# The Impact of Manufacturing Flexibility and Multi-Criteria Optimization on the Sustainability of Manufacturing Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Manufacturing Flexibility

## 3. Multi-Criteria Optimization

^{1}∊ X Pareto is dominated by another solution x

^{2}∊ X in the case where:

^{*}∈ X and the associated output value f (x *) is Pareto optimal if there is no other solution that dominates it. The Pareto group of optimal solutions is called the Pareto front. The Pareto front of multi-objective optimization problems is constrained by two vectors:

- The nadir vector is defined mathematically by Equation (5):$${z}_{i}^{nad}=\underset{x\in X}{\mathrm{sup}}{f}_{i}\left(x\right)\mathrm{for}\text{}\mathrm{all}i=1,\dots ,k$$
- The ideal vector is defined mathematically by Equation (6):$${z}_{i}^{ideal}=\underset{x\in X}{\mathrm{inf}}{f}_{i}\left(x\right)\mathrm{for}\text{}\mathrm{all}i=1,\dots ,k$$

_{1}and f

_{2}. The points in the coordinate system represent possible Pareto solutions where the point Z is not defined as the Pareto optimal solution, because it is dominated by the X and Y points. The points X and Y are not dominated by each other, so we can define both as Pareto optimal solutions.

## 4. Manufacturing Flexibility Modelling

_{i}. The processing time of the operation p

_{jk}may vary, depending on the machine on which it is performed. For the multi-objective flexible job shop scheduling problem, some limitations must be made:

- One machine can process only one job at a time.
- One job can be processed only on one machine at a time.
- When the operation starts it cannot be interrupted until the end of the operation; after completion, the next operation can start.
- All the jobs and operations have equal priorities at the time zero.
- Each machine m is ready at time zero.
- Given an operation O
_{ij}and the selected machine m, the processing time p_{ij}is fixed.

- Makespan (time required to complete all jobs):f
_{1}= max {C_{j}|j = 1, …, n} - Maximum workload (workload of the most loaded machine):$${f}_{2}=max{\displaystyle \sum}_{i=1}^{n}{\displaystyle \sum}_{j=1}^{{n}_{i}}{p}_{ijk}{x}_{ijk},k=1,2,\dots ,m$$
- Total workload of all machines:$${f}_{3}={\displaystyle \sum}_{i=1}^{n}{\displaystyle \sum}_{j=1}^{{n}_{i}}{\displaystyle \sum}_{k=1}^{m}{p}_{ijk}{x}_{ijk},k=1,2,\dots ,m$$
_{j}is the completion time of job J_{i}, and x_{ijk}is a decision variable on which individual machine the operation will be processed.

_{1}), medium (G

_{2}) and large machines (G

_{3}). The price range of operating hours is between 30 to 40 EUR/h for small machines, 40 to 50 EUR/h for medium and 50 to 60 EUR/h for large machines. We assumed real values of fixed costs and recalculated the idle cost of the machines using the literature [22]. A detailed recalculation of machine prices is shown below. The recommendations given in the literature [22] have defined fixed costs as 40% in the case of a small machine, 50% in the case of a medium-sized machine, and 60% of a fixed cost in the case of a large machine. The right column of Table 2 shows the factor between fixed and recalculated idle cost values used by the computer program as a constant value in a mathematical calculation assignment.

- The production system operates in two shifts,
- Financing the purchase of machinery, 50% own funds, 50% loan with 8% interest,
- Electricity value constant 0.2 EUR/kWh,
- 4% maintenance cost,
- Facility costs EUR 100/m
^{2}and - 4% additional operating costs.

