Optimal solutions for stochastic inventory models when the lead-time demand distribution is partially specified

Int. J. Production Economics 59 (1999) 477—485

Optimal solutions for stochastic inventory models when the lead-time demand distribution is partially specified Haim Shore* Department of Industrial Engineering, Faculty of Engineering Science, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel

Abstract Stochastic optimization models used in operations management require that the underlying distributions be completely specified. When this requirement cannot be met, a common approach is to fit a member of some flexibly shaped four-parameter family of distributions (via four-moment matching), and thence use the fitted distribution to derive the optimal solution. However, sample estimates of third and fourth moments tend to have large mean-squared errors, which may result in unacceptable departure of the approximate solution from the true optimal solution. In this paper we develop an alternative approach that requires only first- and second-degree moments in the solution procedure. Assuming that only the first two moments, partial and complete, are known, we employ a new four-parameter family of distributions to derive solutions for two commonly used models in inventory analysis. When moments are unknown and have to be estimated from sample data, the new approach incurs mean-squared errors that are appreciably smaller relative to solution procedures based on three- or four-moment fitting.  1999 Elsevier Science B.V. All rights reserved. Keywords: Distribution fitting; Inventory analysis; Maximum-likelihood estimates; Optimization

1. Introduction Stochastic optimization models used in various fields of operations management typically require that the underlying statistical distributions be completely specified. When this requirement cannot be met, and only partial distribution information is available (possibly via sample observations) two separate approaches may be pursued.

* Fax: #972 7 647 2958; e-mail: [email protected].

First, by using explorative techniques like the standardized P—P (probability—probability) plot or the Q—Q (quantile—quantile) plot, potential candidate distributions are examined, and hence statistically tested for goodness of fit. Once a certain distribution is selected, maximum likelihood procedures are applied to estimate its parameters. While this approach is statistically appealing, the large sample size required for correct distribution identification coupled with the low power of commonly used goodness of fit tests (refer, for example, to [1]) render this approach widely inapplicable. Furthermore, estimating the parameters of the

0925-5273/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 0 3 7 - 1

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H. Shore/Int. J. Production Economics 59 (1999) 477—485

selected distribution via maximum likelihood (the most efficient approach when the underlying distribution is known), requires the use of distributionspecific procedures. The statistical expertise needed to apply these procedures render them, for the common practitioner, largely inaccessible. A second and more widely adopted approach is to fit a member of some flexibly shaped fourparameter family of distributions (like Pearson’s, or [2]), and thence use the fitted distribution to derive the optimal solution. Various procedures may be applied to determine the parameters of the fitted distribution, however, the most common approach is based on four-moment matching. Typical applications of this approach with regard to inventory analysis may be found in [3—5]. Applications in the area of quality control may be found in [6,7]. While the latter approach yields satisfactory accuracy when the first four moments are known, the requirement for specification of skewness and kurtosis (third and fourth moments) may pose serious obstacles when these moments are unknown and have to be estimated from sample data. This is due to the large sampling errors associated with sample estimates of third and fourth moments. Kotz and Johnson [8] make repeated reference to this point when they discuss process capability analysis of non-normal populations where sample estimates are needed. For example, referring to Clements’ method which calculates capability indices by fitting a member of the Pearson family (therein, p. 156), they note that application of this method “requires knowledge of the coefficients (b and b ” (the skewness and kurtosis measures,   respectively) “which may not be easily obtainable. Rather large samples are needed for accurate estimation of these quantities. Alternative approaches seek to avoid this difficulty”. On p. 160, referring to skewness estimation needed for use of the Edgeworth distribution in capability analysis, they warn again that such estimation “may well be subject to quite large sampling variability”. Later on, dealing with large-sample properties of capability indices, they warn again (ibid, p. 170): “Utilization of some of these properties calls for knowledge of shape factors” ((b and b ) “and the   need to estimate the values of these parameters can

