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Simple approximations for the reorder point in periodic and continuous review (s, S) inventory systems with service level constraints
175
Simple approximations for the reorder point in periodic and continuous review (s,S) inventory systems with service level constraints H.C. TIJMS Department of A ctuarial Sciences and Econometrics, Vrije Universiteit, Amsterdam, Netherlands
H. G R O E N E V E L T Graduate School of Business, Columbia University, New York, U.S.A. Received November 1982 Revised March 1983
In this paper we consider a general class of ( s , S ) inventory systems including periodic review and continuous review systems. We allow for stochastic lead times for replenishment orders provided that the probability of orders crossing in time is negligible for the relevant ( s , S ) control rules. In accordance with common practice we emphasize on service level constraints rather than assuming given stockout costs. In particular we consider the service measure requiring that a specified fraction of the demand is met directly from stock on hand. The purpose of this paper is to present practically useful approximations for the recorder point s such that the required service level is achieved. By a simple and direct approach, a unifying treatment of the general class of (s,S) inventory systems considered is given. We obtain for the first time tractable approximations for the continuous review ( s , S ) inventory system with undershoots of the reorder point. Also, we discuss 2-moments approximations obtained by fitting normal respectively gamma distributions to the empirical demand distributions. Extensive numerical experience with the approximations is reported, including results about the sensitivity of the reorder point to the higher moments of the demand distributions.
1. Introduction
Frequently used stochastic inventory control systems are the (s, S) systems in which at each review the inventory position is ordered up to the level S if the inventory position is at or below the reorder point s and no ordering is done otherwise. In these systems the review typically occurs either periodically or continuously. For the periodic review (s, S) inventory system with deterministic lead times and backlogging of excess demand, computational methods for approximately optimal (s, S) control rules are discussed in Ehrhardt [7], Freeland and Porteus [10], Naddor [13], Roberts [171, Schneider [19,20] and Wagner et al [22] amongst others. With the exception of [19,20] these papers assume that shortage costs for unsatisfied demand can be specified and thus focus only on minimization of costs. In practice, however, shortage costs are usually difficult to assign and as an alternative service level requirements (e.g. on the fraction of demand satisfied directly from shelf) are widely used. Moreover, the order quantity and reorder level are often determined separately in practice; this procedure has support in empirical results showing that very often the optimal order quantity is nearly independent of the service level requirement, cf. also Brown [4] and Peterson and Silver [16]. For the periodic review (s,S) inventory system with backlogging of excess demand and deterministic lead times, Schneider [19,20] has obtained approximations for the reorder point when a servic level requirement is specified and the order quantity is predetermined. The purpose of this paper is to extend his results to a general class of (s, S) inventory systems in which the review is not necessarily periodic, and to
North-Holland European Journal of Operational Research 17 (1984) 175-190 0377-2217/84/$3.00 O 1984, Elsevier Science Publishers B.V. (North-Holland)
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H.C. Tijms, H. Groenevelt/ Simple approximationsfor the reorderpoint
present simple approximations that can routinely be used in practice. By a direct approach which is simpler and more general than the approach in [19], we present a unifying treatment of both periodic review and continuous review (s,S) inventory systems. To our knowledge (cf. also [21]), we obtain for the first time tractable approximations for the continuous review (s,S) system taking account of undershoots of the reorder point, and thus improve upon the classical continuous review (s,Q) inventory system ignoring undershoots. Also, our approach covers in a natural way the case of stochastic lead times of replenishment orders provided the probability of orders crossing in time is negligible for the (s, S) policies of interest. In our analysis we assume backlogging of excess demand and focus on the service level defined as the fraction of demand satisfied directly from stock on hand. However, the analysis applies equally well to different service measures like the probability of no stockout during lead time. Also the analysis requires only slight modifications for the (s,S) inventory system in which excess demand is lost. This paper is organized as follows. In section 2 we give the derivation of the approximations for the reorder point. In Section 3 we discuss 2-moments approximations which are suited for routine use in practice. In particular, we consider 2-moments approximations by fitting normal respectively gamma distributions to the empirical demand distributions. Numerical experience with the approximations is given in section 4. We found that the normal approximations give excellent results for the required service level when the coefficient of variation of the demand in the lead time plus review time does not exceed 0.5; otherwise good 2-moments approximations may be obtained by fitting gamma distributions to the demand distributions. Also, we discuss the sensitivity of the reorder point to the higher moments of the demand distributions. Recommendations about the use of the approximations are given in Section 4.