#### The Impact of Cost-Time Profile on Manufacturing Flexibility

- Activities: There are two assumptions about activities. The first is that the cost of the activity is incurred continuously from start to finish of the activity. The second is that the resources must be ready for use before the activity begins. In CTP, activities are represented by a linear function dependence with a positive directional coefficient.
- Resources: In CTP, sources are staged with vertical lines, as they are always available at the time we need them. In addition, their costs are added to the total cost of the contract immediately. When the cost of resources is added to the cost of the product, it is treated as part of the cost that has been spent, and will not be reimbursed until the completion of the order (product sale).
- Waiting: It is defined as the sum of moments during which no activity occurs. It is assumed that, while waiting for the implementation of the activity, its costs do not increase. The fact that waiting costs do not increase in this activity is presented in the CTP as a horizontal line. Activity waiting periods are very important, because it is widely known that there is a considerable amount of time during which there is no added value to production processes. This time does not affect the cost of the order directly, but extends the time before the order is shipped.
- Total cost: Total cost represents the addition of all direct costs incurred in the production of the contract, without already being taken into account in the CTP diagram. The total cost is reflected in the amount at the time the order is completed.
- Cost-value investment: It is represented by a surface area below the CTP line, which represents how and how long the costs have accumulated during the production process. The surface area under the CTP represents the cost-time dimension.
- Direct costs: Represent the total amount of total costs and investment costs.

_{m}. Material costs’ values had to be assigned in order to provide a credible CTP. The material needed to execute the order is identified as the source of the CTP diagram.

_{s}, to 20 pieces of the product per single order in the scenario S

_{2}. The introduction of the simulation scenario method represents the possibility of testing and responding the simulation model to different changes in the production system. In our case, the three simulation scenarios define manufacturing flexibility as, fully customizable production in the R

_{S}scenario, where each order represents one piece of product. By increasing the number of products within the order in scenario S

_{1}(1 to 10 pieces) and S

_{2}(10 to 20 pieces), here, the production is defined as less flexible. The number of product pieces in Table 5, Table 6 and Table 7 is represented by label P

_{q}. The P

_{q}values are assigned numerically according to the distribution function and the interdependence of the production parameters. The simulation scenarios designed in this way allow us to analyze the impact of manufacturing flexibility on the cost-time profile diagram.

## 5. Manufacturing Flexibility Modelling Results

#### Cost-Time-Flexibility Profile Diagram

- The Kacem 5 × 10 low dimensional case shows a rapid increase in costs as production flexibility decreases. An additional feature of the graph is the average cost and flow times in the first third of the diagram, which is attributed to the small number of orders.
- The Kacem 10 × 10 medium dimensional case shows a continuous dependence of three variables, with no additional features detected within the three dimensional diagram. The interdependence in this case shows a linear dependence, the volume of the area below CTFP represents a steady dependence between time-cost and manufacturing flexibility.
- The Mk08 dataset shows a high dependence on cost, processing time and flexibility, which is represented by the slope of the diagram in the upper third of the diagram when flexibility is at its highest. An additional feature is the strong correlation between higher costs and flow time in the mean values. In this case, the production flexibility variable is also located in the middle value range.
- High-dimensional optimization problems (Kacem 15 × 10 and Mk10) show a significant dependence on the three parameters mentioned above to ensure production viability. Cost reduction and shorter processing times are influenced significantly by the flexibility of production, especially when increasing the number of orders.

## 6. Manufacturing Flexibility Case Study

#### 6.1. Manufacturing System Input Data

_{1}to M

_{12}represent the following operations:

- M
_{1}and M_{2}cutting of raw material, - M
_{3}to M_{6}manual welding, - M
_{7}and M_{8}robotic welding, - M
_{9}is a color coating operation, - The assembly operation is performed on two available machines M
_{10}and M_{11}, - The M
_{12}is a final control operation.

#### 6.2. IHKA Multi-Criteria Optimisation

- Min value represents the worst Pareto front position obtained from the state of the subject algorithm which dominates the object algorithm in a certain number of percentages.
- Max value represents the best Pareto front obtained by the subject algorithm, which dominates the object algorithm in a certain number of percentages.
- Mean value represents the performance of the algorithm in the subject column, and its dominance over the algorithm in the object column. The higher the value, the higher the performance of the algorithm in the subject column is.
- Std value represents the performance of the algorithm in the subject column and its stability relative to the algorithm in the object column. The lower the value, the higher the stability of the algorithm in the subject column is.

