System and method for calculation of controlling parameters for a computer based inventory management

A system designed for the calculation of control parameters for a computer based inventory management system according to the present invention comprises a computer program based on mathematical probabilities using statistical distribution functions. In the context of the invention a computer based inventory management system comprises an inventory management program system such as TRITON® as supplied by Baan BV, The Netherlands, running on a computer consisting of at least one processor, memory, input and output.

DETAILED DESCRIPTION OF THE INVENTION Using the values of the parameters calculated by the control system according to the invention, allows the inventory management system to optimize the service levels while minimizing the stock inventory and avoiding excess stock. The logistical parameters calculated by the system are: For the Q-system (R. H. Ballou, Business Logistics Management, Prentice Hall Inc., 1992) the reorder point and the reorder quantity. For the P-system (R. H. Ballou) or fixed order cycle method, the order to level or maximum stock level MSL and the periodic review period Tp. For both Q-system and P-system the break quantity or exceptional demanded quantity threshold and lead times are calculated as well. Currently, the reorder point is calculated as expected demand E(R), the so-called stock reserve SR, to which a quantity is added, the so-called safety stock SS, to cover larger than expected demand, caused by the variability in demand and lead times. See FIG. 3 . The reorder point ROP for Q-systems and order to level MSL for P-systems are calculated in exactly the same manner, the only difference being that ROP depends on the lead time L, whereas MSL depends on Tp&plus;L. Therefore the equations below are given for ROP only. Similar calculations are equally applicable if the items are not re-supplied by ordering from a supplier, but are produced in batch-wise production with a limited production time and production setup time. Current methods of determining reorder points generally use the following mechanism to estimate ROP: The probability distribution of P(R) is assumed to be a normal distribution ( FIG. 4 ). If demand R>ROP, the demand will not be served. This constitutes the area or fraction p. The estimate of ROP is given by ROP&equals;E ( R )&plus; Z×Sd ( R ) Where Z is given by Z&equals;G−1(1−p) resulting in a service level s&equals;(1−p)×100% (G is the standard normal distribution function). Sd(R) is estimated by several methods, the one most commonly used is based on the mean weighted average absolute difference of the consumption and forecast over a number of historical periods: 5 MAD = &Sum; i = 1 n &it; wi × &LeftBracketingBar; Forecast &af; ( period &it; &it; i ) - Real &it; &it; demand &af; ( period &it; &it; i ) &RightBracketingBar; Where the weights wi correspond generally to exponential smoothing, the factor 1.25 is used to convert the mean-absolute-deviation-value into the theoretical desired square-root-mean-squared-deviation Sd ( R )&equals;1.25 ×MAD The value of MAD should be corrected for the difference between L and the forecast time on which MAD is based. A second method is based on an estimation of Sd(R) by direct estimation of the variance of the demand sampled in periodic intervals, to which an additional term is added, allowing for the variance of the lead time. Var ( R )&equals; E ( L )× Var ( D )&plus;&lsqb; E ( D )&rsqb; 2 ×Var ( L ) Both of these methods are based on sampling the total quantity demanded D by periodic intervals only, rather than using the statistics based on individual demands. In this context terms such as “variability in demand” and “variance of demands” are not uncommon, however, this does not imply the use of individual demand statistics. Note that sampling of demands in intervals actually leads to loss of information: In FIG. 5, a bar represents a demand for a quantity q, the length of the bar is proportional to the quantity. A box represents the total demand D within a given period. Two different patterns of demand are represented in FIG. 6 in pattern A1 and B1 respectively, together with the presentation of the same demand patterns sampled in periodic intervals in A2 and B2. Clearly, sampling in a periodic way leads to the false conclusion that the non-identical patterns shown are in fact identical. Moreover, as is depicted in FIG. 7 , small changes in—or shifts of the period of sampling of the same pattern may lead to radically different estimates for the variance of the distribution as can be seen from A2 and A3, both periodic presentations of the same demand pattern A1. (Note that where the variance is highly dependent on the way the pattern is sampled, changing the sampling or shifting the presentation does not influence the mean or average expected quantity for a sampling period.) Only the service level during lead time is important, the service level outside lead time is always at least equal to this. In the context of the invention, ROP is estimated from historic data while avoiding the problems mentioned above. A Rigorous Mathematical Formalism P(R) is derived from P(Q), where P(Q) is the empirical probability distribution, based on historical frequency data ( FIG. 8 .) which may be time-weighted using an empirical weighting scheme wq, down weighting inaccurate data and data obtained from a long time in the past. If the weighted frequency distribution is used, the sum of the corresponding weights is taken for all demanded quantities pertaining to a certain interval, instead of the number of times a demanded quantity pertains to the same interval. The historical frequency distribution is obtained by sampling Qi on a number of intervals q1, q2 . . . qm and counting the frequency of occurrence for each