# Modelling Dynamics of a Log-Yard through Discrete-Event Mathematics

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Discrete Event Modelling (DEM)

#### 1.2. Problem Formulation

- First modelling approach analyses trucks, trains, and ships as different entities which is useful to observe detailed information at every stage of the process, and can be interesting when the company does not record data at those stages. This model is also useful to perform modifications in the logic decision making of the company to evaluate their effect in the process. However, the main disadvantage is that the amount of details in such a model involve a large number of parameters. Thus, defining the specific places that could lead to optimizing the system becomes difficult. Similarly, using mathematical optimization is rather difficult, if not impossible, in such a model.
- Second modelling approach analyses trucks, trains, and ships as a single entity form which is useful to quickly observe the process ability to cope with the incoming logs, and the demands of the pulp mill. It is also useful to observe whether the system can cope with higher amounts of trucks, trains, and ships, or the influence of using new machines. As the number of parameters is small, mathematical optimization is feasible with such a model. However, the main disadvantage is that this model cannot provide the internal details of the decision-making undergoing in the process.

## 2. Materials and Methods

#### 2.1. Data Processing and Normalization

#### 2.2. Model 1

#### 2.3. First Stage

#### 2.3.1. Entity Generation and Attributes

#### 2.3.2. Control over the Size of the Entity Queue and Server Work

#### 2.3.3. Selecting an Storage Queue for Unload

#### 2.4. Second Stage

#### 2.4.1. Server Work in Stage Two

#### 2.4.2. Selecting Logs out of a Storage Area for Log In-Feed to the Mill

#### 2.5. Defining Model Parameters According to Data

#### 2.5.1. Parameters that Can Be Derived from Data

#### 2.5.2. Parameters that Cannot Be Derived from Data

#### 2.6. Model 2

#### 2.7. Stage 1

#### 2.7.1. Entity Generation

#### 2.7.2. Entity Queue Control and Server Work

#### 2.7.3. Storage Queue Selection

#### 2.8. Stage 2

#### 2.8.1. Server Work

#### 2.8.2. Storage Queue Selection for Log in-feed to the Mill

#### 2.8.3. Model Parameter Setting

#### 2.9. Method to Draw Quantitative Comparison between Models and Data

- Case 1: The company data are directly fed into model 1 to record the behaviour of the model according to these data. To this end, the data were organized to provide the necessary intergeneration time, attributes, and id number for each of its entries. After simulation, all the dynamic responses of the data—from stage 1 and 2—were saved.
- Case 2: Model 1 was simulated according to its mathematics presented in Section 2. The simulations are set to produce new values for the PDFs at each run. Thus, each simulation run will be different from the previous. The computer was left to produce 100 simulation results. This process took several hours.
- Case 3: Model 2 was simulated according to its mathematics presented in Section 2. As above, each simulation run is different from the previous. The computer was left to produce 100 simulation results. This process took 1 hour.