- The IHKA algorithm with 95.9% dominates the BBMOPSO algorithm with associated high stability of 6.17%. High performance of the IHKA algorithm is also proved in comparison with the MOPSO algorithm, which IHKA dominates with 85.94%, with stability percentage of 43.33%.
- The BBMOPSO algorithm dominates the IHKA algorithm only with 6.03%, which shows a low degree of dominance, with a corresponding 13.24% stability percent. The BBMOPSO and MOPSO comparative algorithms dominate each other, with 10.28%, and a stability percentage of 20.02%.
- The MOPSO algorithm has a low degree of dominance compared to the proposed IHKA algorithm, and the stability percentage is also high, which shows low stability. Cross-referencing the comparison algorithms shows the dominance of the MOPSO algorithm, with 75.64% versus BBMOPSO and a stability percentage of 29.47%.

_{1}and J

_{2}) of the IHKA output optimization results generated in the MATLAB software environment. The output optimization results of the optimization algorithm assign individual operation to the available machine, a start and finish time, and the machine sequence order in which it performs operations on the machine. The order of performing operations on the machine is performed according to the proposed method of its own decision logic [26], which bypasses the integrated decision logic of the simulation environment. The automated transfer of numerical optimization results to the simulation environment allows the user to determine their own units of measurement according to the real-world manufacturing system.

#### 6.3. Validation of Optimisation Results Using the CTFP Diagram

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Gantt chart of the improved heuristic Kalman algorithm (IHKA) optimization algorithm solutions.

**Figure 6.**Cost-time diagram in relation to manufacturing flexibility of a real-world production system.

Classification | |
---|---|

Level | Description |

Individual resource level | Individual resource level refers to flexibility associated with a resource. Labor flexibility, machine flexibility and material handling flexibility are included. |

Shop floor level | Shop floor level refers to flexibility associated with the shop floor. Routing flexibility and operation flexibility are included. |

Plant level | Plant level refers to flexibility associated with plant. Volume flexibility, mix flexibility, expansion flexibility and product flexibility, modification flexibility and new product flexibility are included. |

Functional level | Functional level describes manufacturing flexibility. |

Group | Operational Cost [EUR/h] | Fixed Cost [%] | Idle Cost [EUR/h] | Factor |
---|---|---|---|---|

G_{1} | 30–40 | 40 | 12–16 | x = 2/5 |

G_{2} | 41–50 | 50 | 20.5–25 | x = 1/2 |

G_{3} | 51–60 | 60 | 30.6–36 | x = 3/5 |

Machine | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | M_{6} | M_{7} | M_{8} | M_{9} | M_{10} |
---|---|---|---|---|---|---|---|---|---|---|

Operational cost [EUR/h] | 43 | 35 | 39 | 53 | 52 | 59 | 36 | 45 | 38 | 45 |

Idle cost [EUR/h] | 21.5 | 14 | 15.6 | 31.8 | 31.2 | 35.4 | 14.4 | 22.5 | 15.2 | 22.5 |

x_{loc} [m] | 0 | 0 | 5 | 5 | 10 | 10 | 15 | 15 | 20 | 20 |

y_{loc} [m] | 0 | 5 | 0 | 5 | 0 | 5 | 0 | 5 | 0 | 5 |

t [min] | 16 | 15 | 50 | 24 | 35 | 38 | 16 | 22 | 18 | 39 |

Data | G_{1} | G_{2} | G_{3} |
---|---|---|---|

Purchase price of the machine [EUR] | 20,000 | 70,000 | 200,000 |

Machine power [kW] | 4 | 10 | 25 |

Workplace surface [m^{2}] | 10 | 20 | 30 |

Depreciation period [year] | 8 | 8 | 8 |

Useful capacity of the machine [h/year] | 3000 | 3200 | 3400 |

Energy costs [EUR/kWh] | 0.40 | 1.00 | 2.50 |

Tool costs [EUR/h] | 2 | 3 | 4 |

Costs of machine [EUR/h] | 3.95 | 8.67 | 18.27 |

Worker gross costs [EUR/h] | 8 | 10 | 12 |

Additional costs [EUR/h] | 0.16 | 0.35 | 0.73 |

Workplace costs [EUR/h] | 12.11 | 19.02 | 31.00 |

Variable costs [%] | 12.8 | 24.6 | 38 |

P_{q} [pcs] | P_{m} [EUR] | P_{q} [pcs] | P_{m} [EUR] | P_{q} [pcs] | P_{m} [EUR] | |||
---|---|---|---|---|---|---|---|---|