interval. F(qm)&equals;frequency Qi in interval qm&equals;sum of 1 (Qi in interval qm) The weighted frequency distribution is obtained by adding the to Qi corresponding weight wq instead of adding 1, when summing the number of occurrences for each interval. Fw(qm)&equals;sum of wq (Qi in interval qm) Given a number of demands, for instance 3 ( FIG. 9 .), the joint probability distribution P3(R) for the total consumption R of 3 demands, can be constructed from the single demand probability distribution P(Q). The demand in the lead time, R, may arise from different numbers of demands j, each with a certain probability of occurring wj, and a probability distribution Pj(R) for the total quantity of the j demands. Therefore P(R) is derived as a series, 6 P &af; ( R ) = &Sum; j = 0 n &it; w &it; &it; j × P &it; &it; j &af; ( R ) ( formula &it; &it; 1 ) Where Pj(R) is the jth selfconvolution of P(Q) and wj is the statistical weight of the corresponding joint probability function for 1 . . . n simultaneous demands in the lead time L. In practice values of n>100 need not be considered, as then the alternative approach given below is valid. Under the assumption of the distribution of Ti being known—e.g. an exponential or truncated normal or Weibull, etc.—the coefficients wj can be calculated directly. This calculation of the coefficients can be modified to include the function P(L). In the case of the distribution of Ti being exponential, for wj a Poisson distribution is the result, with a mean demand number density A equal to the expected number of demands during L divided by L. Calculation of P(R) is then straightforward if the functions Pj are known. The functions Pj are not easily obtained directly, however estimates of sufficient accuracy of these convolutions are obtained by Fourier transformation of P(Q) and back-transforming the jth power of the transform obtained. However, P(R) may be obtained directly by first summing over j the jth powers of the transform of P(Q), 7 L &it; ( R ) = &Sum; j = 1 n &it; w &it; &it; j × [ L &it; ( Q ) ] j ( formula &it; &it; 2 ) using the weights wj, and back-transforming the result just once. P(R)&equals;Fourier transform (L(R))  (formula 3) From this expression, ROP, given a desired service level SLcon, can be calculated easily. The reorder quantity is now calculated in a conventional manner. 8 SL &it; ( ROP ) = &Integral; 0 ROP &it; P &it; ( R ) &it; &it; &dd; R ( formula &it; &it; 4 ) In order to improve the accuracy and avoid series termination effects in Fourier space, the Fourier-transform L(Q) can be multiplied with the, in the reciprocal space defined, weighting factor wL. 9 L &it; ( R ) = &Sum; j = 1 n &it; w &it; &it; j × [ w &it; &it; L × L &af; ( Q ) ] j ( formula &it; &it; 2 &it; B ) Alternatively, for certain probability distributions of P(Q), such as a normal or a gamma distribution, Pj(R) can be found analytically for all values of j. The ROP can be calculated using formula 4 directly or from the cumulative distribution functions Fj of Pj(R) in formula 5. 10 SL &it; ( ROP ) = &Sum; j = 1 n &it; w &it; &it; j × F &it; &it; j &af; ( ROP ) ( formula &it; &it; 5 ) If the number of demands during lead time is sufficiently large, P(R) can be considered to be a normal distribution. In this case P(R) is fully defined by its mean value and variance. The mean may be obtained from E ( R )&equals; A×E ( L )× E ( Q )  (formula 6) And the variance may be obtained from Var ( R )&equals; A×E ( L )× Var ( Q )&plus; A×E ( L )× E ( Q ) 2 &plus;A 2 ×E ( Q ) 2 ×Var ( L )  (formula 7) The advantage is obvious in terms of speed and efficiency. However, it must be stressed that this approach is only valid if de number of demands is high. Note that formula 7 comprises the (first and second moment, mean and variance) moments of the single demanded quantity probability distribution. The E(Q), Var(Q) may be time-weighted using an empirical weighting scheme wq, down weighting inaccurate data and data obtained from sampling a long time in the past. E ( Q )&equals;&Sgr; wq×Qj/&Sgr;wq And Var ( Q )&equals;&Sgr; wq× ( Qj−E ( Q )) 2 /&Sgr;wq If in the Q-system the quantity in stock drops below the reorder point ROP by serving a demand, the calculation does not start at the reorder point ROP but at a point well below the reorder point. In order to compensate for this effect a correction term may be applied to the above mentioned formulae. Variables used in the description, for each item in the inventory: MAD Mean average absolute deviation between forecasted and actual consumption over a number of historical periods D Demand rate, total quantity demanded by period Var(D) Variance of D L The lead time, re-order period or production time P(L) Probability distribution function of L E(L) Expected value of L Var(L) Variance of L R Consumption in units over L P(R) Probability distribution function of R E(R) Expected consumption of item over L Var(R) Variance of R Sd(R) Standard deviation of R Tj Time elapsed between the (j-1)-th and the j-th demand P(T) Probability distribution function of Tj Rj Consumption in units of j demands over L Pj(R) Probability distribution function of Rj wj the statistical weight of the corresponding joint probability function Pj(R) for 1 . . . j simultaneous demands in lead time L. Fj(ROP) The cumulative distribution function of Rj Q(i) Quantity in the i-th single demand during lead time wq Empirical weighting scheme, down weighting inaccurate data P(Q) Probability distribution function of Q for any i E(Q) Expected value for Q for any i Var(Q) Variance of Q A Mean demand number density L(Q) Fourier transform of P(Q) L(R) Fourier transform of P(R) wL Weighting factor in reciprocal space ROP Reorder point SL Service level SL(ROP) Service level as function of the ROP SLcon Desired service level Tp The time between periodic reviews MSL Order-to-level or maximum-stock-level