## 3. Results

#### 3.1. Number of Entities for Model 1 and Model 2

#### 3.2. Detailed Results for Model 1

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Berg, M.; Kinnwall, M.; Heinsoo, K.; Niklasson, M. Så går det för skogsindustrin [This Is How It Is Going for Forest Industry]; Report 1; Skogsindustrierna: Stockholm, Sweden, 2018. [Google Scholar]
- Asikainen, A. Simulation of Logging and Barge Transport of Wood from Forests on Islands. Int. J. For. Eng.
**2001**, 12, 43–50. [Google Scholar] [CrossRef][Green Version] - Karttunen, K.; Lättilä, L.; Korpinen, O.J.; Ranta, T. Cost-efficiency of intermodal container supply chain for forest chips. Silva Fennica
**2013**, 47, 24. [Google Scholar] [CrossRef][Green Version] - Väätäinen, K.; Prinz, R.; Malinen, J.; Laitila, J.; Sikanen, L. Alternative operation models for using a feed-in terminal as a part of the forest chip supply system for a CHP plant. Gcb Bioenergy
**2017**, 9, 1657–1673. [Google Scholar] [CrossRef][Green Version] - Arriagada, R.A.; Cubbage, F.W.; Abt, K.L.; Huggett, R.J., Jr. Estimating harvest costs for fuel treatments in the West. For. Prod. J.
**2008**, 58, 24. [Google Scholar] - Eriksson, A.; Eliasson, L.; Jirjis, R. Simulation-based evaluation of supply chains for stump fuel. Int. J. For. Eng.
**2014**, 1–14. [Google Scholar] [CrossRef] - Berg, S.; Bergström, D.; Nordfjell, T. Simulating conventional and integrated stump- and round-wood harvesting systems: A comparison of productivity and costs. Int. J. For. Eng.
**2014**, 25, 138–155. [Google Scholar] [CrossRef] - Pinho, T.M.; Coelho, J.P.; Moreira, A.P.; Boaventura-Cunha, J. Modelling a biomass supply chain through discrete-event simulation. IFAC-PapersOnLine
**2016**, 49, 84–89. [Google Scholar] [CrossRef] - Chiorescu, S.; Gronlund, A. Assessing the role of the harvester within the forestry-wood chain. For. Prod. J.
**2001**, 51, 77. [Google Scholar] - Salichon, M.C. Simulating Changing Diameter Distributions in a Softwood Sawmill. Master’s Thesis, Graduate School, Oregon State University, Corvallis, OR, USA, 2005. [Google Scholar]
- Mendoza, G.A.; Meimban, R.J.; Araman, P.A.; Luppold, W.G. Combined log inventory and process simulation models for the planning and control of sawmill operations. In Proceedings of the 23rd CIRP International Seminar on Manufacturing Systems, Nancy, France, 6–7 June 1991; p. 8. [Google Scholar]
- Rahman, A.; Yella, S.; Dougherty, M. Simulation model using meta heuristic algorithms for achieving optimal arrangement of storage bins in a sawmill yard. J. Intell. Learn. Syst. Appl.
**2014**, 6, 125–139. [Google Scholar] [CrossRef][Green Version] - Beaudoin, D.; LeBel, L.; Soussi, M.A. Discrete Event Simulation to Improve Log Yard Operations. INFOR Inf. Syst. Oper. Res.
**2013**, 50, 175–185. [Google Scholar] [CrossRef][Green Version] - Puodziunas, M.; Fjeld, D. Roundwood Handling at a Lithuanian Sawmill-Discrete-event Simulation of Sourcing and Delivery Scheduling. Baltic For.
**2008**, 14, 163–175. [Google Scholar] - LeBel, L.; Carruth, J.S. Simulation of woodyard inventory variations using a stochastic model. For. Prod. J.
**1997**, 47, 52. [Google Scholar] - Taha, H.A. Operations Research: An Introduction, 5th ed.; Macmillan: Detroit, MI, USA, 1992; Volume 1. [Google Scholar]
- Lättilä, L. Improving Transportation and Warehousing Efficiency with Simulation-Based Decision Support Systems. Ph.D. Thesis, Faculty of Technology Management, Industrial Management, Lappeenranta University of Technology, Lappeenranta, Finland, 2012. [Google Scholar]
- Aalto, M.; Raghu, K.C.; Korpinen, O.J.; Karttunen, K.; Ranta, T. Modeling of biomass supply system by combining computational methods—A review article. Appl. Energy
**2019**, 243, 145–154. [Google Scholar] [CrossRef] - Zeigler, B.P.; Kim, T.G.; Praehofer, H. Theory of Modeling and Simulation, 2nd ed.; Academic Press, Elsevier: London, UK, 2000. [Google Scholar]
- Silverman, B.W. Density Estimation for Statistics and Data Analysis; Chapman & Hall/CRC: London, UK, 2018. [Google Scholar]
- Väätäinen, K.; Asikainen, A.; Eronen, J. Improving the Logistics of Biofuel Reception at the Power Plant of Kuopio City. Int. J. For. Eng.
**2005**, 16, 51–64. [Google Scholar] [CrossRef] - Laguna, M.; Marti, R. Neural network prediction in a system for optimizing simulations. Iie Trans.
**2002**, 34, 273–282. [Google Scholar] [CrossRef] - Ljung, L. System Identification. In Wiley Encyclopedia of Electrical and Electronics Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2017; pp. 1–19. [Google Scholar] [CrossRef]
- Robinson, S. Simulation: The Practice of Model Development and Use; John Willey and Sons: Chichester, UK, 2004; Volume 1, p. 5. [Google Scholar]
- Jain, A.K.; Murty, M.N.; Flynn, P.J. Data clustering: A review. ACM Comput. Surv. (CSUR)
**1999**, 31, 264–323. [Google Scholar] [CrossRef] - Krishnapuram, R.; Freg, C.P. Fitting an unknown number of lines and planes to image data through compatible cluster merging. Patt. Recogn.
**1992**, 25, 385–400. [Google Scholar] [CrossRef] - Setnes, M.; Babuska, R.; Verbruggen, H.B. Rule-based modeling: Precision and transparency. IEEE Trans. Syst. Man Cybernet. Part C (Appl. Rev.)
**1998**, 28, 165–169. [Google Scholar] [CrossRef][Green Version]