2. The heuristic analysis We consider a stochastic inventory system in which the sequence of successive epochs at which demand events occur can be described by a renewal process, i.e. the successive interoccurrence times of demand events form a sequence of positive, independent and identically distributed random variables. Thus, letting N ( t ) -- the number of demand events in (0, t], the counting process { N(t), t >/0 } is a renewal process. The successive demands for a single commodity are nonnegative, independent random variables with common probability distribution function F with given mean P-1 and standard deviation o~. The successive demands are assumed to be independent of the process ( N(t)} generating the demand events. For ease we suppose that F has a probability density f. The state of the system is only reviewed just after each Rth occurrence of a demand event. Here the review parameter R is a prespecified positive integer. Assuming that excess demand is backlogged, the control of the system is based on the inventory position defined as the stock on hand minus backorders plus stock on order. The control rule is an ( s , S ) rule under which the inventory position is ordered up to the level S if at a review the inventory position is at or below s and no ordering is done otherwise. The lead time of each replenishment order is a nonnegative random variable L with given mean # ( L ) and standard deviation o(L). We assume that L is bounded and discrete-valued. The standard periodic review and continuous review inventory systems can be modelled as special cases within the above class of (s,S) inventory systems. Periodic review inventory system: Take a deterministic process N(t) with unit jumps at equidistant epochs t = 1,2 ..... i.e. the demand process is described by a sequence of (aggregated) demands occurring in periods t - 1,2 ..... The review of the inventory position is only at the beginning of each Rth period. Continuous reoiew inventory system: Take a Poisson process for the process { N(t)} and take R = 1, i.e. the successive times between demand epochs are independent random variables with a common exponential distribution and the inventory position is reviewed just after each demand transaction. In this paper it is assumed that the difference S-s indicating the order quantity is predetermined, e.g. by
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the economic order quantity formula. We concentrate on the determination of the reorder point s under the following service level constraint: fraction of demand to be met directly from stock on hand >1 fl,
(1)
where fl is a specified number. We note that the analysis below applies equally well to different service measures to be discussed in Remark 1. We give a unifying but heuristic treatment for the determination of the reorder point in the class of ( s , S ) inventory systems considered. In order to obtain implementable results, we have to compromise between mathematical and practical standpoints. Of course, each approximate step in the analysis should be based on sound principles and after all the approximate results should be validated. Before deriving the approximations, we introduce the following notation. Denote by F ~k~ and f~k~ the probability distribution function and the probability density of the total demand at k consecutive demand epochs, k >t 1. Let/z k and o k be the mean and standard deviation o f f tk). Note that lak = k/~ 1 and o k = olvrk-. Also, define the review time as the time elapsed between two consecutive reviews and note that the total demand in the review time has probability distribution function F t R). To do the approximate analysis, we make the following assumptions for the ( s , S ) rules achieving the required service level:
Assumptions. (i) the probability that replenishment orders cross in time or arrive simultaneously is negligible, (ii) S - s is sufficiently large compared with the average demand /t R in the review time (say, S - s >/1.5#n), (iii) the stock on hand just after arrival of a replenishment order is positive except for a negligible probability. We now first analyse the service level of a given ( s , S ) rule. Define a replenishment cycle as the time elapsed between two consecutive arrivals of replenishment orders. By a standard result in renewal theory (cf. Ross [18]), we have that the fraction of demand not satisfied directly from stock on hand equals the average amount that goes short per replenishment cycle divided by the average demand per replenishment cycle. Using Assumption (iii), the average amount that goes short per cycle is'approximately equal to the average shortage present just prior to the arrival of a replenishment order. Hence the fraction of demand not satisfied directly from stock on hand = (the average shortage, present just prior to the arrival of a replenishment order)/(the average demand per replenishment cycle).
(2)
To determine the ratio in (2), we tag one of the replenishment orders. Define the random variables Z = the undershoot of the reorder point s at the review at which the tagged replenishment order is placed, = the total demand in the lead time of the tagged replenishment order. Now, we can easily give the denominator of the ratio in (2). Since the average demand per replenishment cycle is equal to the average order size, we have the average demand per replenishment cycle = S - s + E Z .
(3)
We need a tractable approximation for the distribution of Z. Therefore, note that in renewal-theoretic terms Z is the excess variable associated with the random variable representing the review time demand. Recalling that the review time demand has probability distribution function FtR) and using Assumption (ii), we have by a standard result in renewal theory that (cf. [18]), P r { Z < ~ x ) = t a - R'
f0x/
1-F'R'(y)}dy,
x>~O,
(4)
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independently of S - s. This approximate result requires that o~/#~ is not extremely small, i.e. the review time demand is not nearly deterministic. Also,
EZ= (ok + # ~ ) / 2 # , .
