# Computer-Aided Choosing of an Optimal Structural Variant of a Robot for Extracting Castings from Die Casting Machines

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Development of the Set of Possible Structural Variants

#### 2.1. Analysis of the Extracting Process

#### 2.2. Development of a Functional Model of a Casting Extraction Robot

- Casting—this is the casting that was cast during the technological process and must be extracted from the mold;
- Gripper—a tool for gripping the extracted casting;
- Power—source/sources of energy powering the robot’s motors;
- Control signals for the actuators—signals from the control system to the motors to execute the movements of the robot’s cycle.

- Extracted casting—the casting, extracted and placed in a position for further execution of the technological process;
- Noise—the level of noise produced by the industrial robot is usually significant and must be taken into account in view of improving working conditions;
- Signals for executed movement—signals from the position sensors sent to the control system, confirming the movement performed (reaching a programmed position).

#### 2.3. Development of Alternative Variants of Devices for the Implementation of Functions

#### 2.4. A Network Model of the Set of Possible Structural Variants

## 3. Formulation and Analysis of the Problem of Choosing the Optimal Structural Variant

^{2}, mass—kg, payback period of capital investments—years, price—EUR, installed power—kW, duration of work strokes—s, time for readjustment—min, etc. Therefore, it is necessary to perform normalization (norming) of the objective functions. Through normalization, they are converted into a dimensionless form and a uniform scale of measurement.

- choosing a way to normalize (norm) the criteria;
- choosing an optimality principle;
- determining the priority of the criteria;
- choosing an appropriate optimization method, etc.

## 4. Methodology for Choosing the Optimal Structural Variant

- specification of the requirements for the designed system;
- specification of the objectives for the design object;
- building a tree of objectives;
- determination of constraints.

- For objective functions ${f}_{k}\left(x\right)$ to be maximized, one of the following functional transformations can be used:$${w}_{k}\left(x\right)=\frac{{f}_{k}\left(x\right)-{f}_{kmin}}{{f}_{k}^{*}-{f}_{kmin}},\text{}k\in {K}_{1},$$

- 2.
- For objective functions to be minimized, one of the following functional transformations can be used:$${w}_{k}\left(x\right)=\frac{{{f}_{kmax}-f}_{k}\left(x\right)}{{f}_{kmax}-{f}_{k}^{*}},\text{}k\in {K}_{2},$$

^{10}structural variants, and the second is based on a directed search for the optimal solution of problems with a dimension of up to 20

^{20}variants.

- Provides a systematic approach to the problem of choosing an optimal structural variant of a technical system;
- Provides a mathematical model of the problem;
- Proposes ways for criteria normalization;
- Proposes an optimality criterion;
- Generation of large number of possible variants in the initial stages of design;
- Encourages functional thinking during the conceptual design stage;
- Coming from the previous advantage is the benefit of modular design.

- A large amount of data have to be collected and analyzed in order to convert the evaluation criteria into objective functions;
- Generating variants from combinations of devices that implement each function of the system requires the use of software supporting the solution of discrete optimization problems;
- Additional complications can arise when including polyfunctional devices and compatibility between devices. These complications can be related to the ordering of the functions in the network model (Figure 3), difficulties of using the network model as a tool for defining compatibility without a software support (in cases with very complicated relations between the devices), etc. For example, when the compatibility for every device has to be stated regarding all other devices, the network model becomes unpractical. In these cases, only compatibility matrices can be used as representation, and they are not convenient for use by people directly. Therefore, there is a high dependency on software support.

## 5. Dialog System for Computer-Aided Choosing of Optimal Structural Variant

^{20}.

^{10}structural variants. With a larger number of structural variants, the time required to solve the problem becomes extremely large (combinatorial explosion).

^{10}variants. The method is characterized by a directed search for the optimal solution. At the same time, it allows reducing the set of solutions, thus allowing the application of MFC for problems with a very large size as well.

## 6. Solving the Problem

#### 6.1. Solution under Equal Priority of the Objective Functions

- The smallest deviation from its optimum has the energy consumption, and the largest is the production costs;
- The compromise variant achieves small deviations from the optima of the objective functions;
- The optimum is found in the second subset (Table 8).

