Logistics Network Design with Inventory
Stocking, Time-based Service
Allocation and Part Commonality
SCALE
Dissertation Support Award Finalist (First Prize)
Vishv Jeet and Erhan Kutanoglu1
Operations Research and Industrial Engineering Program
The
vjeet_a@mail.utexas.edu, erhank@mail.utexas.edu
We
investigate a multi-part, multi-product logistics network design problem with
inventory stocking for service parts logistics (SPL) systems. The problem is
designing a network of part stocking/service facilities, allocating individual
customer demands to the located facilities and determining part stocking levels
to be maintained at these facilities. Traditionally such problems are solved in
two stages sequentially. The first stage is to solve the location-allocation
problem with a strategic view, and then solve the inventory stocking problem on
open locations with a more tactical view. We present the integrated logistics
network design and inventory stocking problem motivated by challenges in
low-demand systems such as SPL. The interaction (or interdependence) between
network design and inventory decisions arises due to time-based service level
requirements that exist in such systems. In SPL, demands from customers are for
parts that are needed to replace the failed parts in the existing
machines/systems (called products in general) already being used by the
customers. An example of time-based product-oriented service level requirement
could be, “70% of demand for parts in a certain product need
to be satisfied from facilities that are within 4 hours of the demand points.”
Note that the service levels are defined for each product, but the inventory
stocking unit is parts. This leads to a two-dimensional service allocation problem: Allocation of a time-based service
level across facilities, and across parts. The service allocation is the prime
cause of interaction between network design and inventory stocking decisions.
Part commonalities within and across products further complicate the problem.
This along with the consideration of costs associated with locating facilities,
transporting parts from facilities to customers, and stocking inventories (with
inherent part commonality), constitute an extremely complex, intermingled
system of relationships and interactions of management decisions, all of which
should be considered simultaneously if optimization of overall SPL performance
with minimum costs is desired.
To
this end, we introduce a mathematical model2 (in the form of a mixed
integer nonlinear program) for the integrated problem described above. The
model exhibits nonlinearity for two reasons: (1) integration, i.e.,
simultaneously optimizing network design and inventory stocking decisions, and
(2) Poisson distribution of demand for parts, which introduces nonlinear fill
rate calculations. We approximate the exact fill rates using pre-computed
tables and linear interpolation between the table values using binary and
special ordered set (SOS) variables of type 2. This and additional integer
programming (IP) techniques make the model linear and amenable to
state-of-the-art IP solution techniques. However, solving this integrated model
(both stages simultaneously with linearization constraints, table lookups and
time based service level constraints) for any reasonable-size problem still
presents a formidable challenge. Hence, we propose a heuristic that decouples
the two sets of decisions. We further improve the heuristic with an iterative
improvement scheme, which finds good feasible solutions to even larger
problems.
Along
with showing the effects of considering inventory explicitly within the network
design model, we also quantify the effects of considering part commonality
while making the network design decisions. Our results from extensive
experiments with small and large problems show that while network is being
designed, not only considering stock levels and their fill rates but also
considering existing part commonalities is necessary for ultimate optimization
of the overall system costs. Finally, we show that making network design
decisions first and then deciding stock levels for the given network is just a
heuristic for the overall problem. That is, significant savings are possible if
we can find high quality solutions to the integrated model. Moreover,
exploiting part commonality not only improves the inventory decisions, but also
provides opportunities to design better networks with lower total costs.
We
are in the process of designing direct solution techniques that will treat the
integrated model directly by developing tight lower bounds, by investigating
the problem structure and substructures, and by developing feedback mechanisms
for the two-stage heuristic. In terms of modeling, two areas of future research
are (1) investigation of the multi-echelon systems where facilities at two or
more echelons are located, and their stock levels are decided, and (2)
investigation of inventory sharing across facilities through lateral
transshipments that are becoming more common in SPL systems.
1Ph.D. Dissertation Research Advisor. The research is supported in part by NSF CAREER Grant
DMI 0245123.
2A full version of the paper is
available at http://uts.cc.utexas.edu/~jeetv/spl_paper.pdf,
or send an email to the authors.