# Multi-Objective Optimization of Production Objectives Based on Surrogate Model

^{*}

## Abstract

**:**

## 1. Introduction

^{n}) of NP- hard problems is given by both the small velocity of finding and verifying the solution. They belong to the category of problems that cannot be generally solved in polynomial time but exponential time that depends on input size n. As an alternative of metaheuristics, the way to deal with the MOOP can be through finding the adequate surrogate model of a costly evaluated function, which allows solving an MOOP with significant speed-up [13,15]. The challenge is to find a suitable compromise between calculation acceleration and model precision [16].

## 2. Background of Multi-Objective Optimization Based on a Surrogate Model

#### 2.1. Multi-Objective Optimization Problem (MOOP)

**x**∈ D, where the set D is the feasible region in the design space, is constrained by its lower and upper limits

**x**

_{L}≤

**x**≤

**x**

_{U}. The vector

**F**(

**x**) ∈ E

_{k}is the vector of objective functions F

_{i}(

**x**): E

_{n}→ E

_{1}in feasible objective space Z, Z ⊂ E

_{k}. Thus, Z is the forward image of D under the mapping

**F**[8]. The value m defines the number of inequalities, and r defines the number of equalities for the constrained problem. If the objectives of F

_{i}are contradictory, there does not exist only one single optimal solution as a MOOP-solving result. This process leads to the set of tradeoffs that are completely equivalent in a mathematical sense.

#### 2.2. Modeling of Surrogates and Their Application in a Multi-Objective Optimization

#### 2.3. Response Surface Models

_{i}, x

_{j}are input variables that influence the response y, and ε denotes a random error (or a noise) observed in y.

## 3. The Proposed Multi-Optimization Procedure Based on a Surrogate Model Implemented in Production Control

#### 3.1. Procedure of Surrogate-Based MOOP Solving

#### 3.2. Simulation Model of Production System

#### 3.3. Simulation Experiments

#### 3.4. DoE—Proposed Experimental Designs

_{1}–F

_{4}, representing the selected production goals (Table 1). The factorial space was a four-dimensional domain, with lower and upper bounds between values 2 and 10 pcs for the lot size and between 5 and 50 min for the product arrival time of both products. As responses, four selected production performance indicators were considered: the total number of products, average flow time, average machine utilization, and average costs per part unit. Both designs required 25 simulation runs each. The values of responses in each of the experimental points were found by simulation in Witness Horizon by Lanner Group Limited, Houston, TX, USA (see Figure 2 and Figure 3, respectively). The BBD design layout for the actual values of the settings factors with the added responses that resulted from the simulation are shown in Figure 2. Similarly, Figure 3 depicts the experimental scheme with an added four responses for FCD.

#### 3.5. The Particular MOOP Definition and Applied Scalar MOO Methods

**w**. All components w

_{i}have the same value 0.25 to not prefer any production goal.

_{i}entering the scalar multi-objective function. We employed a robust transformation given by Equation (5).

_{i}in Equation (5) is a component of the production goals (individual objectives) vector that resulted from experiments on a discrete-event driven simulation model.

^{2}+ 0.031313 * IATP2

^{2}− 10.209)/(16 − 10.209) + 0.25 * (91.78175 + 0.626283 * LotSizeP1 + 0.301304 * LotSizeP2 − 0.923591 * IATP1 − 0.854483 * IATP2 + 0.048977 * LotSizeP1 * IATP1 + 0.049608 * LotSizeP2 * IATP2 − 84.035)/(60 − 84.035) + 0.25 * (250.66819 + 15.89178 * LotSizeP1 + 14.81886 * LotSizeP2 − 2.28272 * IATP1 − 2.24722 * IATP2 + 0.238889 * LotSizeP1 * IATP1 + 0.237500 * LotSizeP2 * IATP2 − 1.47246 * LotSizeP1

^{2}− 1.34746 * LotSizeP2

^{2}− 289)/(230 − 289) + 0.25 * (585.01423 + 26.38960 * LotSizeP1 + 24.42893 * LotSizeP2 − 2.45189 * IATP1 − 11.04025 * IATP2 + 0.119557 * IATP1

^{2}+ 0.097507 * IATP2

^{2}− 296.309)/(550 − 296.309).

