Annals of Operations Research

, Volume 271, Issue 2, pp 405–427 | Cite as

Evaluating shortfall distributions in periodic inventory systems with stochastic endogenous demands and lead-times

  • Hernan CaceresEmail author
  • Dongchen Yu
  • Alexander Nikolaev
Original Research


This paper addresses a multi-period production/inventory problem with two suppliers, where demand sizes and supplier lead time are stochastic and correlated. A discrete time, single item inventory system is considered, where inventory levels are reviewed periodically and managed using a base-stock policy. At the end of each period, a replenishment order is placed, which enters a queue at the buffer stage and is consequently forwarded to the first available supplier. We present a mathematical model of this inventory system and determine optimal safety stock levels for it, in closed form, using matrix analytic techniques and the properties of phase type distributions. To account for the effect of order crossovers, which occur whenever replenishment orders do not arrive in the sequence in which they were placed, the inventory shortfall distribution is analyzed. Finally, a set of numerical experiments with a system with two suppliers is presented, where the proposed model is compared to other existing models.


Production planning Safety stocks Markov modeling Shortfall Phase type distribution 


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial EngineeringUniversidad Católica del NorteAntofagastaChile
  2. 2.Department of Industrial and Systems EngineeringUniversity at BuffaloBuffaloUSA

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