On the Benefits of Risk Pooling in Inventory Management
Oded Berman†, Dmitry Krass‡, M. Mahdi Tajbakhsh§
† Joseph L. Rotman School of Management, University of Toronto
Tel: +1-416-978-4239; E-mail: berman@rotman.utoronto.ca
‡ Joseph L. Rotman School of Management, University of Toronto
Tel: +1-416-978-7180; E-mail: krass@rotman.utoronto.ca
§ Department of Industrial Engineering, Dalhousie University
Tel: +1-902-494-6173; E-mail: tajbakhsh@dal.ca
Extended Abstract
Inventory pooling which refers to the consolidation of multiple inventory locations into a
single location is generally believed to be beneficial as the centralized location faces less
demand variability than the individual locations [Eppen, 1979; Schwarz, 1981; Chen and
Lin, 1989; Cherikh, 2000].
The results on the benefits of inventory pooling are mostly based on the assumption that
the demand distributions of the individual locations are independent and Normally
distributed. However, when the variability is high it may lead to a significant probability
of negative demands. Silver et al. (1998) introduced a rule of thumb: when the coefficient
of variation of the demand is greater than 0.5, the Normal distribution should not be used.
Several recent publications have cast doubt on the benefits of inventory pooling by
showing that, in contrast to the results for the Normal distribution, the savings due to
pooling may approach zero as the demand variability increases. Gerchak and He (2003)
provide an example where increased variability of the individual demands reduces the
risk pooling benefits. Benjaafar et al. (2005) show that in the production-inventory
systems, inventory pooling leads to little benefit if the variability of the location demands
is very high. They argue that the inconsistency between their results and those obtained
for pure inventory systems is partly due to the assumption of independent demands (or
lead-time demands) in the inventory models. However as we demonstrate in this paper
even when the demands are independent there are vanishing benefits of inventory
pooling. This paradox about the true benefits of risk pooling is investigated in the current
paper.
We consider risk pooling in a multi-location newsboy framework consisting of n identical
locations, facing independent and identically distributed demands. We classify demand
distributions commonly used in the literature into two categories. Category I demand
distributions that allow negative demand values and include the Normal, Uniform,
Laplace and Logistic distributions. The optimal newsboy cost under these distributions
depends only on the standard deviation of the demand. Category II demand distributions
are nonnegative and include the Lognormal, Gamma, and Weibull distributions. The
optimal newsboy cost under these distributions depends on both the mean and the
standard deviation of the demand. We note that Category I distributions can be used only
when the demand variability is low.
Two measures of inventory pooling savings are utilized for the analysis: the absolute
saving (the reduction in the cost of operating the system due to inventory pooling) and
the relative saving (relative to the non-pooled system). We analyze how the benefits of
pooling change with the increase in the variability of demand (at each location) and the
number of locations.
Our main results are:
 We show both analytically (where possible) and numerically (through MonteCarlo simulations), that the absolute benefit of inventory pooling increases with
variability, and the relative benefit stays fairly constant, as long as the demand
variability stays in the low range. However, this behavior changes drastically at
higher variability levels. For all Category II distributions the benefits of pooling
in terms of both measures of savings decrease rapidly once the coefficient of
variation of demand (at each location) exceeds a certain threshold, and disappear
entirely as the variability continues to grow. With respect to the number of
locations being pooled, under a given level of variability the benefits of pooling
do increase with the number of locations. However, under high-variability
conditions, a certain minimum number of locations may need to be pooled before
any appreciable benefits of pooling can be realized.
 These effects are due to the different operating regimes exhibited by the system
under different levels of variability: as the variability is increased, the system
switches from the normal operation to the effective shutdown (the system avoids
taking the risk of too much inventories) and then to the complete shutdown (the
system holds no inventory and accepts the full costs of unmet demand). The
decrease in the benefits of inventory pooling is associated with the two latter
regimes. Pooling allows the system to remain in the normal operation regime
under higher levels of variability compared to the non-pooled system. Thus,
inventory pooling is, indeed, beneficial under all variability conditions.
 We analyze the behavior of the inventory pooling benefits using the distribution
free approximation in which only the first two moments of the demand
distribution are known [e.g., see Gallego and Moon (1993)]. This is very valuable
since, in most cases, the distribution of demand cannot be derived in closed form.
We observed that the behavior of savings due to pooling using the distribution
free approximation is similar to the behavior observed for other distributions in
our numerical tests.
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