introduce substantial errors”. Such concerns are hardly voiced in Operations-Management publications where distribution fitting for non-normal populations serves as a building block for the entire methodology! In this paper we develop and demonstrate a new approach to derive optimal solutions for stochastic inventory models when the underlying distributions are unknown, and merely first- and seconddegree moments (or their sample estimates) are available for use in the solution procedure. The new approach comprises three components: 1. A new flexibly shaped four-parameter family of distributions, that has a quantile function expressed explicitly in terms of P, the associated value of the distribution function. Various allied functions (like the density function or the loss function) are also expressed explicitly in terms of P. As will be demonstrated by two examples in the sequel, this feature is highly valuable in identifying optimal solutions for stochastic inventory models. Most other available four-parameter distributions do not share this feature. For example, the Pearson family does not possess an explicit expression (in terms of P) either for its quantile function or any of its derivatives. 2. New distribution-fitting procedures (initially introduced in [9]), that require specification of at most second-degree moments (partial and complete). In particular, a new fitting procedure that uses both first- and second-degree moments (partial and complete) has been shown to preserve well both the skewness and the kurtosis of the approximated distribution (the originator of the given moments). This procedure will be used in the sequel. 3. A framework to derive optimal solutions for stochastic inventory models. In the following section we present the new family of distributions, and develop the formulae required to identify its parameters by the twomoment (partial and complete) matching procedure. Applying the latter to the normal, the gamma, the lognormal and the Weibull distributions, the skewness and kurtosis measures of the fitted distributions are compared to the exact values for some representative cases. When moments are unknown and only sample estimates are available, a modified procedure to determine the parameters is given.

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The modified procedure yields mean-squared errors for the skewness and kurtosis measures, calculated from the fitted distribution, that are smaller than those of corresponding direct sample statistics [10—12]. In the two sections that follow, we develop general optimal solutions for two commonly used inventory models, and demonstrate the accuracy obtained relative to exact solutions derived under various distributional scenarios. The concluding Section 5 discusses some of the merits of the new methodology.

2. The new distribution and parameters’ estimation Let X be a random variable, and assume that X is bounded by zero from below, and is unbounded, otherwise. Let F(x) be the cumulative distribution function, and denote by k and p the mean and the standard deviation, respectively. Let x be the Pth fractile, namely: F(x)"P. Consider the following inverse distribution function (the quantile function): A [P/(1!P)] , P(1/2, x"  A +[P/(1!P)] !1,#A , P*1/2,  



[1/(A B )]P\ (1!P)> , P(1/2,   [1/(A B )]P\ (1!P)> , P*1/2.  





 [F/(1!F)]G  dF. M\"AG  G  For P*  [M (½)!2(A )M (½)#(1/2)(A )]/     [M (½)!(1/2)A ]!M /M"0 (to find B ),     





A "[M (½)!(1/2)A ]    



(1)

where +A , B , (i"1, 2) are parameters, to be deterG G mined by two-moment (partial and complete) matching. The motivation for introducing this quantile function, and some of its characteristics, are developed in [12]. The density function of X is, from Eq. (1) f (x)"

in Eq. (1): For P(   [p#k!M (½)]/[k!M (½)]!(M\)/(M\)     "0 (to find B ),   A "[k!M (½)] [F/(1!F)]  dF (to find A ),     (3a) where

(2)

Denote by M (X) the ith partial moment of X, G namely: M (X)" xG dP. Note, that we integrate G  with respect to P and not with respect to x. This practice will often be repeated in the sequel where P, rather than x, is used as the variable of integration or differentiation. Adopting the two-moment (partial and complete) distribution fitting approach [9] the following expressions are used to identify the parameters

+[F/(1!F)] !1, dF (to find A ), 

(3b)

 where  M "AG +[F/(1!F)] !1,G dF. G   It is assumed in using Eqs. (3a) and (3b) that k, p and M (X) (i"1, 2) are all known moments. G When moments are unknown, a modified procedure will be developed later on for reasons that will be detailed. Note, that the need to conduct a numeric search for the values of the parameters renders the above procedure cumbersome for practitioners. This is another reason why a modified procedure is highly desirable. For X that is symmetrically distributed and unbounded in both directions, we obtain for the standardized variable z"(x!k)/p





z"

A+[P/(1!P)] !1,,

P(1/2,

!A+[P/(1!P)]\ !1,, P*1/2.

