# Recent Application of Dijkstra’s Algorithm in the Process of Production Planning

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Increasing efficiency: To a large extent, successful business activity depends on planning. For these activities to be as effective as possible, it is necessary that the objectives are set as clearly as possible and that the optimal options for achieving them and the evaluation criteria are defined.
- Risk reduction: Risk in the organization can be eliminated by good planning. Planning makes it possible to identify risks in various business activities and can thus reduce their negative impact on them.
- Successful organizational changes: The better managers’ ideas about organizational change are, the easier it will be for them to cope with the consequences. Successful adaptation in the organization is not possible without a good planning process.

## 2. The Work Methodology

- Assign the vertex v
_{k}:- The value of minimum path MC(v
_{k}) = O(v_{k}), i.e., MC(v_{k}) = JC_{kk}; - The permanent condition (St(v
_{k}) = 1); - The predecessor Pred(v
_{k}) = k.

- Assign to the other vertices v
_{i}, where i = 1, 2, …, m, i $\ne $ k:- The value of minimum path MC(v
_{i}) = $\infty $; - The transient state (St(v
_{i}) = 0); - The predecessor Pred(v
_{i}) = −1.

- Choose the vertex v
_{k}as the working vertex v_{p}(we assigned the value k to the p, i.e., p = k). - For each vertex v
_{i}where i ≠ p, which is adjacent to the working vertex v_{k}, and whose state is transient, calculate the value MC(v_{p}) + O(h_{j}) = MC(v_{p}) + JC_{pi}(hj is the edge connecting the working vertex with the vertex vi), and if this value is lower than MC(v_{i}), then set MC(v_{i}) = MC(v_{p}) + JC_{pi}and Pred(v_{i}) = p. - For all of the vertices v
_{i}whose state is transient, find the vertex v_{l}with the lowest value MC(v_{l}) and- Choose the vertex v
_{l}as the working vertex; - A it a permanent state (St(v
_{l}) = 1).

- Repeat the procedure in points 4 and 5 until all peaks have a permanent state.

## 3. Results and Discussion

- G is a continuous undirected graph without multiple edges and loops;
- V = {v
_{1}, v_{2}, ⋯, v_{m}} is a set of graph vertices that represent mining and processing and consumption points; - H = {h
_{1}, h_{2}, ⋯, h_{n}} is a set of graph edges that represent the transport network (road or rail) between individual places; the vertices of the graph, which correspond to the mining processing, are evaluated, and the evaluation of such a peak v_{i}is defined by [22]:

_{i}) = PTN

_{i}+ NTN

_{i}+ PSN

_{i}+ NSN

_{i}

_{i}is the direct extraction costs per tonne of raw material, NTN

_{i}is the indirect extraction costs per tonne of raw material, PSN

_{i}is the direct processing costs per tonne of material, and NSN

_{i}is the indirect processing costs per 1 tonne ton of material; the vertices of the graph, which correspond to those customer points, are not simultaneously mining–processing points and are not evaluated, and all of the edges of the graph are evaluated. The evaluation of the edge h

_{i}connecting the vertices v

_{k}and v

_{l}is defined by equation [21]:

_{i}) = PN × VZD

_{kl}

_{1}and v

_{2}, using the described algorithm [25,26]. The search procedure and its results are clear in Table 3, in which the states of the vertices of the graph are indicated as follows: the permanent top state is in blue, permanent peaks that are also the working peaks in a given step are in green, and temporary peaks with minimum values are in red.

_{1}, to the value of the price of the raw material after production behind the plant gates, i.e., the work area. We set the predecessor as ourselves, and marked it as permanent (green). The other vertices on the map only form customer sites (white). We marked the transport costs between the individual peaks in blue. A sufficiently significant number of reserves in the deposit is required for a long-term project [6].

_{1}were re-evaluated. The evaluation consisted of summing the price at the top of v

_{1}and the transport price at the top of v

_{1}. Then, we found that out of all of the peaks that were not permanent, the peak with the minimum rating v

_{2}(red) was a temporary peak.

_{2}as permanent and working (green). Then, we repeated the procedure in step 1. Again, we found the vertex with the minimum rating in this step. It was a temporary vertex v

_{3}(red).

_{3}(green) for the permanent and working peak. In this case, the peak with the minimum rating is a temporary peak v

_{4}(red). We can generally say that the Dijkstra algorithm is finite because exactly one node is added to the set of visited nodes in each passage of its cycle. At most, there are as many cycle transitions as there are peaks in the graph peaks.