(5)
To determine the numerator of the ratio in (2), we first ,~bserve that the inventory position just prior to the placing of the tagged order equals s - Z. Also, by Assumption (i), any replenishment order placed earlier than the tagged order has been arrived when the tagged order comes in, while no order placed later than the tagged order arrives earlier than the tagged order. Hence the stock on hand just prior to the arrival of the tagged replenishment order = s - Z - 4. Denoting by
h(x)
the probability density of Z + ~, we have
the average shortage present just prior to the arrival of a replenishment order =ff°(x-s)h(x)dx. In Appendix 1, it is shown after some manipulations that
f~(x-s)h(x)dx=(2#R)-'(f°°(x-s)Z~l(x)dx-f~(x-s)2~(x)dx),
(6)
where (x) = the probability density of the total demand in the lead time plus review time, 4 (x) = the probability density of the total demand in the lead time. Now, by the relations (2)-(3) and (5)-(6), we have approximately determined the service level for a given (s, S) policy. Thus the reorder point s achieving the required service level fl (cf. (1)) can approximately be determined by solving the equation
where in the right side of (7) the difference S - s is given. Note that the relation (7) involves the complete demand densities 77 and 4. In the next section we give solution methods for (7) by using 2-moments approximations. Now we discuss a possible simplification of (7). If the service level/3 is close to 1 and the demand pattern is relatively smooth, it can heuristically be argued that the relation (7) may approximately be simplified to f ~ ( x - s )2r/(x)dx = (1 - / 3 )2~tR( S - s + ( ok + #~ )/2#R ).
(8)
It was pointed out to us by Dr. Helmut Schneider that in the periodic review model the simplified relation (8) arises for the service measure defined as the ratio of the average existing backlog and the average demand per unit time, see [20]. It will intuitively be clear that the latter service measure is very close to the B-service measure for high service levels and smooth demand. Clearly the reorder point determined by (8) is larger than or equal to the reorder point determined by (7) and so we may in general expect that the service level of the (s,S) rule induced by (8) is larger than or equal to the service level of the (s,S) rule induced by (7). Our numerical investigations show that in particular for irregular (erratic) demand the relation (8) is too crude and leads to erroneous results for the service level; this fact was not recognized in the studies [19,20] for the periodic review (s,S) system. Also, we found that for the continuous review system the second term in the left side of (7) plays in general a much more crucial role than in the periodic review system. Detailed recommendations about the use of the equations (7) and (8) for the reorder point are given in Section 4. For the periodic review (s,S) inventory system with deterministic lead times the relations (7) and (8) were already obtained in Schneider [19,20] by using the asymptotic results in Roberts [17]. The above
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approach is simpler and more insightful and it allows for a unifying treatment of a general class of (s,S) inventory systems with stoichastic lead times including periodic review and continuous review systems. We conclude this section with two remarks about different service measures and the (s, S) inventory system in which excess demand is lost.
Remark 1. Different service measures Another widely used service measure requires that the probability of no shortage just prior to the arrival of a replenishment order is at least P1; this criterion indicates the fraction of orders for which expedition might be necessary. By the same analysis as above we find that under this service measure the reorder point s can approximately be determined from
f f (x-s)7(x)dx- f f (x-s)4(x)dx= 1 - p,. The Pi-service measure has the drawback that it does not indicate how often a stockout occurs per unit time. Therefore consider the service measure requiring that the average number of stockouts per unit time is at most P0. This service measure can easily be related to the Pl-service measure. To do this, we note that the average length of a replenishment cycle follows from the renewal-theoretic result that the average demand per unit time equals the average demand per cycle divided by the average lenght of a cycle. The average demand per cycle is given by (3), while the average demand per unit time equals v#~ where v is the long-run average number of demand events per unit time. Thus under the service constraint that the average number of stockouts per unit time is at most P0, the reorder point s can approximately be determined from
ff
ff
+
I
Remark 2. The lost-sales (s,S) inventory system In this Remark we briefly discuss the (s,S) inventory system in which excess demand is lost, cf. also [3,15]. The above results need only slight modifications for the lost-sales case. An examination of the above analysis shows that the average lost demand per replenishment cycle is approximately given by (6). Also, the average demand per cycle is approximately equal to the average lost demand per cycle plus the average order size that is approximately given by the right side of (3). Thus, to achieve that the fraction of demand to be met from stock on hand is at least/3, the reorder point s can approximately be determined from the relation (7) in which 1 - 13 is replaced by (1 - fl)/fl. Clearly, for fl close to 1 the backlogging and lost-sales systems will not differ significantly.