(4)

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H. Shore/Int. J. Production Economics 59 (1999) 477—485

This is a generalization of the five-moment approximation to the quantile function of the standard normal distribution, given in [13]. Therein we have obtained, for the standard normal: A"5.531, B"0.1193. These marginally deviate from the values derived by the two-moment fitting, given in Table 1. Finally, if X is bounded by LB from below and by UB from above (for example, for the standard beta distribution: LB"0, UB"1), Eq. (1) becomes

and the Weibull distributions, for some representative parameters’ values. The resulting parameters, together with the skewness and kurtosis measures (the third and the fourth standardized cumulants), are shown in Table 1. We observe that the twomoment procedure produces fitted distributions with skewness measures that are very close to the exact values, while the kurtosis measures also do not depart appreciably from the exact values. Moreover, quantiles computed from Eq. (1) are highly accurate, as the reader may easily verify.

(5)

Denote by ¸ (P) the loss function of X at F(x)"P, 6 namely:

Henceforth, we will address only Eq. (1), which adequately represents the expected range of variation of lead-time demand for most inventory models. To appreciate the accuracy of Eq. (1), we have fitted it to the normal, the gamma, the lognormal

  ¸ (P)" (t!x) f (t) dt" [1!F(t)] dt. (6) 6 V V The loss function plays an important role in many stochastic optimization models. In particular, this function is useful in the area of inventory

A [P/(1!P)] #LB, P(1/2,  A [(1!P)/P] #UB, P*1/2. 



x"





Table 1 Coefficients for the distributions fitted by Eqs. (1), (3a) and (3b) (two-moment fitting), and the resulting skewness (Sk Y ). Y ) and Kurtosis (Ku The exact values are Sk and Ku, respectively Distribution Normal k"0, p"1 Sk"0, Ku"0 Gamma a"3, b"2, Sk"1.15, Ku"2.00 a"7, b"2, Sk"0.756, Ku"0.857 ¸ognormal k "1.0, p "0.3686   Sk"1.20, Ku"2.66 k "1.0, p "0.5106   Sk"1.80, Ku"6.26 ¼eibull a"2, b"5 Sk"0.631, Ku"0.245 a"1.375, b"5 Sk"1.23, Ku"1.97 a"1.125, b"5 Sk"1.676, Ku"4.04

A1

B1

5.896

0.1105

5.371

0.4026

13.26

0.2380

A2

Sk Y

Ku Y

0.1105

0

0.058

0.05313

1.21

2.46

0.00442

0.773

1.05

B2

5.896

41.94 780.1

2.668

0.1997

8.696

0.07761

1.22

2.97

2.660

0.2809

6.553

0.1454

1.86

6.86

4.274

0.4396

!34.26

0.655

0.364

3.947

0.6292

50.94

0.04434

1.30

2.58

3.726

0.7613

25.84

0.1046

1.82

5.53

!0.0472

H. Shore/Int. J. Production Economics 59 (1999) 477—485

analysis where penalties for shortages constitute a regular component of the objective function. We will now derive an expression for the loss function that is associated with the new family of distrbutions. Define: ¸ (P)"[A B ][F/(1!F)] G(1/F) dF G N G G (i"1 for P(1/2, i"2, otherwise). Then, it may easily be shown from Eq. (1) that the allied loss function is



¸ (P)#[¸ (1/2)!¸ (1/2)], P(1/2,   ¸ (P)"  6 ¸ (P), P*1/2.  (7) This expression will be used in subsequent sections. Suppose now that the moments used to determine the parameters of Eq. (1) are unknown, and have to be estimated from sample observations. As previously noted [10,12] the parameters derived from Eqs. (3a) and (3b) will tend to have large sampling variability due to the fact that B and  B appear as exponents. In particular, B , which   tends to be close to zero, is very susceptible to sampling deviations. Therefore, an alternative approach is desired. To develop the modified procedure, suppose that we impose a continuity requirement on the density function at P"1/2. This implies that (refer to Eq. (2)): A B "A B "C, say, and Eq. (1)     becomes (C/B )[P/(1!P)] , P(1/2,  (C/B )+[P/(1!P)] !1,#A , P*1/2.   (8)



x"

