OPTIMAL PRODUCTION MANAGEMENT WHEN DEMAND DEPENDS ON BUSINESS CYCLES
Abel Cadenillas, Peter Lakner, Michael Pinedo
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
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ISSN 2008-0050 (Print)
ISSN 1927-0097 (Online)