3. Two-moments approximations for the reorder point In practice it will often happen that only the first two moments of the demand densities 7 and 4 are available instead of the complete distributions. In this case one may solve (7) or (8) by fitting suitable probability densities to the demand densities ~ and 4 by matching the first two moments. In practice the normal and gamma densities are often used to fit the empirical demand densities, cf. also [4,5]. In this section we discuss these 2-moments approximations which are suited for routine use in practice. Before doing this, we give formulae for the first two moments of the demand densities 7 and 4. Denote by #(7) and o(7) the mean and the standard deviation of the density ~1. Similarly, for #(4) and 0(4). Routine calculations involving conditional expectations (cf. [18]) yield # ( 7 ) = (R+EN(L)}I~1,
o2(7)=Ro~+EN(L)o?+o2(N(L))#~,
where N(L) denotes the number of demand events in the lead time L of a replenishment order. The formulae for #(/~) and 0(4) follow by putting R -- 0 in the above formulae. In general we can invoke from
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renewal theory approximate expressions for EN(L) and o 2(N(L)) based on the first three moments of the interoccurrence times of the demand events and the first two moments of the lead time, e.g. cf. [23, p. 133]. We omit these general expressions, but we only give the formulae for the most important cases of periodic review and continuous review. In these cases we can easily give exact expressions for EN(L) and a 2(N(L)), since the process ( N ( t ) ) is deterministic for periodic review and is a Poisson process for continuous review. It is no restriction to assume that the lead time L is integer valued for the case of periodic review. We have for the periodic review model
= (R +
o2(,7) = Ro? + ELo? +
whereas for the continuous review model
o2(~)=o~+XELo~+(X2oZ(L)+XEL)#~,
# ( ~ ) = (1 + X E L ) , , ,
with ~, is the occurrence rate of demand transactions. In both models we have ~t(~j)= # ( r l ) - Rp,1 and o2(~) = o2(-q)- Ro~ (note R = 1 for the continuous review model). We now turn to 2-moments approximations for the reorder point. First, we discuss the simplified normal approximation which is obtained by solving the simplified equation (8) in which a normal density is fitted to the demand density rl(x) by matching the first two moments. Clearly, a normal fit can only be used when the coefficient of variation o(~)/~t(~l) is small (say, o(~)/~t(~)~< ½), since negative demand values cannot occur. Let
q ,,( x ,) = - - ~1e -
,'/2
and
~(x)= fx
q~(u)du
(9)
be the probability density and the probability distribution function of the standard normal distribution. Also, define
I(x)= f~(uWe note that
x)ep(u)du
and
J(x)= f~°(u-x)2ep(u)du.
(10)
(cf. [12]),
I(x)=¢(x)-x{1-*(x)}
and
J(x)=(l
+ x2){1-*(x)}-x¢(x).
(11)
Now, if we fit a normal density to the demand density ~ ( x ) by matching the first two moments and use the representation s =#(~) +ko(~),
(12)
then the simplified equation (8) reduces to
o2(
)JIk) =y0,
(13)
where Y0 = (1 - fl)2txR { S - s + (oa2 + #2a ) / 2 # R }.
(14)
The easiest way to compute the safety factor k and so the reorder point s is to approximate the inverse function k = J - 1 ( y o) by a rational function. From [20], we have
k=
a o + alw + a2w 2 + a3w 3 +,(w), bo + blw + b2w2 + b3w3
(15)
where for the case of Y0 ~< 0.5 2
w=(ln(1/y6)) a 3 --- 0,
1/2
b0= 1,
,
a 0 = -0.4188413, b a = 0.2134080,
a I = - 0.2554696, b 2 = 0.04439934,
a 2 ----0.5189103, b 3 = - 0.002639787,
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while for the case of Y0 > 0.5, w -- Y0,
a 0 -- 1.125946,
al = - 1.319002,
b 0 = 1,
b 1 = 2.836738,
b 2 = 0.6559378,
a 2 = - 1.809643, b3 =
a 3 = - 0.1165009,
0.008220435.
