# Supply Sequence Modelling Using Hidden Markov Models

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- (1)
- To propose a method for forecasting the level of subsequent supplies of raw material for production using hidden Markov models.
- (2)
- To identify the hidden states fulfilling the Markov property which describe the dynamics of raw material supplies.
- (3)
- To present a method of modeling using stochastic processes that can be used primarily in relation to irregular deliveries with high randomness.

## 3. Description of the Case Study Company

## 4. Materials and Methods

#### 4.1. Markov Chains

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 4.2. Hidden Markov Models

- $S=\left\{{S}_{1},\dots ,{S}_{m}\right\}\u2014$a set of hidden states;
- $V=\left\{{V}_{1},\dots ,{V}_{h}\right\}\u2014$a set of outputs (set of observed states);
- $\pi =\left({\pi}_{1},{\pi}_{2},\dots ,{\pi}_{m}\right)\u2014$initial distribution, where ${\pi}_{i}\stackrel{def}{=}{\pi}_{{S}_{i}}=P\left({X}_{0}={S}_{i}\right)\ge 0$;
- $P={\left[{p}_{ij}\right]}_{1\le i,j\le m}\u2014$transition probability matrix between the hidden states, where for any $n\in N$ the value ${p}_{i}\stackrel{def}{=}{p}_{{S}_{1}{S}_{j}}=P\left({X}_{n+}={S}_{j}|{X}_{n}={S}_{i}\right)$ denotes the probability of transition from state ${S}_{i}$ at time $n\in N$ to state ${S}_{j}$ at time $n+1$;
- $G={\left[{g}_{ij}\right]}_{1\le i\le m,1\le j\le h}\u2014$output probability matrix, where for any $n\in N$ quantity ${g}_{ij}\stackrel{def}{=}{g}_{{S}_{i}{V}_{j}}=P\left({O}_{n}={V}_{j}|{X}_{n}={S}_{i}\right)$ is the probability of observing the state ${V}_{j}$ at time $n$, provided that the hidden part of the system is in the state ${S}_{j}$.

#### 4.3. The problem of Identifying Hidden Markov Models

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

## 5. Hidden Markov Models in Forecasting the Supply Sequence

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Condition | Range of Supply Volumes [Ton] |
---|---|

O1 | [0, 27] |

O2 | (27, 37] |

O3 | (37, 43] |

O4 | (43, 50] |

O5 | Over 50 |

O1 | O2 | O3 | O4 | O5 | |
---|---|---|---|---|---|

S1 | 0.906 | 0.070 | 0.000 | 0.000 | 0.024 |

S2 | 0.054 | 0.161 | 0.370 | 0.143 | 0.272 |

S3 | 0.139 | 0.410 | 0.311 0.141 | 0.000 | S4 |

S4 | 0.042 | 0.000 | 0.104 | 0.129 | 0.725 |

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**MDPI and ACS Style**

Borucka, A.; Kozłowski, E.; Parczewski, R.; Antosz, K.; Gil, L.; Pieniak, D. Supply Sequence Modelling Using Hidden Markov Models. *Appl. Sci.* **2023**, *13*, 231.
https://doi.org/10.3390/app13010231

**AMA Style**

Borucka A, Kozłowski E, Parczewski R, Antosz K, Gil L, Pieniak D. Supply Sequence Modelling Using Hidden Markov Models. *Applied Sciences*. 2023; 13(1):231.
https://doi.org/10.3390/app13010231

**Chicago/Turabian Style**

Borucka, Anna, Edward Kozłowski, Rafał Parczewski, Katarzyna Antosz, Leszek Gil, and Daniel Pieniak. 2023. "Supply Sequence Modelling Using Hidden Markov Models" *Applied Sciences* 13, no. 1: 231.
https://doi.org/10.3390/app13010231