# Evaluation of system reliability for a stochastic delivery-flow distribution network with inventory

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## Abstract

The primary concern of managers in logistics management is how market demand can be satisfied. The inventory in the transfer center of a distribution network plays a key role in meeting customer demands. This study focuses on evaluating the system reliability of a stochastic delivery-flow distribution network (SDDN) with inventory. An SDDN is composed of nodes and routes, where each node denotes a vendor, transfer center, or market and each route connects a pair of nodes. Along each route, there is a carrier whose available capacity is stochastic. Different from previous issues regarding system reliability, this study does not consider the vendor (source) and market (sink) but also the amount of stocks in transfer center is included. If the market demand cannot be met by inventory in transfer center, then vendors must meet the remaining demand to satisfy the market. Therefore an algorithm is developed in terms of minimal paths to evaluate system reliability, which is defined as the probability that the SDDN with stocks can satisfy the market demand from multiple vendors and transfer centers to the market under budget constraints. A numerical example is given to illustrate this proposed algorithm.

## Keywords

Inventory Transfer center Stochastic delivery-flow distribution network (SDDN) Budget System reliability## List of symbols

**N**Set of nodes including vendors, transfer centers and the market

**A**Set of routes connecting nodes

- (
**N**,**A**) A stochastic delivery-flow distribution network (SDDN)

*n*Number of routes

- \(a_{i}\)
The

*i*th route in (**N**,**A**), \(i = 1, 2, {\ldots }, n\)*r*Number of vendors

- \(v_{e}\)
The

*e*th vendor in (**N**,**A**), \(e = 1, 2, {\ldots }, r\)*o*number of transfer centers

- \(t_{g}\)
The

*g*th transfer center in (**N**,**A**), \(g = 1, 2, {\ldots }, o\)- \(z_{g}\)
The number of stocks in the transfer center, \(g = 1, 2, {\ldots }, o\)

*T*The single market in (

**N**,**A**)- \({\pi }_{i}\)
Number of states that route \(a_{i }\)owns

- \(b_{ih}\)
Capacity #

*h*of route \(a_{i}\), \(h = 1, 2, {\ldots }, {\pi }_{i }\) and \(b_{i1} = 0\)*M*The maximal capacity vector: (\(b_{1\pi 1}\), \(b_{2\pi 2}\), ..., \(b_{n\pi n})\)

- \(c_{i}\)
Cost per unit of the consumed capacity through \(a_{i}\), \(i = 1, 2, {\ldots }, n\)

*B*The total budget

*w*The consumed capacity by per unit of commodity

- (
*i*,*j*) (Source node, target node): a node pair

*d*The demand for the market

- E(
*i*,*j*) Set of MPs from node

*i*to node*j**m*Number of MPs

- \(P_{k}\)
The

*k*th MP, \(k = 1, 2, {\ldots }, m\)- \(f_{k}\)
Flow through \(P_{k}\), \(k = 1, 2, {\ldots }, m\)

*F*(\(f_{1}, f_{2}, {\ldots }, f_{m})\): flow pattern

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