^{2}+ 0.031313 * IATP2

^{2}− 10.209)/(16 − 10.209)) * ((91.78175 + 0.626283 * LotSizeP1 + 0.301304 * LotSizeP2 − 0.923591 * IATP1 − 0.854483 * IATP2 + 0.048977 * LotSizeP1 * IATP1 + 0.049608 * LotSizeP2 * IATP2 − 84.035)/(60 −84.035)) * ((250.66819 + 15.89178 * LotSizeP1 + 14.81886 * LotSizeP2 − 2.28272 * IATP1 − 2.24722 * IATP2 + 0.238889 * LotSizeP1 * IATP1 + 0.237500 * LotSizeP2 * IATP2 − 1.47246 * LotSizeP1

^{2}− 1.34746 * LotSizeP2

^{2}− 289)/(230 − 289)) * ((585.01423 + 26.38960 * LotSizeP1 + 24.42893 * LotSizeP2 − 12.45189 * IATP1 − 11.04025 * IATP2 + 0.119557 * IATP1

^{2}+ 0.097507 * IATP2

^{2}− 296.309)/(550 − 296.309)))) ^ 0.25.

#### 3.6. Numerical Optimization via Maximizing Desirability Function

^{®}version 12 (by Stat-Ease, Minneapolis, MN, USA) where surrogate models for individual objective functions were built. Montgomery [35] describes a multiple response method that employs an objective function D, which is called the Desirability function. It reflects the desirable ranges for each individual response d

_{i}simultaneously. Equation (8) defines the value d

_{i}if the target T for the response y is a maximum value, and Equation (9) defines the value d

_{i}if the target T is a minimum value, respectively.

_{i}:

_{i}are depicted in Figure 4. It shows the graphs of d

_{i}functions in Equations (8) and (9) for specified limit values on individual intervals. Two of the notches on each ramp represent the minimum and maximum values of all response values within the experimental space, and two others are lower and upper limits for the given response. They correspond to L and U values in Equations (8) and (9), respectively (the lower and upper limit in Table 4). At the same time, they correspond to the utopia and nadir points as reference points in other types of applied scalar multi-objective functions.

#### 3.7. Validation of Proposed MOOP Solving Strategy

**Box 1.**Definition of Weighted Sum Method (WSM)-based objective function in a simulator.

**Box 2.**Definition of WPM-based objective function in the simulator.

## 4. Results

#### 4.1. Derivation of Surrogate Models of Production Objectives

^{®}software by Stat-Ease (Minneapolis, MN, USA). It provided the polynomial regression models for four selected production objectives as models of the input–output behavior of the underlying simulation model. Based on the ANOVA analysis by F-test of the overall statistical significance of the model and t-tests of the statistical significance of individual regression coefficients, Equations (11)–(18) in terms of actual factors represent obtained response surface models for both types of design. The Lack of Fit test could not be performed due to no variance in the central point, because the applied simulation model has been deterministic.

**Flow time**= 435.07580 + 125.17042 * LotSizeP1 + 120.09642 * LotSizeP2 − 138.29592 * IATP1 − 133.84925 * IATP2

**Costs per part unit**= 12.51314 + 5.7129 * LotSizeP1 + 5.60725 * LotSizeP2 − 18.85525 * IATP1 − 19.27592 * IATP2 − 11.10425 * LotSizeP1 * IATP1 − 12.24525 * LotSizeP2 * IATP2 + 16.61164 * IATP1

^{2}+ 17.40139 * IATP2

^{2}

**Machine utilization**= 64.56492 + 12.97700 * LotSizeP1 + 11.51400 * LotSizeP2 − 14.87858 * IATP1 − 13.25308 * IATP2

**Number of products**= 274.30769 + 18.33333 * LotSizeP1 + 19.41667 * LotSizeP2 − 7.16667 * IATP1 − 6.75000 * IATP2 + 21.50000 * LotSizeP1 * IATP1 + 21.75000 * LotSizeP2 * IATP2 − 15.80769 * LotSizeP1

^{2}

**Flow time**= 585.01423 + 26.38960 * LotSizeP1 + 24.42893 * LotSizeP2 − 12.45189 * IATP1 − 11.04025 * IATP2 + 0.119557 * IATP1

