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The ‘Hungarian inventory control model’ was initiated by Prékopa [1965. Reliability equation for an inventory problem and its asymptotic solutions. In: Prékopa, A. (Ed.), Application of the Mathematics to Economics. Publication House of the Hungarian Academy of Science, pp. 317–327] and Ziermann [1964. Application of Smirnov's theorems for an inventory control problem. Publications of the Mathematical Institute of the Hungarian Academy of Science Series B 8, 509–518 (in Hungarian)], where the ordered amount is delivered in an interval, rather than at a time epoch according to some stochastic process and consumption takes place in the same interval. The problem is to determine the minimum level of initial safety stock that ensures continuous consumption, without disruption, in the whole time interval with a prescribed high probability. Prékopa [2006. On the Hungarian inventory control model. European Journal of Operational Research 171, 894–914] has formulated a two-stage model with such interval type processes and probabilistic constraints. In this paper we modify the assumptions of those models and formulate simpler, numerically more tractable models. We also present numerical examples.

The term ‘Hungarian inventory control model’ refers to a model system, where both the deliveries of the ordered amounts and consumption take place in an interval according to some random processes, rather than at one time epoch. The problem is to determine the minimum level of initial safety stock that ensures continuous consumption, without disruption, in the whole time interval with a prescribed high probability. The Hungarian inventory control model was initiated by Prékopa (1965) and Ziermann (1964). The original model is of a static and single item type, where the delivered and consumed amounts are assumed to be the same and the mathematical tool used to solve the problem comes from order statistics. In Prékopa (1965) already more general models have been presented and some theorems proved in Prékopa (1973a) have been used to numerically solve the problems. From the later literature in connection with the Hungarian inventory control models we mention the papers by Prékopa and Kelle (1978), Kelle (1984) and the summarizing paper of Prékopa (1980). Recently Prékopa (2006) has shown that interval type delivery and consumption processes can be combined with classical inventory models. The ‘order up to S’ model is taken as an example. He also presented dynamic type (two-stage) inventory models using interval type delivery and consumption stochastic processes which appear in the Hungarian models. However, the solutions of the obtained nonlinear programming problems are computationally intensive. They involve the solutions of nonlinear decomposition type problems, along with the calculation of the multivariate Dirichlet distribution function and gradient values. A normal approximation to the Dirichlet distribution alleviates the numerical difficulties but still further research is needed to come up with efficient numerical solutions for the problems. Those models are hybrid type stochastic programming models, i.e. both probabilistic constraints and penalties for unsatisfied demands are used. In the present paper we keep some of the main characteristics of the new models in Prékopa (2006) but introduce simpler, numerically more tractable formulations. We also discuss the numerical solution methods to our problems and present numerical examples. In Section 2 we recall some earlier results and develop mathematical tools for our model constructions. In Section 3 we formulate a probabilistically constrained multi-item inventory control model with interval type delivery and consumption processes. The consumption process is assumed to be linear with random, normally distributed slope while the expectation of total consumption minus the delivery process is approximated by a Brownian bridge. In Section 4 a two-stage model combined with probabilistic constraints is formulated. We assume that the consumption and delivery processes in connection with the different items are stochastically independent. Research is underway to take stochastic dependence into consideration. Finally, in Section 5 the computational aspects are discussed and numerical examples are presented.

In the Hungarian inventory control model both the deliveries of the ordered amounts and/or consumption take place in intervals, according to some random processes, rather than at discrete time epochs. Inventory control models of this type have been introduced by Prékopa (1965) and Ziermann (1964). It seems to us that in many practical applications one encounters similar situations (see Morris et al., 1987; Segal, 1997) and thus we think that the Hungarian inventory models deserve more attention than that has been paid to these models in the literature so far. The original Hungarian inventory control model is about to calculate an initial safety stock for one period with a prescribed service (reliability) level. No costs are taken into consideration. Since then, however, a few dynamic variants of it have appeared which, in addition, included cost parameters as well. In a recent paper by Prékopa (2006) a class of dynamic type Hungarian inventory models has been introduced where high service level is ensured by probabilistic constraint and various costs are taken into account. The problems in these models are, however, difficult to solve because we need the calculation of the multivariate Dirichlet type c.d.f values along with nonlinear programming algorithm. In this paper we keep some of the main characteristics of the new models in Prékopa (2006) but introduce simplifying assumptions: we assume that the consumption process is linear with random, normally distributed slope and the expectation of total consumption minus the delivery process is approximated by a Brownian bridge. The Brownian bridge is then transformed into a Brownian motion process, of which the probability distribution of the maximum functional is available in closed form (see Bachelier, 1990; Takács, 1967). In our setting this formula provides us with the c.d.f. of the supremum of consumption minus delivery process. First we formulated a static type probabilistically constrained multi-item inventory control model. Then we obtained two-stage type stochastic programming model which is supplemented by probabilistic constraints on the solvability of the second stage problem. The numerical solutions for our problems are convex nonlinear programming problems like in Prékopa (2006). However, under our simplifying assumptions the ‘Hungarian inventory control model’ becomes computationally tractable. We have used a MATLAB implementation of a nonlinear optimization algorithm based on the Sequential Quadratic Programming (SQP) method and we are able to efficiently solve problems of moderate-size. For example, it takes 15 min (wallclock time) to solve a problem with 30 items and 10 scenarios on a PC with Pentium 4 2.402.40 GHz. We have also applied sensitivity analysis by changing the parameters in the numerical examples and demonstrated that the results produced by the model are consistent with the expected behavior of an inventory control system.