 6.40 | −1.78 |

3 | 226.16 | 212.22 | 232.75 | 226.16 | −20.16 | −1.78 |

Author and Year of SR Procedure | Case 1 | Case 2 | ||
---|---|---|---|---|

Iterations to Stop | Best Response | Iterations to Stop | Best Response | |

R. Myers and Khuri (1979) MKSR | 14 | 6.37 | 3 | 232.75 |

Miró-Quezada and Del Castillo (2004) RPR | 5 | 4.62 | 4 | 232.75 |

Del Castillo (2007) RPRE | 5 | 4.62 | 3 | 232.75 |

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**MDPI and ACS Style**

García-Nava, P.E.; Rodríguez-Picón, L.A.; Méndez-González, L.C.; Pérez-Olguín, I.J.C.
A Study of Stopping Rules in the Steepest Ascent Methodology for the Optimization of a Simulated Process. *Axioms* **2022**, *11*, 514.
https://doi.org/10.3390/axioms11100514

**AMA Style**

García-Nava PE, Rodríguez-Picón LA, Méndez-González LC, Pérez-Olguín IJC.
A Study of Stopping Rules in the Steepest Ascent Methodology for the Optimization of a Simulated Process. *Axioms*. 2022; 11(10):514.
https://doi.org/10.3390/axioms11100514

**Chicago/Turabian Style**

García-Nava, Paulo Eduardo, Luis Alberto Rodríguez-Picón, Luis Carlos Méndez-González, and Iván Juan Carlos Pérez-Olguín.
2022. "A Study of Stopping Rules in the Steepest Ascent Methodology for the Optimization of a Simulated Process" *Axioms* 11, no. 10: 514.
https://doi.org/10.3390/axioms11100514