_{4}as the starting point, which is permanent and working (green). After subsequent optimization, v

_{6}becomes a temporary peak with a minimum rating (red). Dijkstra’s algorithm solves the problem of finding the shortest paths in the edge-evaluated and oriented graph. All of its edges must have non-negative weights.

_{6}(green), in which the optimal peak with a minimum rating becomes a temporary peak v

_{5}(red).

_{5}(green), in which the optimal vertex with a minimum rating becomes a temporary vertex v

_{7}(red).

_{7}(green), in which the optimal peak with a minimum rating became a temporary peak v

_{8}(red).

_{8}(green), in which after reassessment, all of the other peaks become optimal peaks with a minimum rating due to a smaller rating than in working peak v

_{8}.

_{1}as a source of building material (Figure 12). At each peak, there is the optimal minimum purchase price as well as the predecessor on the optimal route.

#### Discussion and Limitation

- Methods that deal with editing a graph, representing a graph, or creating auxiliary graphs. This group includes, for example, dual graph representations.
- Methods that modify the crawling algorithm without changing the original graph.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Mula, J.; Poler, R.; García-Sabater, J.; Lario, F. Models for production planning under uncertainty: A review. Int. J. Prod. Econ.
**2006**, 103, 271–285. [Google Scholar] [CrossRef] [Green Version] - Maravelias, C.T.; Sung, C. Integration of production planning and scheduling: Overview, challenges and opportunities. Comput. Chem. Eng.
**2009**, 33, 1919–1930. [Google Scholar] [CrossRef] - Straka, M.; Khouri, S.; Rosova, A.; Caganova, D.; Culkova, K. Utilization of Computer Simulation for Waste Separation Design as a Logistics System. Int. J. Simul. Model.
**2018**, 17, 583–596. [Google Scholar] [CrossRef] - Kuo, Y.-H.; Kusiak, A. From data to big data in production research: The past and future trends. Int. J. Prod. Res.
**2019**, 57, 4828–4853. [Google Scholar] [CrossRef] [Green Version] - Rosova, A. Indices system design of distribution logistics, transport logistics and materials flow as parts of controlling in enterprise’s logistics. Acta Montan. Slovaca
**2010**, 15, 67–72. [Google Scholar] - Wang, F.; Lai, X.; Shi, N. A multi-objective optimization for green supply chain network design. Decis. Support Syst.
**2011**, 51, 262–269. [Google Scholar] [CrossRef] - Erol, I.; Sencer, S.; Sari, R. A new fuzzy multi-criteria framework for measuring sustainability performance of a supply chain. Ecol. Econ.
**2011**, 70, 1088–1100. [Google Scholar] [CrossRef] - Kumar, S.; Putnam, V. Cradle to cradle: Reverse logistics strategies and opportunities across three industry sectors. Int. J. Prod. Econ.
**2008**, 115, 305–315. [Google Scholar] [CrossRef] - Abdallah, T.; Farhat, A.; Diabat, A.; Kennedy, S. Green supply chains with carbon trading and environmental sourcing: Formulation and life cycle assessment. Appl. Math. Model.
**2011**, 36, 4271–4285. [Google Scholar] [CrossRef] - Straka, M.; Rosová, A. Principles of computer simulation design for the needs of improvement of the raw materials combined transport system. Acta Montan. Slovaca
**2018**, 23, 163–174. [Google Scholar] - Bai, C.; Sarkis, J. Integrating sustainability into supplier selection with grey system and rough set methodologies. Int. J. Prod. Econ.