Order 1 | 1 | 22 | Order 2 | 1 | 23 | Order 3 | 1 | 21 |

Order 4 | 1 | 29 | Order 5 | 1 | 25 | Order 6 | 1 | 27 |

Order 7 | 1 | 19 | Order 8 | 1 | 24 | Order 9 | 1 | 25 |

Order 10 | 1 | 20 | Order 11 | 1 | 18 | Order 12 | 1 | 21 |

Order 13 | 1 | 24 | Order 14 | 1 | 30 | Order 15 | 1 | 25 |

P_{q} [pcs] | P_{m} [EUR] | P_{q} [pcs] | P_{m} [EUR] | P_{q} [pcs] | P_{m} [EUR] | |||
---|---|---|---|---|---|---|---|---|

Order 1 | 6 | 132 | Order 2 | 9 | 207 | Order 3 | 6 | 126 |

Order 4 | 1 | 29 | Order 5 | 4 | 96 | Order 6 | 8 | 216 |

Order 7 | 1 | 19 | Order 8 | 5 | 120 | Order 9 | 6 | 150 |

Order 10 | 8 | 160 | Order 11 | 6 | 108 | Order 12 | 8 | 168 |

Order 13 | 2 | 48 | Order 14 | 1 | 30 | Order 15 | 5 | 125 |

P_{q} [pcs] | P_{m} [EUR] | P_{q} [pcs] | P_{m} [EUR] | P_{q} [pcs] | P_{m} [EUR] | |||
---|---|---|---|---|---|---|---|---|

Order 1 | 11 | 242 | Order 2 | 19 | 437 | Order 3 | 20 | 420 |

Order 4 | 15 | 435 | Order 5 | 10 | 240 | Order 6 | 12 | 324 |

Order 7 | 13 | 247 | Order 8 | 19 | 456 | Order 9 | 14 | 350 |

Order 10 | 11 | 220 | Order 11 | 10 | 180 | Order 12 | 17 | 357 |

Order 13 | 16 | 384 | Order 14 | 14 | 420 | Order 15 | 17 | 425 |

Number of Orders | Number of Products [pcs] | Flow Time [h] | Total Cost [EUR/pcs] | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Scenario | R_{S} | S_{1} | S_{2} | R_{S} | S_{1} | S_{2} | R_{S} | S_{1} | S_{2} | |

Kacem 5 × 10 | 5 | 5 | 26 | 75 | 1.76 | 1.63 | 1.55 | 43.8 | 16.3 | 5.3 |

Kacem 10 × 10 | 10 | 10 | 54 | 144 | 1.15 | 1.14 | 1.04 | 24 | 7.2 | 3.4 |

Kacem 15 × 10 | 15 | 15 | 76 | 218 | 1.24 | 1.17 | 1.15 | 12.6 | 2.8 | 0.8 |

Mk08 | 5 | 5 | 26 | 75 | 9.32 | 9.26 | 9.2 | 207 | 65.5 | 15.7 |

Mk10 | 10 | 10 | 54 | 144 | 8.39 | 8.33 | 8.27 | 110.9 | 21.8 | 7.2 |

Product 1 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Machine | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | M_{6} | M_{7} | M_{8} | M_{9} | M_{10} | M_{11} | M_{12} |

