By C.A. Schneeweiss

The time period inventory-production thought isn't good outlined. It com prises e. g. such versions like funds stability types, construction smoothing types and natural stock types. we will right here ordinarily be anxious with stochastic dynamic difficulties and shall supply designated definitions within the subsequent part. such a lot of our paintings will pay attention to funds stability versions. even though, creation smoothing events and natural stock difficulties can also be investigated. because we're confronted in precept with dynamic stochastic situa tions a dynamic programming procedure will be applicable. This procedure, even if, because of computational restraints, is restricted to simply however the easiest types. for this reason, in perform, one ruduces stochastics simply in taking forecasts of call for after which treating the matter as a deterministic optimization challenge. additionally one frequently introduces sure defense shares to defend the process from attainable forecasting blunders. more often than not, this proce dure is suboptimal. even though, there exists one specific situa tion whilst a separation in a forecasting process and a subse quent optimization of the remainder deterministic version isn't suboptimal. this can be referred to as the linear-quadratic version, i. e. a version having linear procedure equations and a quadratic expense crite rion. For this sort of version H. A. Simon ~3J and later H. Theil [25J have proven that the above separation estate holds. in truth, Simon's and Theil's effects are not anything else yet what has later and extra in general turn into recognized to regulate engineers as Kalman's well-known separation principle.

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We shall now show that in cases when this assumption does not hold the effect on our results may in general be disregarded. Let us proceed as follows. First we derive an integral equation for the stationary probability distribution F(x) of inventory X assuming a general (non-Gaussian) demand sequence. This integral equation can in general not be solved analytically. consequently, we represent F(x) by a Gram Charlier expansion. It turns out that generally only the first terms involving mean and variance are of importance.

12). 6a), reduces to C* = (h+v) q,' (y*) a x *+ E±.!! 50) may be illustrated as follows Fig. 2) 50 The parameter ~:may be interpreted as an optimal "dynamic safety stock" [19J. 53) a+B Note that -1 ~ 1<'* ~ 0 In discussing the above results somewhat further let us first consider some special cases of cost parameters. e. the optimal safety inventory should be zero; a result which was reasonably to be expected. ~(~2~)~h~~v__~p__~q (3) p = q = 0 implies u~ ~ x k which reflects the complete summetry of the model.

Not all functions I(·) and P(·) will be admissible. Certain economically reasonable assumptions will be required. (See the derivation of Equs. 2 The general Solution Having stated and discussed the various assumptions of the general model we will be concerned with, let us now develop a general solution procedure. 39 Consider Equ. 3). 2) it follows from the assumption of {r k } being stationary and Gaussian that {uk} and {x k } are also stationary Gaussian processes. e. 1) it follows ~u E {uk} = o~).