#### 6.2. Solving for Different Criteria Priority

#### 6.3. Solving When Considering Floor-Mounted Variants Only

#### 6.4. Solution When Examining the Subsets

## 7. Conclusions

- A concept design of a specialized robot extractor, applying a systematic design approach;
- A large number (168) of possible structural variants of the robot extractor are developed and evaluated;
- The objective functions used for evaluating the developed structural variants are not only based on technical criteria, but also on criteria that are related to economic factors and to the actual exploitation of the equipment;
- Problem analysis using specialized software tools that do not require prior knowledge of the mathematical models used.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Function | Device 1 | Device 2 | Device 3 | Device 4 | Device 5 | Device 6 | Device 7 | Device 8 |
---|---|---|---|---|---|---|---|---|

F1 | 798.74 | 4039.15 | 6094.51 | 6442.25 | 897.18 | 534.18 | 728.58 | 377.47 |

0.22 | 0.32 | 0.32 | 0.32 | 0.06 | 0.04 | 0.06 | 0.04 | |

0.082 | 0.057 | 0.053 | 0.050 | 0.071 | 0.067 | 0.075 | 0.064 | |

0.029 | 0.070 | 0.063 | 0.066 | 0.052 | 0.055 | 0.032 | 0.031 | |

F2 | 2433.17 | 2533.70 | 1155.00 | 2924.00 | ||||

0.79 | 0.28 | 0.09 | 0.28 | |||||

0.033 | 0.037 | 0.042 | 0.029 | |||||

0.078 | 0.082 | 0.053 | 0.060 | |||||

F3 | 2908.42 | 5400.00 | 1243.52 | $C\left(x\right),\text{}\mathrm{E}\mathrm{U}\mathrm{R}$ | ||||

0.11 | 0.08 | 0.21 | $E\left(x\right),\text{}\mathrm{E}\mathrm{U}\mathrm{R}/\mathrm{h}$ | |||||

