Annals of Operations Research

, Volume 277, Issue 1, pp 33–45 | Cite as

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

  • Cheng-Fu HuangEmail author
Reliability and Quality Management in Stochastic Systems


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.


Inventory Transfer center Stochastic delivery-flow distribution network (SDDN) Budget System reliability 

List of symbols


Set of nodes including vendors, transfer centers and the market


Set of routes connecting nodes

(N, A)

A stochastic delivery-flow distribution network (SDDN)


Number of routes


The ith route in (N, A), \(i = 1, 2, {\ldots }, n\)


Number of vendors


The eth vendor in (N, A), \(e = 1, 2, {\ldots }, r\)


number of transfer centers


The gth transfer center in (N, A), \(g = 1, 2, {\ldots }, o\)


The number of stocks in the transfer center, \(g = 1, 2, {\ldots }, o\)


The single market in (N, A)

\({\pi }_{i}\)

Number of states that route \(a_{i }\)owns


Capacity #hof route \(a_{i}\), \(h = 1, 2, {\ldots }, {\pi }_{i }\) and \(b_{i1} = 0\)


The maximal capacity vector: (\(b_{1\pi 1}\), \(b_{2\pi 2}\), ..., \(b_{n\pi n})\)


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


The total budget


The consumed capacity by per unit of commodity

(i, j)

(Source node, target node): a node pair


The demand for the market

E(i, j)

Set of MPs from node i to node j


Number of MPs


The kth MP, \(k = 1, 2, {\ldots }, m\)


Flow through \(P_{k}\), \(k = 1, 2, {\ldots }, m\)


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


  1. Chopra, S. (2003). Designing the distribution network in a supply chain. Transportation Research Part E: Logistics and Transportation Review, 39, 123–140.Google Scholar
  2. Ford, L. R., & Fulkerson, D. R. (1962). Flows in networks. New Jersey: Princeton University.Google Scholar
  3. Hsieh, C. C., & Lin, M. H. (2006). Simple algorithms for updating multi-resource allocations in an unreliable flow network. Computers and Industrial Engineering, 50, 120–129.CrossRefGoogle Scholar
  4. Hsu, C. I., Hung, S. F., & Li, H. C. (2007). Vehicle routing problem with time-windows for perishable food delivery. Journal of Food Engineering, 80, 465–475.CrossRefGoogle Scholar
  5. John, N. E., Etim, J. J., & Ime, T. U. (2015). Inventory management practices and operational performance of flour milling. International Journal of Supply and Operations Management, 1(4), 392–406.Google Scholar
  6. Kumar, S. K., & Tiwari, M. K. (2013). Supply chain system design integrated with risk pooling. Computers and Industrial Engineering, 64, 580–588.CrossRefGoogle Scholar
  7. Lin, Y. K. (2001). A simple algorithm for reliability evaluation of a stochastic-flow network with node failure. Computers and Operations Research, 28, 1277–1285.CrossRefGoogle Scholar
  8. Lin, Y. K., & Chang, P. C. (2012). Reliability evaluation for a manufacturing network with multiple production lines. Computers & Industrial Engineering, 63, 1209–1219.CrossRefGoogle Scholar
  9. Lin, Y. K., & Yeh, C. T. (2013). Determine the optimal carrier selection for a logistics network based on multi-commodity reliability criterion. International Journal of Systems Science, 44, 949–965.CrossRefGoogle Scholar
  10. Lin, Y. K., & Yeh, C. T. (2010). Optimal carrier selection based on network reliability criterion for stochastic logistics networks. International Journal of Production Economics, 128, 510–517.CrossRefGoogle Scholar
  11. Ozsen, L., Coullard, C. R., & Daskin, M. S. (2008). Capacitated warehouse location model with risk pooling. Naval Research Logistics, 55, 295–312.CrossRefGoogle Scholar
  12. Ramezani, M., Bashiri, M., & Tavakkoli-Moghaddam, R. (2013). A new multi-objective stochastic model for a forward/reverse logistic network design with responsiveness and quality level. Applied Mathematical Modelling, 37, 328–344.CrossRefGoogle Scholar
  13. Sarkar, B. (2013). A production-inventory model with probabilistic deterioration in two-echelon supply chain management. Applied Mathematical Modelling, 37, 3138–3151.CrossRefGoogle Scholar
  14. Sarkar, B., Mandal, B., & Sarkar, S. (2015). Quality improvement and backorder price discount under controllable lead time in an inventory model. Journal of Manufacturing Systems, 35, 26–36.CrossRefGoogle Scholar
  15. Shen, S. Y., & Honda, M. (2009). Incorporating lateral transfers of vehicles and inventory into an integrated replenishment and routing plan for a three-echelon supply chain. Computers and Industrial Engineering, 56, 754–775.CrossRefGoogle Scholar
  16. Sicilia, J., González-De-la-Rosa, M., Febles-Acosta, J., & Alcaide-López-de-Pablo, D. (2014). An inventory model for deteriorating items with shortages and time-varying demand. International Journal of Production Economics, 155, 155–162.CrossRefGoogle Scholar
  17. Wang, W., (2012). Safety inventory management: A system dynamics model. In Proceedings of IEEE International Conference on Cyber Technology in Automation, Control, and Intelligent Systems (pp. 762–764).Google Scholar
  18. Whicker, L., Bernon, M., Templar, S., & Mena, C. (2009). Understanding the relationships between time and cost to improve supply chain performance. International Journal of Production Economics, 121, 641–650.CrossRefGoogle Scholar
  19. Yang, C. T. (2014). An inventory model with both stock-dependent demand rate and stock-dependent holding cost rate. International Journal of Production Economics, 155, 214–221.CrossRefGoogle Scholar
  20. Yu, M., & Nagurney, A. (2013). Competitive food supply chain networks with application to fresh produce. European Journal of Operational Research, 224, 273–282.CrossRefGoogle Scholar
  21. Zuo, M. J., Tian, Z., & Huang, H. Z. (2007). An efficient method for reliability evaluation of multistate networks given all minimal path vectors. IIE Transactions, 39, 811–817.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Business AdministrationFeng Chia UniversityTaichungTaiwan, ROC

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