We provide a cost structure that can be used for
decentralized control of a multi-echelon inventory system with a central
warehouse and a number of retailers. This cost structure means that the
warehouse, in addition to its local costs, pays a penalty cost for a delay at
the warehouse to the retailer facing the delay. A basic assumption is that each
installation starts with an initial policy concerning e.g., inventory control
and transportation. The installations then play a Stackelberg game with the
warehouse being the leader.
By
minimizing its local costs according to the suggested cost structure, an
installation can reduce its costs. The total system costs are then reduced by
the same amount. No installation needs to face higher costs due to policy
changes at other installations, since the cost structure satisfies a rationality
constraint. If an installation applies its initial policy the local costs are
the same as in the initial state, even if the other installations change their
policies. If the game is played repeatedly the system will approach a Nash
equilibrium but not necessarily the centralized optimal solution. As an example
we consider a system with one-for-one ordering retailers.
1. Introduction
We consider a two-echelon inventory system consisting of
a central warehouse and a number of retailers. The final demand takes place at
the retailers, who replenish their stocks from the warehouse. The warehouse, in
turn, replenishes its stock from an outside supplier. The different
installations could be owned by the same organization or by different
organizations. The owner, or the different owners together, wish to improve the
performance of the whole supply chain, or in other words, reduce the total
system costs which can be divided into local warehouse costs and local retailer
costs.
The local costs at the warehouse include holding costs at
the warehouse and holding costs during the transportation from the outside
supplier to the warehouse. The warehouse is also normally responsible for
material handling costs at the warehouse and for the costs for transportation of
goods from the outside supplier to the warehouse.
Similarly, the local costs at a retailer include the
holding costs at the retailer and holding costs under transportation between the
warehouse and the retailer, material handling costs at the retailer, and costs
for transportation between the warehouse and the retailer. We shall also assume
that a retailer has some type of shortage costs associated with late deliveries
to its customers.
It is obvious that some of the decisions regarding the
considered supply chain can be completely decentralized to a certain site
because they concern exclusively a single installation and do not affect other
installations at all. Examples can be decisions concerning material handling.
There are also decisions, however, which will affect more than one installation.
Decisions regarding inventory control policies are obvious examples. It is
therefore not feasible to let each installation minimize its own local costs.
For example, the warehouse would then not like to carry any stock. However, if
the warehouse stock is too small, this will cause long delays for the retailer
orders due to stockouts at the warehouse and lead to increased total costs for
the supply chain. Some decisions concerning a site are mainly local but will
still have a small impact on other sites. Assume, for example, that the
warehouse considers a different type of transportation that is faster but more
expensive. This means that the transpo rtation costs will increase while the
holding costs under transportation will decrease. These effects have nothing to
do with the other sites. But the warehouse lead-time will become shorter and it
may also be necessary to adjust the ordering policy at the warehouse. This means
that the retailers will be affected. When considering changes that mainly have a
local impact it would to be highly attractive to be able to take the decisions
locally.