Since B is expected to be close to zero (observe  representative values in Table 1), and since for nonnegative u: Lim [(uH!1)/k]"Log(u), we may H rewrite (8) as A [P/(1!P)] , P(1/2,  A Log[P/(1!P)]#B , P*1/2.  



x"

(9)

Two new parameters are introduced for P*1/2 to allow for a two-moment fitting. The “price” we pay is the loss of continuity at P"1/2. The resulting expressions to determine the parameters in Eq. (9)

481

are (find details in [12]) B "1.7099+0.5k (º)![k (º)], ,    A "exp+2[k (º)#0.6931B ],,    A"+M (X)!2[M (X)],/(0.6840),    B "2[M (X)!0.6931A ],    where º is the log transformation



Log(X), X(M,

º"

X*M,

0,

(10)

(11)

k (º)"E[º ] and M is the median. Note, that all G the parameters in Eq. (10) may be calculated directly from the given expressions, and no numeric search routine is required. When moments are unknown, however sample estimates are available, all moments in Eq. (10) are replaced by their estimates. It is numerically demonstrated in [12] that for small to medium-sized samples (say n(100), the skewness and kurtosis estimates, derived from the distribution fitted by Eq. (10), have mean-squared errors smaller than those of corresponding direct sample statistics. The loss function associated with Eq. (9) is



¸ (P)#[¸ (1/2)!¸ (1/2)], P(1/2,   ¸ (P)"  6 !A ¸og(P), P*1/2.  (12) Finally, to estimate the parameters of Eq. (1) by maximum likelihood, assume that there are n available observations +x , (k"1, 2, n), where x is I I the kth-order-statistic (there are k!1 smaller observations). The likelihood function (LF) is, from Eq. (2) L LF"[1/(A B )]L “ [P\ (1!P )> ]   I I I L ;[1/(A B )]L “ [P\ (1!P )> ],   I I IL> (13) where: P " F(x ) and n is assumed to be an even I I number. Introducing from Eq. (1) for P in terms I of x , Eq. (13) is expressed in terms of the given I observations. The maximum likelihood estimates for +A , B , (i"1, 2) may now be identified by G G

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maximizing Ln(LF) either via differentiation or by a numerical search. Similar expressions for truncated data may easily be developed. Other fitting procedures, including procedures for right- and left-censored observations, are developed in [12]. Note, that the case of truncated observations may be particularly interesting for models with lost sales. The reader is referred to the above reference for the appropriate fitting procedures. In the numerical examples that follow, we assume that all moments are known, and use either Eqs. (3a) and (3b) or Eq. (10) in the solution procedure.

most inventory models it is expected that P*'1/2, this interval should be searched first for the optimal solution. Also note, that in Eq. (15) Q* and P* are the exact expressions, and only ¸ (P*) is particular V to the new solution procedure. An alternative procedure may be derived by using the modified expressions (9) and (12). For P*(1/2 the solution is given by Eq. (15). However, for the common case of P**1/2, we obtain the simpler optimality conditions

3. The continuous-review (Q, R) model

P*"1!(Q*h)/(Dp).

This is the most common continuous-review model, which, given its “old” age (refer, for example, to [14]), still attracts relatively large research effort. A most recent example is [15]. We will address here the approximate backorder version, as treated by Lau and Lau. The long-run average cost per period is

Only a routine root-finding routine is required to identify Q*.

