**OPTIMAL PRODUCTION
MANAGEMENT WHEN DEMAND DEPENDS ON BUSINESS CYCLES **

Abel
Cadenillas, Peter Lakner,
Michael Pinedo

**Abstract **

In
this paper we assume that consumer demand for an item follows a Brownian motion
with a drift that is modulated by a continuous-time Markov chain that
rep-resents the regime of the economy. The economy may be in either one of two
regimes; one regime may represent a recession period with a low demand rate and
the other may represent an expansion period with a high demand rate. The
economy remains in one regime for a random amount of time that is exponentially
distributed with a given rate and then moves to the other regime and remains
there for an exponentially distributed amount of time with another rate. The
management of the company would like to maintain the inventory level of the
item as close as possible to a target inventory level and would also like to
produce the items at a rate that is as close as possible to a target production
rate. The company is penalized by the deviations from the target levels and the
objective is to minimize the long term total expected discounted penalty costs.
We consider two models. In the first model the management of the company
observes the regime of the economy at all times (case of full information),
whereas in the second model the regime of the economy is unobserved (case of
limited information). We solve both problems by applying the technique of
“completing squares” and obtain formulas for the optimal production policy as
well as for the minimal total expected discounted cost. Our analytical results
show, among various other results, that in both models the optimal production
policy depends on factors that are based on short term concerns as well as
factors that are based on long term concerns. We analyze how the impacts of
these factors depend on the values of the parameters in the model. We
furthermore compare the total expected discounted costs of the two models and
determine the value of information concerning the current regime of the
economy. Among other results we show that the difference of the costs in the
two models is proportional to the total discounted expected squared error of
estimate of the regime. We show that under a quadratic utility in both the full
and the limited information cases the optimal production rate is a linear
function of the current inventory level. In the full information case the
coefficients in this linear function depend on the current state of the
economy, and we present an explicit representation for the optimal production
rate and the minimal level of penalty cost. In the limited information case the
coefficients are the conditional expectations of the corresponding coefficients
in the full information case, given the available information at the time. We
compute a more explicit formula than just this conditional expectation, using
filtering theory applied to Hidden Markov Models. In this case our solution is
not as explicit as in the full information case as it is a function of the
solution of a two-dimensional linear stochastic differential equation (SDE).
Numerical solution of SDE's is not a trivial matter. However, we can replace
these SDE's by a system of linear homogeneous ordinary differential equations (ODE)
of the first order, and the numerical solution of such system of ODE's is
standard. There is an extensive literature on the theory of optimal production
control. In the continuous-time case, Bensoussan, Sethi, Vickson and Derzko (1984), Sethi and Thompson
(2000), and Khmelnitsky, Presman and Sethi (2010) consider the case when the demand is constant.
Fleming, Sethi and Soner
(1987) allow the demand to be stochastic, and model it as a continuous-time
Markov chain with a finite state space. While a continuous-time Markov chain
model for the demand is more realistic than a deterministic model, it still is
not ideal since it assumes that the demand can take only a finite number of
values, and that it remains constant for long periods of time. It does not appear
to be suitable for modeling the demand for commodity items such as wine, oil,
and electricity. This motivates us to study a production problem in which the
demand process changes continuously over time, and is allowed to take a
continuum of values. In the present model we allow the demand to depend on
macro-economic conditions that remain in effect for ex-tended periods of time.

Lecture
Notes in Management Science (2011) Vol. 3: 121-122

3rd
International Conference on Applied Operational Research, Proceedings

©
Tadbir Operational Research Group Ltd. All rights reserved.

www.tadbir.ca

ISSN
2008-0050 (Print)

ISSN
1927-0097 (Online)

**ARTICLE OUTLINE**

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**References **