SRINAGESH GAVIRNENI [1]
SRIDHAR TAYUR [2]
Received January 1999 and accepted January 2000
We present an efficient solution method -- Direct
Derivative Estimation (DDE) - for computing optimal order-up-to levels for a
discrete time non-stationary inventory control model. We generalize a number of
non-stationary inventory control models (which underlie larger models of supply
chains) that include forecast updates, seasonality, information sharing, and
exchange-rate fluctuations. This procedure is different from the existing ones
in that it computes, in a recursive manner,
the
derivative of the cost function and not the cost function itself. It can also
handle a much wider variety of fluctuations in the problem parameters. In our
computational testing it was found to be considerably faster than Dynamic
Programming and Infinitesimal Perturbation Analysis.
1. Introduction
Underlying many models dealing with supply chain
management issues is a basic discrete-time production-inventory model that
incorporates random demand in a non-stationary environment. These
non-stationaries arise from a variety of different causes: a perusal of recent
literature provides many reasons including: (i) changes in economic conditions
(Song and Zipkin, 1993); (ii) seasonal effects (Karlin, 1960a; Zipkin, 1939);
(iii) Bayesian updates of demand distributions (Aviv and Federgruen, 1998;
Lovejoy, 1990, 1992), (iv) information flow (Gavirneni et al., 1999), and (v)
exchange rate fluctuations (Scheller-Wolf and Tayur, 1997). It is well known
that, in many cases, the optimal policy is order-up-to (equivalently, base
stock), and the order-up-to level in a period depends on the state of the system
in that period; (Karlin, 1960b). For other cases the optimal policy is typically
very complex, and consequently one restricts analysis to the restrictive class
of order-up-to policies, or uses this class as a bou nd (see Aviv and Federgruen
(1998) for example). In this paper, we present an efficient solution procedure,
which we call the Direct Derivative Estimation (DDE) procedure, to find these
optimal order-up-to levels.
Existing solution procedures for non-stationary inventory
control are either very complex and computationally expensive or specific to the
setting they address and not easily generalized. Iglehart and Karlin (1962)
developed a solution procedure for a discrete time non-stationary inventory
control problem that involves solving systems of integral equations. This
procedure requires considerable computational effort. Song and Zipkin (1993)
present solution procedures for the continuous time non-stationary problem under
the additional assumption of Poisson demands. Karlin (1960a) and Zipkin (1989)
present solution procedures for the discrete time problem when the demands are
cyclic. These solution procedures are very specific to the non-stationarity for
which they were developed.