C(Q, R)"KD/Q#h[Q/2#R!k]#pD¸ (P)/Q, V (14) where Q (the lot size) and R (the reorder point) are decision variables, ¸ (P) is the loss function at V P"F(R) (X is the random lead-time demand with distribution function F(x)), K is the (fixed) order cost, h and p are the unit carrying cost per period and the unit shortage cost, respectively, D is the average demand per period and k is the average lead-time demand (average of X). Introducing for R from Eq. (1) and for ¸ (P) V from Eq. (7), and differentiating with respect to Q and to P, we obtain the optimality conditions (Q*)"(2DK)/h#[(2Dp)/h]¸ (P*), V P*"1!(Q*h)/(Dp),



(15)

¸ (P*)#[¸ (1/2)!¸ (1/2)], P*(1/2,   ¸ (P*)"  V ¸ (P*), P**1/2.  From these expressions, the optimal (Q*, P*) may be identified, and thence R* from Eq. (1). Since in

(Q*)"(2DK)/h#[(2Dp)/h][!A Log(P*)]  "(2DK)/h#[(2Dp)/h] +!A Log[1!(Q*h)/(Dp)],, P**1/2,  (16)

3.1. A numerical example For the cost and demand parameters (taken from Lau and Lau’s examples): D"1000; K"10, 200; h"1; p"5, 30, Table 2 shows exact and approximate solutions, derived for some of the cases depicted in Table 1. The three distributions selected for Table 2 represent mild skewness (Weibull), medium skewness (Gamma) and extreme skewness (Lognormal). Yet, we observe that for all three cases, the new approach yields highly accurate solutions. For the normal case, a corresponding solution procedure that employs approximations for the standard normal quantile function is presented and numerically demonstrated in [13]. To better appreciate the effectiveness of the new approach, suppose now that the actual lead-time distribution is either lognormal or Weibull, with parameters as given in Table 2. However, we have mistakenly identified the underlying lead-time distribution as gamma, with parameters that preserve the actual mean and standard deviation, namely: a"(k/p); b"p/k,

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H. Shore/Int. J. Production Economics 59 (1999) 477—485

Table 2 Comparison of exact solution (Q*, P*) and approximate solution (Qa*, Pa*), for the (Q, R) model. Approximate solutions using either Eq. (15) (upper entry) or Eq. (16) (lower entry) are given. The lowest two entries for the lognormal and Weibull cases display the solutions derived under the gamma assumption Distribution

K"10 p"5 Q* P*

Gamma: a"3, b"2, Sk"1.15, M"5.3

144.0

0.9704

¸ognormal: k "1.0, p "0.5106   Sk"1.8, M"2.72

143.1

0.9714

¼eibull: a"2, b"5 Sk"0.631, M"4.2

142.6

0.9715

K"200 p"5

p"30 Qa* ----Pa*

Q* P*

Qa* ----Pa*

Q* P*

p"30 Qa* ----Pa*

Q* P*

Qa* ----Pa*

144.2 143.9 144.0 ---------0.9712 0.9952 0.9712

143.9 635.2 144.0 ---------0.9952 0.8729 0.9952

635.2 635.0 635.1 ---------0.8729 0.9788 0.8730

635.2 635.0 ---------0.9788 0.9788

143.2 143.4 142.8 ---------0.9713 0.9952 0.9714 142.6 0.9683

143.2 634.0 142.8 ---------0.9952 0.8732 0.9952 142.6 0.9914

634.0 634.2 633.9 ---------0.8732 0.9789 0.8732 633.8 0.8859

634.3 633.8 ---------0.9789 0.9789 633.6 0.9749

142.7 143.4 142.9 ---------0.9714 0.9952 0.9714 143.1 0.9779

142.4 633.9 142.9 ---------0.9952 0.8732 0.9952 142.9 0.9984

633.9 633.6 634.0 ---------0.8732 0.9789 0.8732 634.3 0.8656

633.7 633.9 ---------0.9789 0.9789 634.1 0.9853

where k and p are the actual mean and standard deviation of the lead-time demand distribution. The lowest two entries of Table 2 for the lognormal and the Weibull cases present the optimal solutions under this scenario. Note, that P* stands for the actual value of P associated with the optimal R*, where the latter had been identified under the gamma assumption. Examining Table 2, we realize that although both the mean and the variance are preserved, the wrongly assumed gamma distribution consistently results in solutions that are worse than those obtained under the new approach. Although the differences seem to be negligible in this example, cumulative experience with the new approach suggests that a wrongly selected distribution (that preserves the first two moments) will

generally yield worse results than those derived by the new approach. Numerical evidence may be found in [11,16]. Alternatively, we may wish to use some other four-parameter family of distributions (like Pearson’s), which will be fitted by four-moment matching. When the exact moments are unknown and only sample estimates are available, this approach will also provide less accurate solutions (relative to the new approach) since sample estimates of skewness and kurtosis, required for four-moment matching, have mean-squared errors that are larger than those associated with the new two-moment fitting procedure (refer for details to [10,12]). Thus, relative to the two options addressed in the introduction, namely: identifying the lead-time demand