^{2}+ 0.097507 * IATP2

^{2}

**Costs per part unit**= 11.47091 + 6.53901 * LotSizeP1 + 7.91294 * LotSizeP2 − 1.42610 * IATP1 − 1.40960 * IATP2 − 0.150095 * LotSizeP1 * IATP1 − 0.176751 * LotSizeP2 * IATP2 + 0.030254 * IATP1

^{2}+ 0.031313 * IATP2

^{2}

**utilization**= 91.78175 + 0.626283 * LotSizeP1 + 0.301304 * LotSizeP2 − 0.923591 * IATP1 − 0.854483 * IATP2 + 0.048977 * LotSizeP1 * IATP1 + 0.049608 * LotSizeP2 * IATP2

**Number of products**= 250.66819 + 15.89178 * LotSizeP1 + 14.81886 * LotSizeP2 − 2.28272 * IATP1 − 2.24722 * IATP2 + 0.238889 * LotSizeP1 * IATP1 + 0.237500 * LotSizeP2 * IATP2 − 1.47246 * LotSizeP1

^{2}− 1.34746 * LotSizeP2

^{2}

#### 4.2. Results of Numerical Optimizations

^{®}by Stat-Ease (Minneapolis, MN, USA). The results are presented in Table 7, Table 8 and Table 9.

#### 4.3. Validation of Approximate MOOP Solving Strategies

## 5. Discussion

- We assume that an adaptable simulation model of the production system must be built.
- We assume that the controlled production system does not involve too fast inner parameters and structural changes. The inner parameters and structure can be flexible, but they need to be changed in hours, not in minutes. Even if the simulation model could reflect changes almost immediately, the steps of the designed MOOP procedure take some time to derive the surrogate model and consequently find the solutions.
- We must consider that some of the production goals are too sensitive to the change of input parameters; therefore, the approximate solution is not satisfactory.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Scheme of the implementation of metamodel to the multi-objective or vector optimization problem (MOOP) solving.

**Figure 2.**Box–Behnken Design (BBD) layout defined in actual factors values. The responses in the red box are obtained from experiments on the simulation model.

**Figure 3.**Face-Centered Design (FCD) design layout defined in actual factors values. The responses in the red box are obtained from experiments on the simulation model.

**Figure 4.**Ramp graphs with r = 1 show minimum, maximum, and lower and upper limits for individual responses d

_{i}.

Factor | Response |
---|---|

A—LotSizeP1 | F_{1}—Average flow time |

B—LotSizeP2 | F_{2}—Average costs per part unit |

C—IATP1 | F_{3}—Average machine utilization |

D—IATP2 | F_{4}—Total number of products |

Production Objective | $\mathbf{Utopia}\text{}\mathbf{Point}\text{}{\mathit{F}}_{\mathit{i}}^{\mathit{U}}$ | $\mathbf{Nadir}\text{}\mathbf{Point}\text{}{\mathit{F}}_{\mathit{i}}^{\mathit{N}}$ |
---|---|---|