Process time [min] | 20 | 24 | 40 | 45 | 38 | 47 | 20 | 25 | 11 | 22 | 20 | 12 |

Usage cost [EUR] | 45 | 45 | 35 | 35 | 35 | 35 | 52 | 52 | 59 | 43 | 43 | 35 |

Idle cost [EUR] | 22.5 | 22.5 | 14 | 14 | 14 | 14 | 31.2 | 31.2 | 35.4 | 21.5 | 21.5 | 15 |

x_{loc} [m] | 8 | 8 | 12.5 | 18.5 | 24.5 | 30.5 | 36 | 36 | 24.5 | 19.5 | 27.5 | 20 |

y_{loc} [m] | 9.5 | 4.5 | 0 | 0 | 0 | 0 | 5.5 | 10.5 | 16.5 | 12 | 12 | 7 |

Setup time [min] | 10 | 10 | 15 | 15 | 15 | 15 | 8 | 8 | 18 | 7 | 7 | 3 |

**Table 10.**C-metric performance measures of IHKA, multi-objective particle swarm optimization (MOPSO) and bare bones multi-objective particle swarm optimization (BBMOPSO) algorithms.

Subject | Object | Min [%] | Max [%] | Mean [%] | Std [%] |
---|---|---|---|---|---|

IHKA | BBMOPSO | 77.78 | 100 | 95.9 | 6.17 |

IHKA | MOPSO | 0 | 100 | 58.94 | 43.33 |

BBMOPSO | IHKA | 0 | 50 | 6.03 | 13.24 |

BBMOPSO | MOPSO | 0 | 66.67 | 10.28 | 20.02 |

MOPSO | IHKA | 0 | 100 | 32.98 | 39.74 |

MOPSO | BBMOPSO | 11.11 | 100 | 75.64 | 29.47 |

Algorithm | Mark | Optimization Parameter | ||
---|---|---|---|---|

MC [min] | TW [min] | MW [min] | ||

IHKA | x | 392.45 | 1800.73 | 217.98 |

BBMOPSO | + | 422.21 | 1807.1 | 210.76 |

MOPSO | * | 400.48 | 1800.76 | 214.48 |

Order | Operation | Machine | Start Time [min] | Finish Time [min] | Machine Sequence |
---|---|---|---|---|---|

J_{1} | O_{1,1} | M_{1} | 102 | 124 | 6 |

O_{1,2} | M_{4} | 131 | 174 | 3 | |

O_{1,3} | M_{8} | 177 | 200 | 5 | |

O_{1,4} | M_{9} | 211 | 223 | 10 | |

O_{1,5} | M_{10} | 223 | 248 | 4 | |

O_{1,6} | M_{12} | 268 | 282 | 10 | |

J_{2} | O_{2,1} | M_{1} | 82 | 102 | 5 |

O_{2,2} | M_{5} | 116 | 154 | 3 | |

O_{2,3} | M_{7} | 159 | 179 | 4 | |

O_{2,4} | M_{9} | 200 | 211 | 9 | |

O_{2,5} | M_{11} | 216 | 236 | 5 | |

O_{2,6} | M_{12} | 239 | 251 | 8 |

Number of Orders | Number of Products [pcs] | Flow Time [h] | Total Cost [EUR/pcs] | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Scenario | R_{S} | S_{1} | S_{2} | R_{S} | S_{1} | S_{2} | R_{S} | S_{1} | S_{2} | |

RW_PS | 15 | 15 | 76 | 218 | 0.34 | 0.23 | 0.21 | 356.6 | 251.5 | 218.61 |

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**MDPI and ACS Style**

Ojstersek, R.; Buchmeister, B.
The Impact of Manufacturing Flexibility and Multi-Criteria Optimization on the Sustainability of Manufacturing Systems. *Symmetry* **2020**, *12*, 157.
https://doi.org/10.3390/sym12010157

**AMA Style**

Ojstersek R, Buchmeister B.
The Impact of Manufacturing Flexibility and Multi-Criteria Optimization on the Sustainability of Manufacturing Systems. *Symmetry*. 2020; 12(1):157.
https://doi.org/10.3390/sym12010157

**Chicago/Turabian Style**

Ojstersek, Robert, and Borut Buchmeister.
2020. "The Impact of Manufacturing Flexibility and Multi-Criteria Optimization on the Sustainability of Manufacturing Systems" *Symmetry* 12, no. 1: 157.
https://doi.org/10.3390/sym12010157