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H. Shore/Int. J. Production Economics 59 (1999) 477—485

distribution (and risking mis-identification), or using some flexibly shaped four-parameter distribution fitted by four-moment matching, the new approach seems to perform consistently better in cases where sample data are needed in the solution procedure.

4. Safety lead-times for purchased components in assembly systems This problem has been initially addressed and formulated in [17], where an approximate solution procedure, assuming normality for lead times, is developed. An alternative solution procedure, extended for non-normal cases and numerically demonstrated for the normal distribution, is given in [18]. However, in the solution procedure developed therein the fitted distribution is derived from another family of distributions that is unbounded in both directions (find details in [9]). Therefore, optimal solutions that lie in the far left tail of the lead-time distribution tend to be inaccurate. Here, we use the new family of distributions (Eq. (1)) to derive a general solution for this problem, and demonstrate its accuracy for non-normal populations. Only the first of the two formulations given in [17] will be addressed, however extension to the other formulation is straightforward. Consider the problem of setting safety lead times for purchased components, where the only manufacturing operation is the final assembly. There are n components, and delivery times (for each component) are independent random variables with cumulative density function (for component i) of F (x) and mean k (i"1, 2, 2, n). Denote by l the G G G lead time used to order the component (a decision variable), and assume that the holding cost per day, for a component that arrives too early, is: h "rp /365, where r is the annual inventory carryG G ing cost rate, and p is the cost of component i. We G also assume that assembly cannot be performed early, namely: assembly does not start when the last component has arrived, but according to a pre-planned schedule. However, if at least one component is late, assembly is delayed at a timeindependent cost of C. Denote by ¼(l ) the average G waiting time for component i, namely,

¼(l )"JG (l !x) dF (x). Then the cost equation G  G G is (find more details in [17]):





L L Z" h ¼(l )#C 1! “ P , (17) G G G GJ GJ where P "F (l ). It may easily be verified that: G G G ¼(l )"¸ (P )#l !k . Introducing this into G V G G G Eq. (17), and then ¸ (P ) from Eq. (7) and l from V G G Eq. (1), we obtain on differentiating with respect to P and equating to zero G (A B )[P*/(1!P*)] I>(h )!CH"0 I I G G G (i"1, 2, 2, n) (k"1 or 2), (18) where P*"F (l*), l* is the optimal lead time for G G G G component i, and H"“L P*. From Eq. (18) and G G the expression for H, the values +P*, may be comG puted.

4.1. A numerical example Suppose that we have two components, and delivery times are independently and identically gamma distributed with parameters: a"7, b"2 (second case in Table 1). Further assume that F(l*)'1/2 (namely, we use k"2 in Eq. (18)). The G cost data are taken from Case 1 in [17]: p "979,  p "298, r"0.25. These yield: h "0.6705,   h "0.2041. For C we take C"150 (instead of  C"500 in [17]; Their numerical example comprises 20 components, and employing their C will for our case yield a trivial solution of: P*"P* 1).   For the above gamma distribution: A "780.1,  B "0.00442. Introducing into Eq. (18) we obtain  the three equations: (3.4480)[P*/(1!P*)] (0.6705)   [!1#1/(1!P*)]!150H"0  (for the first component), (3.4480)[P*/(1!P*)] (0.2041)   [!1#1/(1!P*)]!150H"0  (for the second component), P*P*"H.  

