Dept. of Industrial and Systems Engineering Lehigh University Bethlehem, PA, USA
Zuo-Jun Max Shen
Dept. of Industrial Engineering and Operations Research University of California Berkeley, CA, USA
May 27, 2008
ABSTRACT We investigate optimal system design in a One-Warehouse Multiple-Retailer system in which supply is subject to disruptions. We examine the expected costs and cost variances of the system in both a centralized and a decentralized inventory system. We show that using a decentralized inventory design reduces cost variance through the risk-diversiﬁcation eﬀect, and that when demand is deterministic and supply may be disrupted, a decentralized inventory system is optimal. This is in contrast to the classical result that when supply is deterministic and demand is stochastic, centralization is optimal due to the risk-pooling eﬀect. When both supply may be disrupted and demand is stochastic, we demonstrate that a risk-averse ﬁrm should typically choose a decentralized inventory system design.
As supply chains expand globally, supply risk increases. Classical inventory models have generally focused on demand uncertainty and established best practices to mitigate demand risk. However, supply risk can have very diﬀerent impacts on the optimal inventory management policies and can even reverse what is known about best practices for system design. In this paper, we focus on the impact of supply uncertainty on the One-Warehouse MultipleRetailer (OWMR) system, and compare two policies: centralization (stocking inventory at the warehouse only) and decentralization (stocking inventory at the retailers only). While most research 1
on the OWMR model allows inventory to be held at both echelons, we allow inventory to be held at only one echelon in order to consider two opposing eﬀects that can occur: risk pooling and risk diversiﬁcation. The risk pooling eﬀect occurs when inventory is held at a central location, which allows the demand variance at each retailer to be combined, resulting in a lower expected cost . The risk diversiﬁcation eﬀect occurs when inventory is held at a decentralized set of locations, which allows the impact of each disruption to be reduced, resulting in a lower cost variance . Whereas the risk-pooling eﬀect reduces the expected cost but (as we prove) not the cost variance, the risk-diversiﬁcation eﬀect reduces the variance of cost but not the expected cost. We prove that the risk diversiﬁcation eﬀect occurs in systems with supply disruptions. We also consider systems with both supply and demand uncertainty, in which both risk pooling and risk diversiﬁcation have some impact, and numerically examine the tradeoﬀ between the two. We employ a risk-averse objective to determine which eﬀect dominates the system and drives the choice for optimal inventory system design. Speciﬁcally, comparing centralized and decentralized inventory policies, we contribute the following: • The exact relationship between optimal costs and inventory levels when demand is deterministic and supply may be disrupted • The exact relationship between optimal cost variances when: – demand is deterministic and supply may be disrupted – supply is deterministic and demand is stochastic • Formulations of the expected cost and cost variance when supply is disrupted and demand is stochastic • Evidence that decentralization is usually optimal under risk-averse objectives The remainder of the paper is organized as follows. In Section 2 we review the relevant literature. In Section 3 we analyze the risk-diversiﬁcation eﬀect in the OWMR system with deterministic demand and disrupted supply. We consider stochastic demand in Section 4. In Section 5 we consider both demand uncertainty and disrupted supply and again compare...