0.060 | 0.044 | 0.047 | $V\left(x\right)$ | |||||

0.045 | 0.071 | 0.039 | $T\left(x\right)$ | |||||

F4 | 2058.86 | 1798.18 | 715.90 | 1243.52 | ||||

0.02 | 0.02 | 0.05 | 0.21 | |||||

0.036 | 0.082 | 0.034 | 0.035 | |||||

0.044 | 0.036 | 0.053 | 0.041 |

## References

- Zaharinov, V.V.; Malakov, I.K.; Nikolov, S.N.; Reneta, K.D. Classification of parts used in mechatronic products and produced by permanent-mold casting methods. IOP Conf. Ser. Mater. Sci. Eng.
**2020**, 878, 012063. [Google Scholar] [CrossRef] - Chougule, R.G.; Ravi, B. Casting cost estimation in an integrated product and process design environment. Int. J. Comput. Integr. Manuf.
**2006**, 19, 676–688. [Google Scholar] [CrossRef] - Cornacchia, G.; Dioni, D.; Faccoli, M.; Gislon, C.; Solazzi, L.; Panvini, A.; Cecchel, S. Experimental and Numerical Study of an Automotive Component Produced with Innovative Ceramic Core in High Pressure Die Casting (HPDC). Metals
**2019**, 9, 217. [Google Scholar] [CrossRef] - Rodrigues, D.; Godina, R.; da Cruz, P.E. Key Performance Indicators Selection through an Analytic Network Process Model for Tooling and Die Industry. Sustainability
**2021**, 13, 13777. [Google Scholar] [CrossRef] - Rosnitschek, T.; Erber, M.; Hartmann, C.; Volk, W.; Rieg, F.; Tremmel, S. Combining Structural Optimization and Process Assurance in Implicit Modelling for Casting Parts. Materials
**2021**, 14, 3715. [Google Scholar] [CrossRef] [PubMed] - Ružbarský, J.; Gašpár, Š. Analysis of Selected Production Parameters for the Quality of Pressure Castings as a Tool to Increase Competitiveness. Appl. Sci.
**2023**, 13, 8098. [Google Scholar] [CrossRef] - Małysza, M.; Żuczek, R.; Wilk-Kołodziejczyk, D.; Ja’skowiec, K.; Głowacki, M.; Długosz, P.; Dudek, P. Technological Optimization of the Stirrup Casting Process with the Use of Computer Simulations. Materials
**2022**, 15, 6781. [Google Scholar] [CrossRef] [PubMed] - RELBO. Arcofast Product Brochure; RELBO: Rezzato, Italy, 2023. [Google Scholar]
- SPESIMA. Brochure 2020, Bulgarisch-Deutsche Gesellschaft; SPESIMA: Sofia, Bulgaria, 2020. [Google Scholar]
- Bühler. DC Brochure Carat 2020; Bühler: Uzwil, Switzerland, 2020. [Google Scholar]
- BORUNTE. BORUNTE Product Catalogue 2020, Vertical Sprayer; Guangdong BORUNTE Intelligent Equipment Co., Ltd.: Dongguan, China, 2020. [Google Scholar]
- Hsieh, T.; Yeon, S.; Herr, H. Energy Efficiency and Performance Evaluation of an Exterior-Rotor Brushless DC Motor and Drive System across the Full Operating Range. Actuators
**2023**, 12, 318. [Google Scholar] [CrossRef] - Galabov, V.; Slavkov, V.; Savchev, S.; Slavov, G.; Todorov, G.; Nikolov, N.; Sofronov, Y.; Stoyanova, Y. Selection of schematic solution for integrated implementation of casts extractor with pneumatic actuation. In Proceedings of the International Conference Automatics and Informatics′10, Sofia, Bulgaria, 3–7 October 2010; pp. III-537–III-542. [Google Scholar]
- Galabov, V.; Nikolov, N.; Savchev, S.; Slavkov, V.; Slavov, G.; Stoyanova, Y. Synthesis of primary kinematic chains of specialized robots using a matrix method. In Proceedings of the 6th International Conference “Research and Development in Mechanical Industry” RaDMI 2006, Budva, Serbia and Montenegro, 13–17 September 2006; pp. 41–49. [Google Scholar]
- Bian, Z.; Ye, Z.; Mu, W. Kinematic analysis and simulation of 6-DOF industrial robot capable of picking up die-casting products. In Proceedings of the 2016 IEEE International Conference on Aircraft Utility Systems (AUS), Beijing, China, 10–12 October 2016. [Google Scholar] [CrossRef]
- Stachera, A.; Stolarski, A.; Owczarek, M.; Telejko, M. A Method of Multi-Criteria Assessment of the Building Energy Consumption. Energies
**2023**, 16, 183. [Google Scholar] [CrossRef] - Borcherding, K.; Schmeer, S.; Weber, M. Biases in multiattribute weight elicitation. In Contributions to Decision Making; Caverni, J.-P., Ed.; Elsevier: Amsterdam, The Netherlands, 1995. [Google Scholar]
- Luque, M.; Ruiz, F.; Miettinen, K. GLIDE—General Formulation for Interactive Multiobjective Optimization; Working Papers W-432; University of Malaga and Helsinki School of Economics Department of business Technology: Helsinki, Finland, 2007; ISSN 1235-5674. [Google Scholar]
- Shanmugasundar, G.; Kalita, K.; Cep, R.; Chohan, J. Decision Models for Selection of Industrial Robots—A Comprehensive Comparison of Multi-Criteria Decision Making. Processes
**2023**, 11, 1681. [Google Scholar] [CrossRef] - Hagag, A.M.; Yousef, L.S.; Abdelmaguid, T.F. Multi-criteria decision-making for machine selection in manufacturing and construction: Recent trends. Mathematics
**2023**, 11, 631. [Google Scholar] [CrossRef] - Pahl, G.; Beitz, W. Konstruktionslehre. Methoden und Anwendung; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Haik, Y.; Shahin, T. Engineering Design Process, 2nd ed.; Cengage Learning: Belmont, CA, USA, 2011. [Google Scholar]
- Miettinen, K. Nonlinear Multi-Objective Optimisation; Kluwer Int. Series: Dordrecht, The Netherlands, 1999. [Google Scholar]
- Jakob, W.; Blume, C. Pareto Optimization or Cascaded Weighted Sum: A Comparison of Concepts. Algorithms
**2014**, 7, 166–185. [Google Scholar] [CrossRef] - Efrani, T.; Utyuzhnikov, S. Directed search domain: A method for even generation of the Pareto frontier in multiobjective optimization. Eng. Optim.
**2011**, 43, 467–484. [Google Scholar] - Rai, R.; Allada, V. Modular product family design: Agent-based Pareto-optimization and quality loss function-based post-optimal analysis. Int. J. Prod. Res.
**2003**, 41, 4075–4098. [Google Scholar] [CrossRef] - Chinchuluun, A.; Pardalos, P.M. A survey of recent developments in multiobjective optimization. Ann. Oper. Res.
**2007**, 154, 29–50. [Google Scholar] [CrossRef] - Ruzika, S.; Wiecek, M.M. Approximation methods in multiobjective programming. J. Optim. Theory Appl.
**2005**, 126, 473–501. [Google Scholar] [CrossRef] - Saaty, T. Fundamentals of Decision Making and Priority Theory with the AHP; RWS Publications: Pittsburgh, PA, USA, 1994. [Google Scholar]
- Malakov, I.; Zaharinov, V.V. Interactive software system for multicriteria choosing of the structural variant of complex technical systems. In Proceedings of the 23rd DAAAM International Symposium on Intelligent Manufacturing and Automation 2012, Zadar, Croatia, 24–27 October 2012; pp. 199–204. [Google Scholar]
- ISO/IEC 9899:2011; Information Technology—Programming Languages—C. ISO Publishing: Geneve, Switzerland, 2011.
- Carabin, G.; Scalera, L. On the Trajectory Planning for Energy Efficiency in Industrial Robotic Systems. Robotics
**2020**, 9, 89. [Google Scholar] [CrossRef]

