Application of Semi-Markov Drift Processes to Logistical Systems Modeling and Optimization

Abstract

In our paper, a general scheme for modeling different logistical systems functioning under uncertainty is presented. It is defined in the terms of so-called semi-Markov drift processes with several continuous components (random walks in non-negative orthant of Eucleadean space). Some examples of this scheme application for modeling and optimization of logistical systems are given: optimal lot sizing taking into account the irregularity of product delivery from a vendor to wholesaler in the multi-echelon distribution system under fixed demand; optimal distribution of a manufactured product to a set of retailers under random demand.

Appendix: Definition of Semi-Markov Drift Process and Its Properties

Consider the stochastic process $(Y(t),\xi_{1} (t), \ldots ,\xi_{M} (t))$ with the phase space ${\varvec{\Omega}} = {\mathbf{D}} \times {\mathbf{R}}_{M}^{ + }$ where D is a finite set, ${\text{R}}_{M}^{ + }$ is the non-negative orthant of M-dimensional Euclidean space. Here Y(t) is the semi-Markov process with the phase space D and semi-Markov matrix $\left\| {\pi_{ij} A_{i} (t)} \right\|,i,j \in {\mathbf{D}},$ where $\pi_{ij}$ are transition probabilities of embedded Markov chain, $A_{i} (t)$ is d.f. of sojourn-time of Y(t) in the state $i \in {\mathbf{D}}.$ Let us assume that continuous components $\xi_{m} (t)$, m = 1,2,…, M satisfy the following differential equations (with probability 1):

$$\xi^{\prime}_{m} (t) = \sum\limits_{{i \in {\mathbf{D}}}} {v_{im} } I(Y(t) = i) - \sum\limits_{{i \in {\mathbf{D}}_{m}^{ - } }} {v_{im} } I(Y(t) = i,\xi_{m} (t) = 0),\quad t \ge 0,$$

(23)

where I(A) is the indicator of an event A; $v_{im} ,i \in {\mathbf{D}} ,$ are the given values (velocities); ${\mathbf{D}}_{m}^{ - } = \{ i :\;v_{im} < 0,i \in {\mathbf{D}}{\text{\} }} \ne \emptyset ,m = 1,2, \ldots, M.$ In accordance with Eq. (23) $\vec{\xi }(t) = (\xi_{1} (t), \ldots, \xi_{M} (t))$ is M-dimensional random walk in non-negative orthant ${\mathbf{R}}_{M}^{ + }$ with the sticky bounds. The stochastic process $(Y(t),\vec{\xi }(t))$ defined above is a kind of semi-Markov drift process (Postan 2006a). Let $\{ t_{n} \} ,n \ge 1$ be the sequence of moments, when Y(t) changes its state. Denote $\left\langle {\varPhi_{i} } \right.\left. {(\vec{x}),i \in {\text{D}},\vec{x} \in {\mathbf{R}}_{M}^{ + } } \right\rangle$ and $\left\langle {F_{i} } \right.(\vec{x}),i \in {\mathbf{D}},\vec{x} \in {\mathbf{R}}_{M}^{ + } \left. {} \right\rangle$ the stationary probabilistic measures of homogeneous Markov chain $(Y(t_{n} ),\vec{\xi }(t_{n} ))$ and process $(Y(t),\vec{\xi }(t))$ correspondingly. Taking into account Eq. (23), it may be proven (Postan 2006a) that functions $\varPhi_{i} (\vec{x})$ satisfy the following system of convolution type integral equations

$$\varPhi_{i} (\vec{x}) = \sum\limits_{{k \in {\mathbf{D}}}} {\pi_{ki} } \int\limits_{0}^{{T_{k} (\vec{x})}} {\varPhi_{k} } (\vec{x} - \vec{v}_{i} \tau ){\text{d}}A_{k} (\tau ),\quad i \in {\mathbf{D}},\quad \vec{x} \in {\mathbf{R}}_{M}^{ + } ,$$

(24)

where $T_{k} (\vec{x}) = \min_{1 \le m \le M} (x_{m} /v_{km}^{ + } ),v_{km}^{ + } = \hbox{max} (0,v_{km} ).$ From the semi-Markov processes theory of (Korolyuk and Turbin 1976; Gnedenko and Kovalenko 2005), it follows that

$$F_{i} (\vec{x}) = \sigma \int\limits_{0}^{{T_{i} (\vec{x})}} {(1 - A_{i} (\tau ))\varPhi_{i} } (\vec{x} - \vec{v}_{i} \tau ){\text{d}}\tau ,\quad i \in {\mathbf{D}},\quad \vec{x} \in {\mathbf{R}}_{M}^{ + } ,$$

(25)

where $\sigma = \left( {\sum\nolimits_{{i \in {\mathbf{D}}}} {\alpha_{i} } p_{i}^{*} } \right)^{ - 1} ;\alpha_{i} = \int\nolimits_{0}^{\infty } {(1 - A_{i} (\tau )){\text{d}}\tau < \infty } ;\;{\text{and}}\;p_{i}^{*} = \varPhi_{i} (\infty ),i \in {\text{D}},$ are the stationary probabilities of the Markov chain $Y(t_{n} ).$