**Figure 4.**Layout representing three different transport methods, going into an algorithm to produce a single entity out of them.

**Figure 5.**After an entity has been selected (see Figure 4), the remaining of the operation is to define its attribute (volume of logs), and an id tag to differentiate them throughout the simulation process.

**Figure 7.**Block scheme of controlling queue length of incoming entities and number of working servers.

**Figure 9.**Data fitting of intergeneration time (

**a**) and attribute for delivered log volume (

**b**). Where the grey area is a empirical data from company and bold line is showing fitting of the PDF.

**Figure 13.**Total numbers of generated and delivered entities based 100 simulation runs representing one year of the log yard’s operations using Model 1 (

**a**,

**b**) and Model 2 (

**c**,

**d**).

**Figure 14.**Number of waiting entities to be unloaded at Stage 1. The dark solid line represents results obtained using Model 1, while the grey line shows company data.

**Figure 15.**One example result of total inventory level of entities as predicted by model 1 and indicated by the company data.

Parameter | Value |
---|---|

Solver | Fixed Step Discrete |

Fixed Step Size (fundamental sample time), s | 300 |

Model 1 Total Simulated Time, s | 2.7388 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{7}$ |

${\eta}_{1}$ | 36 |

${\eta}_{2}$ | 132 |

${k}_{1}$ | 0.3680 |

${\sigma}_{1}$ | 0.0078 |

${\theta}_{1}$ | 0.00361 |

${\mu}_{2}$, m${}^{3}$ | 45.4302 |

${\sigma}_{2}$, m${}^{3}$ | 4.5819 |

${a}_{3}$, m${}^{3}$ | 1 |

${b}_{3}$, m${}^{3}$ | 30 |

${W}_{1}$ | 0.78 |

${W}_{2}$ | 0.22 |

${q}_{1}$, entities | 3 |

${q}_{2}$, entities | 5 |

${q}_{3}$, entities | >7 |

${\mu}_{3}$, sec | 319 |

${\sigma}_{3}$, sec | 75 |

${\mu}_{4}$, sec | 832.9049 |

${\sigma}_{4}$, sec | 249.8715 |

$IN{V}_{A}$, Storage A, m${}^{3}$ | 40,000 |

$IN{V}_{B}$, Storage B, m${}^{3}$ | 15,000 |

$IN{V}_{C}$, Storage C, m${}^{3}$ | 7000 |

$IN{V}_{D}$, Storage D, m${}^{3}$ | ∞ |

$\mu \left({\omega}_{1)}\right)$, % | 99.2 |

$\mu \left({\omega}_{2)}\right)$, % | 0.3 |

$\mu \left({\omega}_{3)}\right)$, % | 0.5 |

$\mu \left({\omega}_{4)}\right)$, % | 55 |

$\mu \left({\omega}_{5)}\right)$, % | 8 |

$\mu \left({\omega}_{6)}\right)$, % | 35 |

$\mu \left({\omega}_{7)}\right)$, % | 2 |

Parameter | Value |
---|---|

Solver | Fixed Step Discrete |

Fixed Step Size (fundamental sample time), s | 300 |

Total Model 2 Simulation Time, s | 1.8576 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{7}$ |