Here maxl{(w)l < 2.3 x 10 -4 for - 4 ~< k < 4. The computation of the simplified normal approximation by (12)-(15) is very easy and can routinely be done in practice. Next we discuss the computation of the normal approximation which is obtained by solving the relation (7) in which normal densities are fitted to the demand densities 71(x) and ~(x) by matching the first two moments. Therefore we first note that the equation (7) has a unique solution since the left side of (7) is decreasing in s and approaches 0 as s ~ ~ . Denoting by M ( s ) the left side of (7), we easily derive for the normal fit
while the derivative of M ( s ) is given by
M'(s)
=
-2o(7)1(s-
#(*/)
s
We can easily evaluate the function M ( s ) and its derivative by using the relation (11) and so we can effectively solve the equation (7) by the standard N e w t o n - R a p h s o n method. Since the function M ( s ) is convexly decreasing this iteration method converges for each starting point; as suggested in [19] a good starting point is the value s o determined by (12)-(15). Thus the normal approximation can be computed by the following algorithm:
Step O. Compute s o by (12)-(15). Let i ,= 0. Step 1. Given the current value s , compute the new value si÷ ~ from (with Y0 defined by (14)), s'+l '= si
(M(si) -Y0) M'(s,) "
using the relation (11) for the evaluation of the functions l ( x ) and J(x). The standard normal distribution function ~ ( x ) required in (11) could be calculated by using the approximation formula ~ ( x ) = 1 O(X)~.5=lCktk-st-C.(X) for x>~0 and O ( x ) = 1 - O ( - x ) for x~<0, where I{(x)[ < 7 . 5 x 10 -8, t = 1/(1 + ax), a = 0.2316419, c 1 = 0.319381530, c: = -0.356563782, c 3 = 1.781477937, c 4 = - 1.821255978 and c 5 = 1.330274429, cf. [1]. Step 2. If Is,+~ - si] ~< 0.1, then stop, otherwise i '-= i + 1 and go to Step 1. This algorithm converges very rapidly and in all cases tested it required at most 4 iterations. Also, this computational procedure for the normal approximation is easy to implement and suited for routine use. The normal demand distribution is typically adequate for representing the demand of fast-moving items with relatively smooth demand patterns. If the coefficient of variation of the demand in the lead time plus review time is larger than 0.5 (say), it becomes inadequate to fit normal demand distributions since negative demand cannot occur. The difficulty of negative demand is avoided when using g a m m a demand distributions. In m a n y practical inventory applications a g a m m a distribution fits well the empirical demand distribution, cf. also [5]. Also the g a m m a distribution is quite tractable for numerical calculations. For g a m m a densities the left side of (7) can be written in terms of incomplete g a m m a functions for which easily implementable numerical procedures are available. The incomplete g a m m a function is defined by
1 £
P( a,z ) = F( a---") e 't ~ 'at,
a > O,
where the term F ( a ) in the denominator is the complete g a m m a function, cf. [1,5]. The incomplete g a m m a
H. C Tijms, 14. Groenevelt / Simple approximations for the reorder point
182
function can easily be calculated from a series expansion whose positive terms can recursively be computed, see [1; 5 Appendix 2] for details. An excellent Fortran program for the incomplete gamma function is stated in [11]. If gamma densities are fitted to the demand densities 7(x) and ~(x) by matching the first two moments, we easily derive that the left side of (7) is given by M(s).= o 2 ( 7 ) ( a ( 7 ) + 1){1 - P(a(7) + 2,b(7)s)} - 2s~(7){1 - P(a(11) + 1,b(r/)s)} + s2{1 - P(a(7),b(7)s } - o2(l~)(a(l~) + 1)(1 - P(a(~) + 2,b(~j)s) } +2s~(~){1-P(a(~)+l,b(/~)s}-sZ{1-P(a(~),b(~)s)},
s>O,
where a(7) = 1~t2(7)/o2(7) and b(7) = ~(7)/o:(7). Similarly, for a(~j) and b(~). For s ~
= o2(7)
+
- o2(
) -
_ 2s {
-
}•
Moreover, the derivative of M(s) is given by
M'(s) = - 2 # ( 7 ) { 1 - P(a(7) + 1,b(~)s} + 2s{1 - P(a(~l),b(7)s)) +2#(~)(1-P(a(~)+l,b(l~)s}-2s(1-P(a(~),O(~)s)),
s>0.
From a numerical standpoint it is important to note that for the computation of M(s) and M'(s) we need only two evaluations of the incomplete gamma function in view of the recurrence relation (cf. [1]), + 1,z)
=
+ 1),
where F(a ÷ 1) = aF(a). Also, for the gamma fit equation (7) can effectively be solved by the standard Newton-Raphson method taking s o determined by (12)-(15) as starting point. The numerical procedure for the gamma approximation is less simple than the one for the normal approximation, but it is still tractable enough for practical purposes. None the less we give in Appendix 2 a simple approximate method to solve (7) for gamma distributed demand. We conclude this section by remarking that other popular demand distributions (cf. also [9]) for which (7) can effectively be solved by Newton-Raphson include the lognormal and the Weibull distributions. However, noting the role of the tail probabilities in (7), it should be pointed out that for larger values of the coefficients of variation of the demand densities 7(x) and ~(x) (i.e. for more erratic demand), the choice of the distributions to describe the empirical demand distributions gets more important, in particular when the required service level is close to 1.
