The Q(s,S) control policy for the joint replenishment problem extended to the case of correlation among item-demands

Abstract

We develop an algorithm to compute an optimal Q(s,S) policy for the joint replenishment problem when demands follow a compound correlated Poisson process. It is a non-trivial generalization of the work by Nielsen and Larsen (2005). We make some numerical analyses on two-item problems where we compare the optimal Q(s,S) policy to the optimal uncoordinated (s,S) policies. The results indicate that the more negative the correlation the less advantageous it is to coordinate. Therefore, in some cases the degree of correlation determines whether to apply the coordinated Q(s,S) policy or the uncoordinated (s,S) policies. Finally, we compare the Q(s,S) policy and the closely connected P(s,S) policy. Here we explain why the Q(s,S) policy is a better choice if itemdemands are correlated.

A Stochastic Joint Replenishment Problem with Dissimilar Items

“…Though the Q(s, S) policy also sends out an order when the cumulative demand reaches Q, it only replenishes an item if its inventory position is less than, or equal to, its reorder point. Nielsen and Larsen (2005) and Larsen (2009) work out an analytical solution procedure for the Q(s, S) policy. Mustafa Tanrikulu, Ş en, and Alp (2010) relax the Q(s, S) policy through dropping S. They propose the (s, Q) policy, by which an order with size Q is triggered whenever any item's inventory position falls to its reorder point.…”

“…It is not hard to show stochastically increasing and linear continuous time demand mentioned by Rao (2003), such as compound Poisson or Brownian motion with drift, could be discretized to discrete time demand. Some recent papers consider demand correlations among different items, for instance, Larsen (2009) and Feng, Wu, Muthuraman, and Deshpande (2015).…”

A systematic literature review on the joint replenishment problem solutions: 2006-2015

“…In this case, the demand was considered as a compound Poisson distribution for each type of product to be replenished in the same order. In addition, decomposition methods were also present, in which the JRP was decomposed in several single-item problems one for each type of product, as in Huang & Chen (2007), Kayiş et al (2008), andLarsen (2009), and the solution comprehended the merge of each sub-solution obtained. Huang & Chen (2007) 1 , Eynan & Kropp (2007) 1 , Moon et al (2008) 1,2 , Yao (2010) 1 , Narayanan & Robinson (2010) 2 , Roushdy et al (2011) 1 , Karalli & Flowers (2011) …”

Constrained Stochastic Joint Replenishment Problem with Option Contracts in Spare Parts Remanufacturing Supply Chain

“…In the stochastic review period they found (s, S) policy for a fixed order size Q, and the solution of sub-problems could be obtained by a heuristic algorithm [14]. Larsen proposed algorithms to solve the JRP based on optimal Q(s, S) policy when the demands of items followed a compound correlated Poisson process [15]. Mustafa Tanrikulu et al proposed a new optimal replenishment policy; namely, (s, Q) policy.…”