Semi-Markovian capacities in production network models

Göttlich, Simone; Knapp, Stephan

doi:10.3934/dcdsb.2017090

Cited by 4 publications

(12 citation statements)

References 28 publications

(61 reference statements)

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“…In [17,23,24], stochastic effects are introduced into macroscopic production models, where externally given stochastic capacity functions to model machine failures (or capacity drops) are used. The randomness in capacities strongly influences the dynamics of the production and leads to interesting system behavior.…”

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confidence: 99%

Load-Dependent Machine Failures in Production Network Models

Göttlich¹,

Knapp²

2019

SIAM J. Appl. Math.

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In this paper, a production model based on (hyperbolic) differential equations with stochastic and load-dependent machine failures is introduced. We derive the model on the base of a well-established deterministic model and show its well-posedness. To do so, we make use of the theory of piecewise deterministic Markov processes and fuse it with the theory of the underlying deterministic production model. Finally, we compare the load-dependent model to the already established loadindependent model and highlight the new properties in numerical examples.

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confidence: 99%

Load-Dependent Machine Failures in Production Network Models

Göttlich¹,

Knapp²

2019

SIAM J. Appl. Math.

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“…Macroscopic production models have been widely studied in the literature, see [3] for an overview. Since in production capacity drops occur due to machine failures or human influences, deterministic models have been extended to stochastic production models, see [4,8,9,10]. Therein, a random flux function in the form of f (ρ) = min{vρ, µ} has been chosen with a deterministic production velocity v > 0, a stochastic capacity µ for a production density ρ.…”

Section: Applications and Numerical Resultsmentioning

confidence: 99%

“…The latter corresponds to the variable u in our context. In [4,9] the capacity µ is a Continuous Time Markov Chain, in [8] a semi-Markov process and in [10] a PDMP construction has been developed.…”

Section: Applications and Numerical Resultsmentioning

confidence: 99%

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Piecewise deterministic Markov processes driven by scalar conservation laws

Knapp¹

2019

Preprint

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We investigate piecewise deterministic Markov processes (PDMP), where the deterministic dynamics follows a scalar conservation law and random jumps in the system are characterized by changes in the flux function. We show under which assumptions we can guarantee the existence of a PDMP and conclude bounded variation estimates for sample paths. Finally, we apply this dynamics to a production and traffic model and use this framework to incorporate the well-known scattering of flux functions observed in data sets.

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“…This can be interpreted as follows: even if we have debts of −0.2283 · 10 3 , the probability to face bankruptcy at T = 365 is lower than 0.1. The more pessimistic risk measure, Average Value at Risk, leads to a best choice N 1 = 8 and N 2 = 10 with a AV@R 0.1 (Π(T )) of 0.6255 · 10 3 , i.e., we should have (10,12) a surplus of 0.6255 · 10 3 for these cluster sizes.…”

Section: Distribution Parameter Planning In the Load-dependent Casementioning

confidence: 99%

Uncertainty quantification with risk measures in production planning

Göttlich

Knapp

2020

J.Math.Industry

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This paper is concerned with a simulation study for a stochastic production network model, where the capacities of machines may change randomly. We introduce performance measures motivated by risk measures from finance leading to a simulation based optimization framework for the production planning. The same measures are used to investigate the scenario when capacities are related to workers that are randomly not available. This corresponds to the study of a workforce planning problem in an uncertain environment.Considering monetary quantities, appropriate performance measures have been introduced in the literature of finance [8] and are based on so-called risk measures. Famous examples are the expectation, Value at Risk and Average Value at Risk (also called conditional Value at Risk). They are originally introduced in the finance area to quantify financial risk. Since risk measures are fairly general, they can be also used in other contexts. In work [4], risk measures have been introduced for the optimization of oil production. Therein, the focus is the combination and comparison of risk measures in optimization formulations. However, our major goal is to focus on risk measures, their evaluation and impact for the stochastic production network model. Therefore, we examine how the distribution rates for an optimal routing should be chosen on the base of the introduced performance measures in a simulation based optimization. Furthermore, we introduce a setting, where the capacity is determined as a sum of individual capacities, which can be on or off, respectively. This allows for a numerical analysis of the maximal possible capacity, e.g. number of workers, with respect to the performance measures.This paper is organized as follows: in section 2 we introduce the stochastic production network model and its extensions. Section 3 is devoted to the performance measures. Numerical simulation results are studied in section 4, where mainly two cases are considered: the optimal routing and the workforce planning. Stochastic model equationsThe core model is a production network model consisting of a coupled system of partial and ordinary differential equations. We start with the introduction of the deterministic model and present then two possible stochastic extensions.

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Semi-Markovian capacities in production network models

Cited by 4 publications

References 28 publications

Load-Dependent Machine Failures in Production Network Models

Load-Dependent Machine Failures in Production Network Models

Piecewise deterministic Markov processes driven by scalar conservation laws

Uncertainty quantification with risk measures in production planning

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