Multi-location Inventory Models for Retail Supply Chain Management

Abstract

In this chapter, we provide a review of recent research on multi-location inventory systems that is related to retail supply chain management. Our review is restricted in scope in a number of ways. First, the focus is on papers that model multi-level inventory systems, since virtually all retail supply chains are multi-level. Second, attention is restricted to papers after 1993, and the reader is referred to the reviews in other papers for articles prior to 1993. Third, certain model formulations that are not typical of retail inventory management are excluded, such as serial systems, since they are not representative of typical retail chains, and are a special case of general multi-location multi-echelon systems. Also excluded are papers that assume deterministic demand, since demand uncertainty is a key aspect of most retail systems. Finally, the primary focus is on periodic review systems. We conclude with suggestions for future research in this area.

Appendix: Continuous Review Inventory Systems

Many of the results in this research area, particularly for centrally controlled continuous review systems, grew out of the METRIC approximation derived in the seminal work done by Sherbrooke (1968). Consider a one-warehouse multi-retailer system where inventory is managed using a one-for-one (S − 1, S) inventory policy. Further, let the demand distribution at each retailer i be independent and Poisson (λ_i). Then, it follows that the demand faced by the warehouse is Poisson (λ₀ = Σ_i=1..N λ_i ). Using Palm’s theorem, it then follows that the number of outstanding orders at the warehouse has a Poisson distribution with mean λ₀ L ₀, where L ₀ is the replenishment lead time at the warehouse. Then, for a given order up to level S ₀, expressions for expected backorders (B ₀), waiting time (W ₀) as well as inventory levels (I ₀) can be derived as follows:

$$ E\left({B}_0\right)={\displaystyle \sum_{j={S}_0+1}^{\infty}\left(j-{S}_0\right)\frac{{\left({\lambda}_0{L}_0\right)}^j}{j!} \exp \Big(-}{\lambda}_0{L}_0\Big), $$

$$ E\left({W}_0\right)=E\left({B}_0\right)/{\lambda}_0, $$

$$ E\left({I}_0\right)={\displaystyle \sum_{j=0}^{S_0-1}\left({S}_0-j\right)\frac{{\left({\lambda}_0{L}_0\right)}^j}{j!} \exp \Big(-}{\lambda}_0{L}_0\Big). $$

While the actual lead time is random, the average lead time for retailer orders now equals the shipping lead time plus the average delay time due to shortages at the warehouse. The problem is that the random replenishment lead times for retailers are not independent, since they all depend upon the inventory situation at the warehouse. The METRIC approximation ignores this correlation, and replaces the random lead time with its expected value. This allows results similar to the ones for the warehouse to be derived for the retailers as well. Thus, cost expressions can be derived and optimized.

Exact expressions can be obtained by characterizing the steady state distributions of inventory levels. While the previous papers focused on characterizing the distribution of the retailer lead times, an alternate approach was taken by Axsater (1990) to develop an exact evaluation methodology for the costs directly. In particular, he observed that any unit ordered by facility i will be used to fill the S _i -th unit of demand at this facility following that particular order, where S _i is the order up to level. Therefore, the distribution of the time elapsed between an order and the occurrence of the unit of demand that it will satisfy will have an Erlang (λ _i , S _i ) distribution, with the following density function:

$$ {g}_i^{S_i}(t)=\frac{{\left({\lambda}_i^{S_i}{t}^{S_i-1}\right)}^j}{\left({S}_i-1\right)!} \exp \left(-{\lambda}_it\right). $$

Now, conditioning on the delay at the warehouse (which also has an Erlang distribution similar to the one above), cost expressions for that unit can be derived (consisting of holding and backordering costs). Axsater derived a recursive procedure for evaluating the resulting costs. Thus, this method primarily focuses on keeping track of costs associated with arbitrary supply units.

Such procedures and results become ineffective when we consider general systems where one-for-one policies are replaced by batch ordering policies (R, Q) due to fixed ordering costs. In this case, the demand arising from retailers is no longer Poisson, but Erlang instead. Consequently, the demand process at the warehouse is the sum of N Erlang processes, which is more complicated to analyze.

This generalization is considered in Axsater (1993b), where the author considers a one warehouse multi-retailer inventory system, with N identical retailers facing independent Poisson demand. However, all locations are allowed to order in batches using a (R, Q) policy, and the policies at the warehouse are defined in terms of retailer batches. Lead times are assumed to be constant. Unmet demand is assumed to be backordered, and costs include proportional holding as well as backordering costs. The basic idea stems from a similar observation in Axsater (1990). In this case, a sub-batch ordered at the warehouse will fill the (R _w  + 1)th subsequent order for a retailer batch at the warehouse. Of course, this will happen after a random number of system demands. The costs are then derived by conditioning on which subsequent demand triggers an order. Exact as well as approximate evaluation procedures are derived.

Following a similar logic, in Axsater (1997), the results are further generalized to a two-level inventory system with one warehouse N retailers and constant lead times (transportation times), but where the retailers face different compound Poisson demand processes. All facilities apply continuous review echelon stock (R, Q) policies and backorder unmet demands. They provide a method for exact evaluation. Note however that echelon stock based policies may not always dominate installation stock based policies.

The third approach to solving such problems is based on characterizing the steady state distribution of inventory levels. For example, Graves (1985) fitted a two parameter Negative Binomial distribution to the number of outstanding orders for the basic METRIC model. In a similar manner, Chen and Zheng (1997) consider a one warehouse N retailer system where the retailers face different but independent compound Poisson demands, lead times are fixed, and orders are restricted to be batches of some specified lot size. They too assume installation stock based replenishment policies. For the case of simple Poisson demands, exact results are possible. The inventory level at the warehouse can be determined easily, since its echelon inventory position has a uniform distribution. The distribution of the inventory level at the retailer locations is more complicated, for which the authors determine an exact procedure. For the case of compound Poisson demand, approximate evaluation methods are derived.