4. Numerical results
In this section we discuss both for periodic review and continuous review (s,S) inventory systems with backlogging of excess demand the performance of the approximations for the B-service level defined as the fraction of demand to be met directly from stock on hand. We concentrate on the following approximations for the reorder point: (i) the simplified normal approximation (snor.) resulting from (8) in which a normal density is fitted to 7(x) by matching the first two moments, (ii) the normal approximation (norm.) resulting from (7) in which normal densities are fitted to 7(x) and ~(x) by matching the first two moments, (iii) the gamma approximation (gamma) resulting from (7) in which gamma densities are fitted to 7(x) and ~(x) by matching the first two moments, (iv) the true approximation (true) resulting from (7) in which the actual demand densities 7(x) and/j(x) are used. Also, denote by C~ = o(7)/E7 and C~ = o(~)/E~ the coefficients of variation of the demand densities 7(x) and ~j(x) respectively. Extensive numerical investigations reveal that in many practical inventory applica-
H.C Tijms, H. Groenevelt / Simple approximations for the reorder point
183
tions the following 2-moments approximations for the reorder point yield an excellent performance for the /3-service level.
Recommendationsfor 2-moments approximationsfor the reorderpoint (A) Periodic review (s,S) inventory system. For the case of Cn < 0.5, use the simplified normal approximation provided both fl >/0.90 and C~ - C, >/0.01, otherwise use the normal approximation. For the case of Cn > 0.5, use the gamma approximation. (B) Continuous review (s,S) inventory system. For the case of Cn ~<0.5, use the simplified normal approximation provided both fl >/0.97 and C~ - Cn >/0.01, otherwise use the normal approximation. For the case of Cn > 0.5, use the gamma approximation. We wish to make two side-notes to these recommendations. First, the approximations are not warranted when the ordering frequency #~1(S - s) is less than 1½ and when the probability of orders crossing in time is not small (~< ~ , say). The effect of possible cross-overs of orders diminishes if fl gets closer to 1. Secondly, the reorder point becomes increasingly sensitive to more than the first 2 moments of the demand densities ~(x) and ~(x) when the coefficients of variation C, and C~ get larger and when the service level fl gets very close to 1. This phenomenon for erratic demand was also observed in [2]; the insensitivity conjectured in [14] is typical for rather smooth demand patterns provided the service level is not very close to 1. Thus for erratic demand some care should be exercised in using 2-moments approximations, in particular when the service level fl is close to 1. To illustrate this, we include in the tables 2 and 3 the lognormal approximation (logn.) resulting from (7) in which lognormal densities are fitted to ~(x) and ~(x); the fact that the lognormal distribution has a longer tail then the gamma distribution is typically reflected in the reorder point. Clearly, for erratic demand and for very high service levels it is always safer to use the true approximation whenever possible. Nevertheless in many practical inventory applications with erratic demand, gamma distributions give an excellent fit to the empirical demand distributions (cf. also [5]) and the associated 2-moments approximation show a good performance for the service level. Also, for most items in inventory control the demand "patterns are relatively smooth so that for those items one can in general safely use 2-moments approximations based on normal demand densities. In the test examples below the probability density f(x) of the demand transaction sizes is given by a negative binomial distribution or a Poisson distribution. These discrete distributions have as possible values the nonnegative integers. The negative binomial distribution is completely specified by t~l and otz and requires o~/#1 > 1, while the Poisson distribution is completely specified by #1 and has the property o~/~1 = 1. Also, we consider the following two distributions for the lead time L of a replenishment order, (a) P r { L = 2 } = l (b) P r { L = I } = ¼ ,
(EL=2, o2(L)=O), Pr{L=2}=½, Pr{L=3}=¼
(EL= 2 , 0 2 ( L ) =½).