(19)

H. Shore/Int. J. Production Economics 59 (1999) 477—485

Solving these equations, the approximate optimal solution is P*"0.9842, P*"0.9951. This solution   yields the cost Z*"15.83. The exact solution, searched numerically with the exact statistical expressions (including repeated numerical integration), is: P*"0.9855, P*"0.9958. This solution yields   the cost Z*"15.82. We note that a highly accurate solution has been obtained. By deriving the optimal solution using an incorrect lead time distribution whilst preserving the actual mean and standard deviation, the reader may wish to examine the sensitivity of this solution, as was done for the previous example.

5. Discussion A general approach to solve stochastic inventory models where lead time demand distribution is only partially specified has been developed and numerically demonstrated. The major motivation for developing the new approach has been the realization that the likelihood of selecting an incorrect lead time demand distribution, given the typical sample sizes available to practitioners, is relatively high. Pursuing the alternative approach (namely, fitting a four-parameter distribution without attempting to identify the correct lead time demand distribution) requires estimates of skewness and kurtosis, which tend to be highly inaccurate. By contrast, the new methodology developed here is characterized by the minimal distributional information required for distribution fitting. Two numerical examples have clearly demonstrated that specification of only first and second moments (partial and complete) suffice to yield highly accurate solutions, even in cases where the lead time demand distribution is highly skewed. A comprehensive study, that will study the new methodology in a more systematic fashion with respect to bounds, worst-case scenarios and sensitivity to extremely skewed lead time demand distributions, may provide the desired evidence pertaining to the general applicability of the new approach.

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References [1] H. Shore, Identifying a two-parameter distribution by the first two sample moments (partial and complete), Journal of Statistical Computation and Simulation 52 (1995a) 17—32. [2] B.W. Schmeiser, S.J. Deutsch, A versatile four parameter family of probability distributions suitable for simulation, AIIE Transactions 9 (1977) 176—181. [3] E.A. Silver, A safety factor approximation based on Tukey’s Lambda distribution, Operational Research Quarterly 28 (1977) 743—746. [4] J. Kottas, H. Lau, The use of versatile distribution families in some stochastic inventory calculations, Journal of the Operational Research Society 31 (1980) 393—403. [5] A.H. Lau, H.S. Lau, A simple cost minimization procedure for the (Q, R) inventory model: Development and evaluation, IIE Transactions 25 (1993) 45—53. [6] J.A. Clements, Process capability calculations for non-normal distributions, Quality Progress 22 (1989) 49—55. [7] S. Kocherlakota, K. Kocherlakota, S.N.U.A. Kirmani, Process capability indices under non-normality, International Journal of Mathematical Statistics 1992, 1. [8] S. Kotz, N.L. Johnson, Process Capability Indices, Book. Chapman and Hall, London, 1993. [9] H. Shore, Fitting a distribution by the first two moments (partial and complete), Journal of Computational Statistics and Data Analysis 19 (1995b) 563—577. [10] H. Shore, A new estimate of skewness with MSE smaller than that of the sample skewness, Communication in Statistics (Simulation and Computation) 25 (1996a) 403—414. [11] H. Shore, A general formula for the failure-rate function when distribution information is partially specified, IEEE Transactions on Reliability 46 (1997) 116—121. [12] H. Shore, Approximating an unknown distribution when distribution information is extremely limited, Communication in Statistics 27 (2) (1998). [13] H. Shore, Simple approximations for the inverse cumulative function, the density function and the loss integral of the normal distribution, Applied Statistics 31 (1982) 108—114. [14] G. Hadley, T. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963. [15] D.E. Platt, L.W. Robinson, R.B. Freund, Tractable (Q, R) heuristic models for constrained service levels, Management Science 43 (1997) 951—965. [16] H. Shore, Optimum schedule for preventive maintenance: A general solution for a partially specified time-to-failure distribution, Journal of Production and Operations Management 5 (1996b) 148—162. [17] W.J. Hopp, M.L. Spearman, Setting safety lead-times for purchased components in assembly systems, IIE Transactions 25 (1993) 2—11. [18] H. Shore, Setting safety lead-times for purchased components in assembly systems: a general solution procedure, IIE Transactions 27 (1995c) 634—637.