**Figure 5.**Main GUI of PolyOptimizer. (1) Main menu with common program menus; (2) Buttons ribbon for fast access to commonly used software functions such as saving, opening, and solving problems; (3) Objective function table selection and choice of optimization criterion (minimum/maximum) for the selected objective function; (4) Data input ribbon for consecutive input of values in the table, accelerating data input; (5) Display of table data for the currently selected objective function.

**Figure 10.**Graphical interpretation of solutions ${x}_{1}^{*}$, ${x}_{2}^{\mathrm{*}}$, and ${x}_{3}^{\mathrm{*}}$.

**Figure 12.**Variant ${x}_{1}^{*}$. (1) Pneumatic cylinder; (2) Driving arm; (3) Driven arm; (4) End plate; (5) Rotary pneumatic motor; (6) Parallelogram linkage mechanism; (7) End of arm plate.

**Figure 13.**Variant ${x}_{1}^{*}$ kinematic diagram. (1) Pneumatic cylinder; (2) Driving arm; (3) Driven arm; (4) End plate; (5) Rotary pneumatic motor; (6) Parallelogram linkage mechanism; (7) End of arm plate.

**Figure 14.**Pose of the robot in position for extracting the casting from the mold. (1) Pneumatic cylinder; (2) Driving arm; (3) Driven arm; (4) End plate; (5) Rotary pneumatic motor; (6) Parallelogram linkage mechanism; (7) End of arm plate.