Although our primary goal is to test the service level of the approximate (s,S) rules, we assume in the examples a fixed setup cost K = 36 and linear holding cost h = 1. Thus in our test examples we predetermined S - s as the nearest integer to ~ (periodic review) and ~/2KXt~l/h (continuous review) respectively, where ~, is the occurrence rate of demand transactions. Also, for the 2-moments approximations the reorder point s was rounded to the nearest integer. For the true approximation, we used a discretized version of equation (7) and determined the reorder point as the largest integer s for which
(k-s)2,7(k) - E k~s
kss
with Y0 given by (14). To test the quality of the approximations we computed by an analytical method the actual value/3 (s,S) (say) of the service level of the approximate (s,S) rule, using the original (negative binomial or Poisson) distribution for the demand transaction sizes. We shall not discuss here this analytical method but refer to [8] for details. Also, for testing purposes, we have applied a Lagrangian method which
H.C. Tijm~, H. Groenevelt / Simple approximations for the reorder poin~
184
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~k
,4
Z
#. mm~
,#
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H.C. Tijms, H. Groenevelt / Simple approximations for the reorder point
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11. C. Tijms, 14. Groenevelt / Simple approximations for the reorder point
187
searches for (but not necessarily finds) an (s, S)rule minimizing the average holding and setup costs within the class of (s,S) rules satisfying the t-service level constraint. In this Lagrangian method a fixed penalty cost ~r per unit shortage is introduced and is varied until its smallest value is found for which the associated average cost optimal (s,S) rule still satisfies the service level constraint, cf. [8] for details. Incidentally, it may be interesting to note that our numerical experiments support the empirical result that the average holding and setup costs achieved by choosing S - s equal to the economic order quantity are often quite close to the minimal average costs; e.g. in our test examples the relative difference percentage in costs of the true (s,S) rule and the Lagrangian (s,S) rule is typically between 0 and 4% and is occasionally as large as 10%. We note that the Lagrangian method and the analytical method to compute the exact service level are both exact up to neglecting the probability of orders crossing in time when the lead times are stochastic. In the test examples with stochastic lead times the actual values fl(s,S) for the service level have been validated by computer simulation. In the tables we give the various approximate (s,S) rules and the Lagrangian (s, S) rule (Lagr.) and the associated actual values fl(s, S) of the service level. The Tables 1 and 2 deal with the periodic review (s,S) inventory system, where in all examples we have chosen the review period R = 1. Table 3 deals with the continuous review (s,S) inventory system. It is indicated in the tables how the different parameters are varied.
Appendix
1. The d e r i v a t i o n of t h e b a s i c relation (6)
In this appendix we derive the relation (6). We first introduce some notation. For each t >/0, let
p~( t ) = Pr{ N( t ) = n }, n = 0,1 .... (withp0(0)
= 1,
by convention),
i.e. p~(t) is the probability that n demand events occur in (0,t]. Also, put for abbreviation ge(x) =
x>_.0.
Then, by (4), the probability density of the undershoot variable Z is approximately given by ge(x). Using the independence of the random variables Z and ~ and conditioning on the lead time L of the tagged order and on the number of demand events during this lead time, we find Pr{Z+~
Y~.Pr{Z + ~<~xIL=I}Pr{ L = I } 1
= Y~.Pr{L=I} Y'. P r { Z + ~ < ~ x l L = l , N ( I ) = n } p . ( I ) I
n>~O
= ]~Pr{L=/} I
[:o Po(I)
ge(y)dy+
E p.(I) n~l
::
Ft")(x-y)ge(Y)dy
The probability density h(x) of Z + ~j follows after differentiation (use thatF ~") (0)= O, n >/1),
h ( x ) = Y'.Pr{L=I I
'[
po(l)ge(x)+
Using the change of variable u = x - y
Y'.p.(l n~l
':0
f'")(x-y)g~(y)dy
]
, x>O.
in the latter integral and using the convolution formula
f ' " + ' ) ( x ) = foXf'")(u)f'm)(x-u)du,
n,m>: l,
]
.
H.C. Tijms, H. Groenevelt / Simple approximations for the reorderpoint
188
we find after differentiation
h'(x)
= • P r { L = l} [--po(I)g-Rlf(R)(X) 1 [
+ Y'. p,(l)ft")(x)ge(O) n>>.l
The first term between brackets in the last relation gives the probability density ~ ( x ) of the lead time plus review time demand, while the second term gives the probability density 4(x) of the lead time demand for x > 0 (note that 4(0)= E/Pr{ L -- l}Y..~op.(l)F(")(O), where F (°) (0) = 1 and F (') (0) -- 0, n >/1). Hence we can write
h'(x)= -g;~'[,7(x)-4(x)],
x>0.
Finally we obtain
f;(x s)h(x)d --
=
--
)2=
__ 1
f; ( X - - s ) 2 h . ' ( x ) d x
Here it is assumed that the demand d e n s i t y f ( x ) has a finite third moment so that lira x ~ ~x:'h ( x ) = 0. This completes the verification of (6).