**2010**, 124, 252–264. [Google Scholar] [CrossRef] - Chen, R.J.; Gotman, C. Efficient fastest-path computations for road maps. Comput. Vis. Media
**2021**, 7, 267–281. [Google Scholar] [CrossRef] - Žic, J.; Zic, S. Multi-criteria decision making in supply chain management based on inventory levels, environmental impact and costs. Adv. Prod. Eng. Manag.
**2020**, 15, 151–163. [Google Scholar] [CrossRef] - Delling, D.; Goldberg, A.V.; Pajor, T.; Werneck, R.F. Customizable Route Planning in Road Networks. Transp. Sci.
**2017**, 51, 566–591. [Google Scholar] [CrossRef] - Nagyová, A.; Pačaiová, H.; Markulik, Š.; Turisová, R.; Kozel, R.; Džugan, J. Design of a Model for Risk Reduction in Project Management in Small and Medium-Sized Enterprises. Symmetry
**2021**, 13, 763. [Google Scholar] [CrossRef] - Ahluwalia, P.K.; Nema, A.K. Multi-objective reverse logistics model for integrated computer waste management. Waste Manag. Res. J. Sustain. Circ. Econ.
**2006**, 24, 514–527. [Google Scholar] [CrossRef] - Yanjun, L.; Xiaobo, L.; Osamu, Y. Traffic engineering framework with machine learning based meta-layer un software-defined networks. In Proceedings of the 2014 4th IEEE International Conference on Network Infrastructure and Digital Content, Beijing, China, 19–21 September 2014; pp. 121–125. [Google Scholar]
- Malindzakova, M.; Straka, M.; Rosova, A.; Kanuchova, M.; Trebuna, P. Modeling the process for incineration of municipal waste. Przem. Chem.
**2015**, 94, 1260–1264. [Google Scholar] [CrossRef] - Wicher, P.; Staš, D.; Karkula, M.; Lenort, R.; Besta, P. A computer simulation-based analysis of supply chains resilience in industrial environment. Metalurgija
**2015**, 54, 703–706. [Google Scholar] - Sinay, J.; Balážiková, M.; Dulebová, M.; Markulik, Š.; Kotianová, Z. Measurement of low-frequency noise during CNC machining and its assessment. Measurement
**2018**, 119, 190–195. [Google Scholar] [CrossRef] - Sobrino, D.R.D.; Košťál, P.; Cagáňová, D.; Čambál, M. On the Possibilities of Intelligence Implementation in Manufacturing: The Role of Simulation. Appl. Mech. Mater.
**2013**, 309, 96–104. [Google Scholar] [CrossRef] - Kleijnen, J. Supply chain simulation tools and techniques: A survey. Int. J. Simul. Process Model.
**2005**, 1, 82. [Google Scholar] [CrossRef] [Green Version] - Pačaiová, H.; Sinay, J.; Turisová, R.; Hajduová, Z.; Markulik, Š. Measuring the qualitative factors on copper wire surface. Measurement
**2017**, 109, 359–365. [Google Scholar] [CrossRef] - Wielki, J.; Grabara, J. The Impact of Ad-Blocking on the Sustainable Development of the Digital Advertising Ecosystem. Sustainability
**2018**, 10, 4039. [Google Scholar] [CrossRef] [Green Version] - Drastich, A. Optimization of material flow by simulation methods. Acta Logist.
**2017**, 4, 23–26. [Google Scholar] [CrossRef] - Hadi, M.A.; Brillinger, M.; Wuwer, M.; Schmid, J.; Trabesinger, S.; Jäger, M.; Haas, F. Sustainable peak power smoothing and energy-efficient machining process thorough analysis of high-frequency data. J. Clean. Prod.
**2021**, 318, 128548. [Google Scholar] [CrossRef] - Cvitić, I.; Peraković, D.; Perisa, M.; Gupta, B.B. Ensemble machine learning approach for classification of IoT devices in smart home. Int. J. Mach. Learn. Cybern.
**2021**, 12, 3179–3202. [Google Scholar] [CrossRef]

