Contracting in manufacturing networks with inventories

An important issue in the management of supply chains and manufacturing systems is to achieve the
desired customer service level while respecting the individual economic objectives of the partners.
These objectives are generally antagonistic and can lead to contradictory choices in the context of
a network with a high degree of local decisional autonomy. The challenge involved is to determine
a coordination mechanism that avoids global loss of economic efficiency and to solve nonlinear
optimization problems that capture the key dynamics of a complex production/inventory system.
This study investigates a two-stage serial supply chain with one manufacturer and one supplier.
The manufacturer, producing goods to serve a Poisson demand process, procures a key component
from his supplier. At each production facility, the successive processing times of the units are
independent exponential random variables. In the decentralized setting considered, each firm
manages the local production/inventory control system while attempting to maximize his own
steady state expected profit. The main operational decision at each stage is the level of the
inventory control parameter. In the game theory framework, the partners play a two-stage game of
the Stackelberg type. The manufacturer leads the game by setting the component purchasing price
as a function of the observed lateness of delivery. In the considered make-to-stock queuing system,
the production facility of the supplier behaves as an M/M/1 queue. However, the departure of
units from the finished-goods inventory of the supplier is not a Poisson process. In order to derive
closed form results, the method of Lee and Zipkin (1992) is used to approximate the limiting
probabilities of the second stage production process. The supplier’s profit function is concave
and the optimal action of the supplier can be correctly anticipated by the manufacturer. The
manufacturer’s constrained optimization problem consists of determining the optimal values of
the contract parameters and the second stage base-stock level. It is shown that using the defined
contracting scheme, the system optimal solution is obtained as the Stackelberg equilibrium.