Sira Allende Alonso
Universidad de La Habana

Carlos Bouza Herrera
Universidad de La Habana

Abstract The determination of the optimum lot size is a stochastic problem because of the randomness of the demands. The usual approaches consider that the involved distributions are known. We consider the case in which they are unknown. The optimization problem is probabilistic constraint program. The demands are modeled by an autoregressive process and the needed quantiles are derived. The newsboy problem is revisited using the derived results. Resumen La determinación del tamaño óptimo del lote es un problema estocástico dada la aleatoriedad de las demandas. Usualmente se consideran conocidas las funciones de distribución que entran en la modelación. Consideramos el caso en que ellas son desconocidas. El. problema de optimización es uno programa con restricciones probabilísticas. Utilizando los resultados obtenidos el problema del newsboy es reanalizado.

1.- INTRODUCTION    In many applications we need to determine the optimum lot size [ols]. Large inventories determine increases in the management costs. Therefore the problem to be solved is an ols one. When the demands are considered random the involved optimization problem is stochastic. Lasserre-Bes-Roubellat [1985] studied it considering that the distribution is known. The problem is to determine the lot size during k periods. Assuming the standardization of the involved variables they denoted, for a fixed period t, the inventory level [lot size] by the corresponding cost function. Similarly is the production level [control variable] and the corresponding cost function. The demand is a random variable . They assumed that the demands are iid and that the system is an open loop. The program is:    [1.2] is the evolution function of the Stochastic Dynamic Model P1. The constraints [1.4] and [1.5] establish that it is a probabilistic constrained problem. A deterministic equivalent program is:       is the quantile of order p of the known distribution function F. Suppose that it is unknown and that the DM is able to fix F*={F1,...,Fn,.... } as the family of distribution functions where F belongs. Then we need to estimate M. If is an estimate of M derived from a sample the approximation error [AE] is a non decreasing function. Using this assumptions Allende-Bouza [1998] derived the deterministic equivalent of P1 and the convergence of was obtained using results of Birgé [1991].    In this paper we drop the hypothesis that the demands are i.i.d and assume that the sums of the demands conform a linear autoregressive process. An approximation by Edgeworth Series [ES] is used for deriving an approximation to the involved quantiles. A discretization of the interval where they supposedly belong permit to compute a solution using scenario analysis. The accuracy is related with the interval´s width. The newsboy problem [NP] is studied as a particular case: a single item inventory problem with k-periods. 2.- MAIN RESULTS    Consider that the demand at moment j+1 is modeled by the linear autoregressive process where is an idd sequence of random errors with null expectation . It reflects the responses and the model establishes that the demand at time t is related with at most I previous periods. The are unknown parameters such that: belongs to the zero unit circle. The counterpart of [1.4] is:       The following proposition establishes the conditions needed for using the normal distribution for obtaining adequate approximations for the unknown quantiles.    Proposition 2.1. Take U* as a class of Borel sets of R such that for and for some a>0       Proof:    The sequence is Markovian, see Friedst and Gray (1997), and only an initial distribution is admitted by it because all the zeros of [2.1] belong to Z*. Hence is a stationary ergodic [SE] Markov process resulting that is a SE sequence. From the boundness conditions of the expected value of some absolute values of and the Cramer´s condition fixed by 3) the hypothesis of the example (1.1) of Götze-Hipp (1983) hold. Therefore the Edgeworth expansion for is valid, see Bhattacharya (1987).    It is clear that this constraint set is more restrictive than its counterpart in the original problem. Because of the convergence of the Cornish-Fisher expansion we expect that the solution of this problem be close to the real one.    The DM fixes the intervals These intervals can be discretized and a smaller number of quantiles needs to be computed. The nature of this approach suggests that we can use scenario analysis, within the theoretical frame as proposed by Rockafellar-Wets (1991) for example. For the obtention of a ´well hedged´ solution to the underlying problem. 3.- AN APPLICATION: THE RISK AVERSE NEWSBOY PROBLEM.    The newsboy problem is a single-item inventory problem. We can model this problem for k periods. The original problem deals with the determination of the number of newspapers to buy. If he buys a small quantity a profit is missed out. When the quantity is too large a penalty is charged. The newsboy may want to maximize the expected profit. This model fit many economic problems. See Dohoi-Watanabe-Osaki (1994) and Eechoudt-Gollier-Schlesinger (1995) for a detailed discussion.    The risk averse newsboy problem is a single item is considered in each period. is the amount and the demand of the items at a fixed period. is the production. The sale is is ordered and delivered at a cost C per unit and the selling price is R. If the seller sells at a price V at the final of the period. We assume that R>C>V>0 and V´ is the shortage penalty per unit.    An important feature is that is an unimodal function of with only one solution if the newsboy is risk averse. See Dohoi-Watanabe-Osaki (1994) for examples. Take:   Then the approach proposed in Section 2 may be used by the seller for fixing an optimal strategy for k periods. References Allende, S. and Bouza C. (1998): Minimum lot size determination under unknown distribution of the demands. Read at the International Congress of Mathematics, Berlin. Bhattacharya, R. N. (1987): Some aspects of Edgeworth expansions in statistics and probability. In "New perspectives in Theoretical and Applied Statistics". Wiley, N. York. Dohoi T., Watanabe, A. and Osaki S. (1994): A note on the risk averse newsboy. Recherche Opèrationalle, 28, 198-202. Eechoudt, L., Gollier, C. and Schelesinger, H. (1995): The risk averse (and prudent) newsboy. Managament Sc. 41, 786-94. Friesdt, B. and Gray, L. (1997): A Modern Approach to Probability Theory. Birkhäuser, Boston. Götze, F.N. and Hipp, C. (1983): Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrsch. Verw. Gebeite. 64, 211-239. Laserre, J. B. , Bess, C. and Roubellat, F. (1985):The stochastic discrete dynamic lot size problem: an open loop solution . Operations Res. 33, 684-689. Rockafellar, R. and Wets, R.J.B. (1991): Scenario and policy aggregation in optimization under uncertainty. Mathematics of Operations Res. 16, 139-147. . About the Authors: Autor: Sira Allende Alonso Dirección: Universidad de La Habana. San Lázaro y L. Vedado, La Habana. CP 10400 Correo electrónico: sira@matcom.uh.cu Autor: Carlos Bouza Herrera Dirección: Universidad de La Habana. San Lázaro y L. Vedado, La Habana. CP 10400 Correo electrónico: bouza@matcom.uh.cu

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