Inputs | Outputs | ||||
---|---|---|---|---|---|

Material | Energy | Information | Material | Energy | Information |

Casting | Power | Control signals for the actuators | Extracted casting | Noise | Signals for executed movement |

Gripper |

Designation | Function | Description |
---|---|---|

F1 | Extract the casting from the tool | A movement to extract the casting from the mold |

F2 | Remove the casting from the working area of the machine | A movement to take the casting out of the overall dimensions of the machine plates |

F3 | Move the casting to the unloading position | Movement to reach unloading position |

F4 | Orient the casting for unloading | Movement to orient the casting into the unloading position so that it unloads properly |

No. | Metrics | Value | Unit |
---|---|---|---|

1 | Payload | 1.5 | kg |

2 | Mounting | Floor/On the die casting machine | - |

3 | Unloading position | 90 | deg |

4 | Extraction force | 400 | N |

5 | Stroke for extraction from the tool | 100 | mm |

6 | Stroke for removing from the working area (for floor mounted structures) | 1300 | mm |

7 | For die casting machines with locking force of | 80–130 | t |

8 | Cycle time | 8 | s |

Variant No. | Figure | Description |
---|---|---|

1 | Translational pneumatic module: the last link in the kinematic chain of the extractor | |

2 | Translational electrically driven module with ballscrew: first link in the kinematic chain of the extractor | |

3 | Translational electrically driven module with toothed rack and pinion: first link in the kinematic chain of the extractor | |

4 | Translational electrically driven module with chain gear: first link in the kinematic chain of the extractor | |

5 | Articulated mechanism with input rotation with electric drive: last link in the kinematic chain of the extractor | |

6 | Articulated mechanism with input translation with electric drive: last link in the kinematic chain of the extractor | |

7 | Articulated mechanism with input rotation with pneumatic drive: last link in the kinematic chain of the extractor | |

8 | Articulated mechanism with input translation with pneumatic drive: last link in the kinematic chain of the extractor |

**Table 5.**Variants for performing the function “remove the casting from the working area of the machine”.

Variant No. | Figure | Description |
---|---|---|

1 | Translational telescopic pneumatically driven module | |

2 | Translational telescopic electrically driven module | |

3 | Articulated mechanism with input translation, horizontal. This device performs F3 and F4 | |

4 | Articulated mechanism with input rotation, vertical. This device performs F3 and F4 |

Variant No. | Figure | Description |
---|---|---|

1 | Rotary pneumatic module | |

2 | Rotary electrically driven module | |

3 | Articulated mechanism with input translation and pneumatic drive |

Variant No. | Figure | Description |
---|---|---|

1 | Rotary pneumatic module, compatible with telescopic translation modules only | |

2 | Rotary pneumatic module: for installation as the penultimate link in the kinematic chain of the extractor | |

3 | Rotary electrically driven module: compatible with telescopic translation modules only | |

4 | Articulated mechanism with input translation |

Subset | ${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | ${\mathit{X}}_{3}$ | ${\mathit{X}}_{4}$ | Number of Variants |
---|---|---|---|---|---|

1 | ${x}_{1}^{1};\text{}{x}_{1}^{2};\text{}{x}_{1}^{3};\text{}{x}_{1}^{4};$ ${x}_{1}^{5};\text{}{x}_{1}^{6};\text{}{x}_{1}^{7};\text{}{x}_{1}^{8}$ | ${x}_{2}^{1};\text{}{x}_{2}^{2}$ | ${x}_{3}^{1};\text{}{x}_{3}^{2};\text{}{x}_{3}^{3}$ | ${x}_{4}^{1};\text{}{x}_{4}^{2};\text{}{x}_{4}^{3}$ | 144 |

2 | ${x}_{1}^{1};\text{}{x}_{1}^{2};\text{}{x}_{1}^{3};\text{}{x}_{1}^{4};$ ${x}_{1}^{5};\text{}{x}_{1}^{6};\text{}{x}_{1}^{7};\text{}{x}_{1}^{8}$ | ${x}_{2}^{3}$ | ${x}_{4}^{2}$ | 8 | |

3 | ${x}_{1}^{1};\text{}{x}_{1}^{2};\text{}{x}_{1}^{3};\text{}{x}_{1}^{4};$ ${x}_{1}^{5};\text{}{x}_{1}^{6};\text{}{x}_{1}^{7};\text{}{x}_{1}^{8}$ | ${x}_{2}^{4}$ | ${x}_{4}^{2};\text{}{x}_{4}^{4}$ | 16 |

$\mathbf{Structural}\text{}\mathbf{Variant}\text{}{\mathit{x}}_{1}^{\mathit{*}}$ | ||||
---|---|---|---|---|