${\eta}_{1}$ | 36 |

${\eta}_{2}$ | 132 |

${k}_{2}$ | 0.4243 |

${\sigma}_{6}$ | 0.0076 |

${\theta}_{2}$ | 0.002 |

${a}_{2}$, m${}^{3}$ | 45.9942 |

${b}_{2}$, m${}^{3}$ | 9.65723 |

${a}_{3}$, m${}^{3}$ | 1 |

${b}_{3}$, m${}^{3}$ | 30 |

${a}_{4}$ | 0 |

${b}_{4}$ | 0.0007 |

${W}_{3}$ | 0.83 |

${W}_{4}$ | 0.17 |

${W}_{5}$ | 0.64 |

${W}_{6}$ | 0.36 |

${q}_{1}$, entities | 3 |

${q}_{2}$, entities | 5 |

${q}_{3}$, entities | >7 |

${\mu}_{3}$, sec | 319 |

${\sigma}_{3}$, sec | 75 |

${\mu}_{5}$, sec | 564.8994 |

${\sigma}_{5}$, sec | 169.4698 |

Main Storage, m${}^{3}$ | >62,000 |

**Table 3.**Total number of generated and delivered entities over one year period based on 100 simulation runs using models 1 and 2. * Confidence intervals based on the final values of the simulations.

Difference from Company Data, % | Standard Deviation | Standard Error | T-Score | Confidence Interval 5% * | ||||
---|---|---|---|---|---|---|---|---|

Lower | Upper | Lower | Upper | |||||

Generated Entities (INPUT) | ||||||||

Model 1 | 22,223 | 2.5 | 6332 | 42.4828 | −1.9601 | 1.9601 | 22,138 | 22,307 |

Model 2 | 21,738 | 0.2 | 6230 | 42.2557 | −1.9601 | 1.9601 | 21,654 | 21,822 |

Company Data | 21,685 | - | - | - | - | - | - | - |

Delivered Entities (OUTPUT) | ||||||||

Model 1 | 21,666 | 2.4 | 6216 | 42.2336 | −1.9601 | 1.9601 | 21,583 | 21,750 |

Model 2 | 21,742 | 2.7 | 6323 | 42.9702 | −1.9601 | 1.9601 | 21,659 | 21,825 |

Company Data | 21,167 | - | - | - | - | - | - | - |

Percentage of Unloaded Entities per Server Combination | ||||
---|---|---|---|---|

1 Server | 2 Servers | 3 Servers | 4 Servers | |

Model 1 | 71.0 | 2.5 | 0.5 | 26.0 |

Company Data | 71.0 | 1.5 | 0.5 | 27.0 |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Kons, K.; La Hera, P.; Bergström, D.
Modelling Dynamics of a Log-Yard through Discrete-Event Mathematics. *Forests* **2020**, *11*, 155.
https://doi.org/10.3390/f11020155

**AMA Style**

Kons K, La Hera P, Bergström D.
Modelling Dynamics of a Log-Yard through Discrete-Event Mathematics. *Forests*. 2020; 11(2):155.
https://doi.org/10.3390/f11020155

**Chicago/Turabian Style**

Kons, Kalvis, Pedro La Hera, and Dan Bergström.
2020. "Modelling Dynamics of a Log-Yard through Discrete-Event Mathematics" *Forests* 11, no. 2: 155.
https://doi.org/10.3390/f11020155