Avg. flow time (min.) | 296.309 | 550 |

Avg. costs per part unit (€) | 10.209 | 16 |

Avg. machine utilization (%) | 84.035 | 60 |

Total number of products (pcs) | 289 | 230 |

Production Objective | Constraint |
---|---|

Average flow time (min.) | F_{1} ≤ 550 |

Average costs per part unit (€) | F_{2} ≤ 16 |

Average machine utilization (%) | F_{3} ≥ 60 |

Total number of products (pcs) | F_{4} ≥ 230 |

Optimization Goal | Lower Limit | Upper Limit |
---|---|---|

A is in range | 2 | 10 |

B is in range | 2 | 10 |

C is in range | 5 | 50 |

D is in range | 5 | 50 |

F_{1}—minimize | 296.309 | 550 |

F_{2}—minimize | 10.209 | 16 |

F_{3}—maximize | 60 | 84.035 |

F_{4}—maximize | 230 | 289 |

Response | R^{2} | Adj. R^{2} | Pred. R^{2} | Std. Dev. |
---|---|---|---|---|

Flow time | 0.887 | 0.864 | 0.821 | 71.67 |

Costs per part unit | 0.931 | 0.896 | 0.766 | 7.82 |

Number of products | 0.768 | 0.672 | 0.368 | 16.35 |

Machine utilization | 0.901 | 0.881 | 0.842 | 6.80 |

Response | R^{2} | Adj. R^{2} | Pred. R^{2} | Std. Dev. |
---|---|---|---|---|

Flow time | 0.991 | 0.988 | 0.985 | 22.75 |

Costs per part unit | 0.945 | 0.918 | 0.858 | 9.37 |

Number of products | 0.968 | 0.952 | 0.942 | 10.19 |

Machine utilization | 0.943 | 0.924 | 0.906 | 5.48 |

Applied Model | Algorithm | Factors/Input Parameters in Design Space | Responses/Production Objectives in Objective Space | ||||||
---|---|---|---|---|---|---|---|---|---|

A | B | C | D | Flow Time | Costs Per Part Unit | Number of Products | Machine Utilization | ||

FCD model | Evolutionary alg.Module Solver (Excel ^{1})U = 0.333 | 6 | 7 | 29 | 34 | 391.15 | 10.21 | 286.21 | 62.14 |

Simulation model | Simulation-optimization All Combinations Module Experimenter (Witness ^{2})U = 0.269 | 3 | 3 | 10 | 30 | 305.13 | 10.58 | 288 | 60.99 |

Verification | Simulation (Witness ^{2}) | 6 | 7 | 29 | 34 | 446.98 | 10.77 | 288 | 60.31 |

^{1}by Microsoft, Redmond, WA, USA;

^{2}by Lanner Group Limited, Houston, TX, USA.

Applied Model | Algorithm | Factors/Input Parameters in Design Space | Responses/Production Objectives in Objective Space | ||||||
---|---|---|---|---|---|---|---|---|---|

A | B | C | D | Flow Time | Costs Per Part Unit | Number of Products | Machine Utilization | ||

FCD model | Evolutionary alg. Solver module (Excel ^{1})U = 0.0271 | 8 | 7 | 42 | 31 | 406.5 | 10.22 | 287.50 | 60.84 |

Simulation model | Simulation–Optimization All Combinations Module Experimenter (Witness ^{2})U = 0.051 | 2 | 5 | 7 | 46 | 335.59 | 10. 216 | 287 | 60. 093 |

Verification | Simulation (Witness ^{2}) | 8 | 7 | 42 | 31 | 459.16 | 10.93 | 286 | 60.22 |

^{1}by Microsoft, Redmond, WA, USA;

^{2}by Lanner Group Limited, Houston, TX, USA.

Applied Model | Algorithm | Factors/Input Parameters in Design Space | Responses/Production Objectives in Criterion Space | ||||||
---|---|---|---|---|---|---|---|---|---|

A | B | C | D | Flow Time | Costs Per Part Unit | Number of Products | Machine Utilization | ||

FCD model | Numerical optimization Multi-response method (Design-Expert® ^{1}) D = 0.631 | 6 | 2 | 26 | 11 | 433.20 | 10.21 | 274.49 | 70.98 |

Verification | Simulation (Witness) | 6 | 2 | 26 | 11 | 428.55 | 10.87 | 287 | 60.84 |

^{1}by Stat-Ease, Minneapolis, MN, USA.

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**MDPI and ACS Style**

Červeňanská, Z.; Kotianová, J.; Važan, P.; Juhásová, B.; Juhás, M.
Multi-Objective Optimization of Production Objectives Based on Surrogate Model. *Appl. Sci.* **2020**, *10*, 7870.
https://doi.org/10.3390/app10217870

**AMA Style**

Červeňanská Z, Kotianová J, Važan P, Juhásová B, Juhás M.
Multi-Objective Optimization of Production Objectives Based on Surrogate Model. *Applied Sciences*. 2020; 10(21):7870.
https://doi.org/10.3390/app10217870

**Chicago/Turabian Style**

Červeňanská, Zuzana, Janette Kotianová, Pavel Važan, Bohuslava Juhásová, and Martin Juhás.
2020. "Multi-Objective Optimization of Production Objectives Based on Surrogate Model" *Applied Sciences* 10, no. 21: 7870.
https://doi.org/10.3390/app10217870