${\mathit{x}}_{1}^{\mathit{*}}=\left\{{\mathit{x}}_{1}^{7};{\mathit{x}}_{2}^{3};{\mathit{x}}_{4}^{2}\right\}$ | ||||

No. | Objective Function | Value | Deviation from the Optimum for the Objective Function | Upper and Lower Boundary |

1 | $C\left(x\right)$, EUR | 3681.76 | ${w}_{1}=0.0268$ | $3330.65\le C\left(x\right)\le \mathrm{16,434.81}$ |

2 | $E\left(x\right)$, EUR/h | 0.17 | ${w}_{2}=0.0164$ | $0.15\le E\left(x\right)\le 1.37$ |

3 | $V\left(x\right)$ | 0.123 | ${w}_{3}=0.0221$ | $0.120\le V\left(x\right)\le 0.256$ |

4 | $T\left(x\right)$ | 0.121 | ${w}_{4}=0.0190$ | $0.118\le T\left(x\right)\le 0.276$ |

$\mathbf{Structural}\text{}\mathbf{Variant}\text{}{\mathit{x}}_{2}^{\mathit{*}}$ | ||||
---|---|---|---|---|

${\mathit{x}}_{2}^{\mathit{*}}=\left\{{\mathit{x}}_{1}^{8};{\mathit{x}}_{2}^{3};{\mathit{x}}_{4}^{2}\right\}$ | ||||

No. | Objective Function | Value | Deviation from the Optimum for the Objective Function | Upper and Lower Boundary |

1 | $C\left(x\right)$, EUR | 3330.65 | ${w}_{1}=0.0000$ | $3330.65\le C\left(x\right)\le \mathrm{16,434.81}$ |

2 | $E\left(x\right)$, EUR/h | 0.15 | ${w}_{2}=0.0000$ | $0.15\le E\left(x\right)\le 1.37$ |

3 | $V\left(x\right)$ | 0.129 | ${w}_{3}=0.0662$ | $0.120\le V\left(x\right)\le 0.256$ |

4 | $T\left(x\right)$ | 0.120 | ${w}_{4}=0.0127$ | $0.118\le T\left(x\right)\le 0.276$ |

$\mathbf{Structural}\text{}\mathbf{Variant}\text{}{\mathit{x}}_{3}^{\mathit{*}}$ | ||||
---|---|---|---|---|

${\mathit{x}}_{3}^{\mathit{*}}=\left\{{\mathit{x}}_{1}^{1};{\mathit{x}}_{2}^{3};{\mathit{x}}_{4}^{2}\right\}$ | ||||

No. | Objective Function | Value | Deviation from the Optimum for the Objective Function | Upper and Lower Boundary |

1 | $C\left(x\right)$, EUR | 3751.92 | ${w}_{1}=0.0322$ | $3330.65\le C\left(x\right)\le \mathrm{16,434.81}$ |

2 | $E\left(x\right)$, EUR/h | 0.33 | ${w}_{2}=0.1475$ | $0.15\le E\left(x\right)\le 1.37$ |

3 | $V\left(x\right)$ | 0.120 | ${w}_{3}=0.0000$ | $0.120\le V\left(x\right)\le 0.256$ |

4 | $T\left(x\right)$ | 0.118 | ${w}_{4}=0.0000$ | $0.118\le T\left(x\right)\le 0.276$ |

**Table 12.**Structural variant with equal priority of objective functions for floor-mounted extractors.

$\mathbf{Structural}\text{}\mathbf{Variant}\text{}{\mathit{x}}_{4}^{\mathit{*}}$ | ||||
---|---|---|---|---|

${\mathit{x}}_{4}^{\mathit{*}}=\left\{{\mathit{x}}_{1}^{7};{\mathit{x}}_{2}^{4};{\mathit{x}}_{4}^{2}\right\}$ | ||||

No. | Objective Function | Value | Deviation from the Optimum for the Objective Function | Upper and Lower Boundary |