**Figure 1.**The relationship between important functions in the company management [Authors own processed].

${\mathit{v}}_{1}$ | ${\mathit{v}}_{2}$ | ${\mathit{v}}_{3}$ | ${\mathit{v}}_{4}$ | ${\mathit{v}}_{5}$ | ${\mathit{v}}_{6}$ | ${\mathit{v}}_{7}$ | ${\mathit{v}}_{8}$ | |
---|---|---|---|---|---|---|---|---|

${\mathit{v}}_{1}$ | 0 | 5 | 9 | - | 22 | - | - | - |

${\mathit{v}}_{2}$ | 5 | 0 | 5 | 8 | - | - | - | - |

${\mathit{v}}_{3}$ | 9 | 5 | 0 | - | 13 | - | - | - |

${\mathit{v}}_{4}$ | - | 8 | - | 0 | - | 8 | - | - |

${\mathit{v}}_{5}$ | 22 | - | 13 | - | 0 | 9 | - | 25 |

${\mathit{v}}_{6}$ | - | - | - | 8 | 9 | 0 | 25 | - |

${\mathit{v}}_{7}$ | - | - | - | - | - | 25 | 0 | 3 |

${\mathit{v}}_{8}$ | - | - | - | - | 25 | - | 3 | 0 |

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

1 | 7.20 | 0.75 | 1.35 | ∞ | 3.30 | ∞ | ∞ | ∞ |

2 | 0.75 | 7.00 | 0.75 | 1.20 | ∞ | ∞ | ∞ | ∞ |

3 | 1.35 | 0.75 | 0 | ∞ | 1.95 | ∞ | ∞ | ∞ |

4 | ∞ | 1.20 | ∞ | 0 | ∞ | 1.20 | ∞ | ∞ |

5 | 3.30 | ∞ | 1.95 | ∞ | 0 | 1.35 | ∞ | 3.75 |

6 | ∞ | ∞ | ∞ | 1.20 | 1.35 | 0 | 3.75 | ∞ |

7 | ∞ | ∞ | ∞ | ∞ | ∞ | 3.75 | 0 | 0.45 |

8 | ∞ | ∞ | ∞ | ∞ | 3.75 | ∞ | 0.45 | 0 |

Peak | |||||||||
---|---|---|---|---|---|---|---|---|---|

Step | Parameter | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

0 | The length of the journey | 7.20 | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ |

Condition | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

Predecessor | 1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | |

1 | The length of the journey | 7.20 | 7.95 | 8.55 | ∞ | 10.50 | ∞ | ∞ | ∞ |

Condition | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

Predecessor | 1 | 1 | 1 | −1 | 1 | −1 | −1 | −1 | |

2 | The length of the journey | 7.20 | 7.95 | 8.55 | 9.15 | 10.50 | ∞ | ∞ | ∞ |

Condition | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |

Predecessor | 1 | 1 | 1 | 2 | 1 | −1 | −1 | −1 | |

3 | The length of the journey | 7.20 | 7.95 | 8.55 | 9.15 | 10.50 | ∞ | ∞ | ∞ |

Condition | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | |

Predecessor | 1 | 1 | 1 | 2 | 1 | −1 | −1 | −1 | |

4 | The length of the journey | 7.20 | 7.95 | 8.55 | 9.15 | 10.50 | 10.35 | ∞ | ∞ |

Condition | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | |

Predecessor | 1 | 1 | 1 | 2 | 1 | 4 | −1 | −1 | |

5 | The length of the journey | 7.20 | 7.95 | 8.55 | 9.15 | 10.50 | 10.35 | 14.10 | ∞ |

Condition | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | |

Predecessor | 1 | 1 | 1 | 2 | 1 | 4 | 6 | −1 | |

6 | The length of the journey | 7.20 | 7.95 | 8.55 | 9.15 | 10.50 | 10.35 | 14.10 | 14.25 |

Condition | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | |

Predecessor | 1 | 1 | 1 | 2 | 1 | 4 | 6 | 5 | |

7 | The length of the journey | 7.20 | 7.95 | 8.55 | 9.15 | 10.50 | 10.35 | 14.10 | 14.25 |

Condition | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | |

Predecessor | 1 | 1 | 1 | 2 | 1 | 4 | 6 | 5 | |

8 | The length of the journey | 7.20 | 7.95 | 8.55 | 9.15 | 10.50 | 10.35 | 14.10 | 14.25 |

Condition | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

Predecessor | 1 | 1 | 1 | 2 | 1 | 4 | 6 | 5 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Behún, M.; Knežo, D.; Cehlár, M.; Knapčíková, L.; Behúnová, A.
Recent Application of Dijkstra’s Algorithm in the Process of Production Planning. *Appl. Sci.* **2022**, *12*, 7088.
https://doi.org/10.3390/app12147088

**AMA Style**

Behún M, Knežo D, Cehlár M, Knapčíková L, Behúnová A.
Recent Application of Dijkstra’s Algorithm in the Process of Production Planning. *Applied Sciences*. 2022; 12(14):7088.
https://doi.org/10.3390/app12147088

**Chicago/Turabian Style**

Behún, Marcel, Dušan Knežo, Michal Cehlár, Lucia Knapčíková, and Annamária Behúnová.
2022. "Recent Application of Dijkstra’s Algorithm in the Process of Production Planning" *Applied Sciences* 12, no. 14: 7088.
https://doi.org/10.3390/app12147088