1 | $C\left(x\right)$, EUR | 5450.76 | ${w}_{1}=0.1618$ | $3330.65\le C\left(x\right)\le \mathrm{16,434.81}$ |

2 | $E\left(x\right)$, EUR/h | 0.36 | ${w}_{2}=0.1721$ | $0.15\le E\left(x\right)\le 1.37$ |

3 | $V\left(x\right)$ | 0.155 | ${w}_{3}=0.2574$ | $0.120\le V\left(x\right)\le 0.256$ |

4 | $T\left(x\right)$ | 0.134 | ${w}_{4}=0.1013$ | $0.118\le T\left(x\right)\le 0.276$ |

**Table 13.**Structural variant, optimal for subset 1, under equal priority of the objective functions.

$\mathbf{Structural}\text{}\mathbf{Variant}\text{}{\mathit{x}}_{5}^{\mathit{*}}$ | ||||
---|---|---|---|---|

${\mathit{x}}_{5}^{\mathit{*}}=\left\{{\mathit{x}}_{1}^{8};{\mathit{x}}_{2}^{2};{\mathit{x}}_{3}^{3};{\mathit{x}}_{4}^{1}\right\}$ | ||||

No. | Objective Function | Value | Deviation from the Optimum for the Objective Function | Upper and Lower Boundary |

1 | $C\left(x\right)$, EUR | 6213.55 | ${w}_{1}=0.2200$ | $3330.65\le C\left(x\right)\le \mathrm{16,434.81}$ |

2 | $E\left(x\right)$, EUR/h | 0.55 | ${w}_{2}=0.3279$ | $0.15\le E\left(x\right)\le 1.37$ |

3 | $V\left(x\right)$ | 0.184 | ${w}_{3}=0.4706$ | $0.120\le V\left(x\right)\le 0.256$ |

4 | $T\left(x\right)$ | 0.196 | ${w}_{4}=0.4937$ | $0.118\le T\left(x\right)\le 0.276$ |

$\mathbf{Structural}\text{}\mathbf{Variant}\text{}{\mathit{x}}_{6}^{\mathit{*}}$ | ||||
---|---|---|---|---|

${\mathit{x}}_{6}^{\mathit{*}}=\left\{{\mathit{x}}_{1}^{6};{\mathit{x}}_{2}^{4};{\mathit{x}}_{4}^{4}\right\}$ | ||||

No. | Objective Function | Value | Deviation from the Optimum for the Objective Function | Upper and Lower Boundary |

1 | $C\left(x\right)$, EUR | 4702.33 | ${w}_{1}=0.1047$ | $3330.65\le C\left(x\right)\le \mathrm{16,434.81}$ |

2 | $E\left(x\right)$, EUR/h | 0.53 | ${w}_{2}=0.3115$ | $0.15\le E\left(x\right)\le 1.37$ |

3 | $V\left(x\right)$ | 0.131 | ${w}_{3}=0.0809$ | $0.120\le V\left(x\right)\le 0.256$ |

4 | $T\left(x\right)$ | 0.156 | ${w}_{4}=0.2405$ | $0.118\le T\left(x\right)\le 0.276$ |

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## Share and Cite

**MDPI and ACS Style**

Malakov, I.; Zaharinov, V.; Nikolov, S.; Dimitrova, R.
Computer-Aided Choosing of an Optimal Structural Variant of a Robot for Extracting Castings from Die Casting Machines. *Actuators* **2023**, *12*, 363.
https://doi.org/10.3390/act12090363

**AMA Style**

Malakov I, Zaharinov V, Nikolov S, Dimitrova R.
Computer-Aided Choosing of an Optimal Structural Variant of a Robot for Extracting Castings from Die Casting Machines. *Actuators*. 2023; 12(9):363.
https://doi.org/10.3390/act12090363

**Chicago/Turabian Style**

Malakov, Ivo, Velizar Zaharinov, Stiliyan Nikolov, and Reneta Dimitrova.
2023. "Computer-Aided Choosing of an Optimal Structural Variant of a Robot for Extracting Castings from Die Casting Machines" *Actuators* 12, no. 9: 363.
https://doi.org/10.3390/act12090363