E(Z
+ 4): < ¢¢ and so
Appendix 2. An approximate method for gamma distributed demand This appendix gives for gamma distributed demand an approximate method to solve the equation (7) avoiding the evaluation of incomplete gamma integrals. We can always approximate the gamma density by a simpler probability density having the same first two or three moments, cf. also [6]. Consider a gamma density with mean m and coefficient of variation C. Distinguish between 0 < C ~< 1 / v ~ and C > 1/vr2.
Case 1, 0 < C ~< 1 / v ~ . Then 1/k <~C: < 1/(k - 1) for a unique integer k >/2, and the following mixture of E k_l and E L densities with the same scale parameters has the same first two moments as the gamma density, xk-2
xk-1
k(x)=pb~-'--e-b"+(1-p)bk--e
-h', x>~O,
where
-11--[ k C : _ ( k ( l + C : ) _ k 2 C : ) , / 2 ]
k-p.,
P= 1+C 2
Case 2, C > 1 / v ~ . Then the following probability density gamma density, k(x)=pble-b''+(1-p)b2e -h2',
x>~0,
k(x)
has the same first
three moments
as the
189
H.C. Tijms, H. Groenevelt / Simple approximations for the reorder point Table A.1 The reorder points s G and SME (~: = 8 and L = 2) o]/~
5
8
10
15
25
40
SG,
SME
$G'
$ME
SG'
SME
$G'
SME
SG'
SME
SG"
$ME
0.90 0.95 0.99
26, 32, 44,
26 32 44
30, 38, 55,
30 38 55
34, 43, 63,
33 42 61
42, 54, 81,
41 54 82
57, 75, 117,
58 76 118
81, 108, 170,
82 108 169
(P)
0.90 0.95 0.99
31, 39, 58,
30 38 55
35, 45, 68,
35 45 68
38, 50, 75,
38 50 76
46, 60, 92,
46 61 95
61. 81, 126,
61 82 129
84, 112, 177,
85 113 178
(C)
where 2+ b12• = m -
2 { C. . .2. .- ½ } 1/2 I+C 2 and
m
=
p
b:(b2m-1) b2_bl
Note that b 1 > b2 > 0. Also, 0 ~
Simple approximations for the reorder point in periodic and continuous review (s,S) inventory systems with service level constraints H.C. TIJMS Department of A ctuarial Sciences and Econometrics, Vrije Universiteit, Amsterdam, Netherlands
H. G R O E N E V E L T Graduate School of Business, Columbia University, New York, U.S.A. Received November 1982 Revised March 1983
In this paper we consider a general class of ( s , S ) inventory systems including periodic review and continuous review systems. We allow for stochastic lead times for replenishment orders provided that the probability of orders crossing in time is negligible for the relevant ( s , S ) control rules. In accordance with common practice we emphasize on service level constraints rather than assuming given stockout costs. In particular we consider the service measure requiring that a specified fraction of the demand is met directly from stock on hand. The purpose of this paper is to present practically useful approximations for the recorder point s such that the required service level is achieved. By a simple and direct approach, a unifying treatment of the general class of (s,S) inventory systems considered is given. We obtain for the first time tractable approximations for the continuous review ( s , S ) inventory system with undershoots of the reorder point. Also, we discuss 2-moments approximations obtained by fitting normal respectively gamma distributions to the empirical demand distributions. Extensive numerical experience with the approximations is reported, including results about the sensitivity of the reorder point to the higher moments of the demand distributions.
1. Introduction
Frequently used stochastic inventory control systems are the (s, S) systems in which at each review the inventory position is ordered up to the level S if the inventory position is at or below the reorder point s and no ordering is done otherwise. In these systems the review typically occurs either periodically or continuously. For the periodic review (s, S) inventory system with deterministic lead times and backlogging of excess demand, computational methods for approximately optimal (s, S) control rules are discussed in Ehrhardt [7], Freeland and Porteus [10], Naddor [13], Roberts [171, Schneider [19,20] and Wagner et al [22] amongst others. With the exception of [19,20] these papers assume that shortage costs for unsatisfied demand can be specified and thus focus only on minimization of costs. In practice, however, shortage costs are usually difficult to assign and as an alternative service level requirements (e.g. on the fraction of demand satisfied directly from shelf) are widely used. Moreover, the order quantity and reorder level are often determined separately in practice; this procedure has support in empirical results showing that very often the optimal order quantity is nearly independent of the service level requirement, cf. also Brown [4] and Peterson and Silver [16]. For the periodic review (s,S) inventory system with backlogging of excess demand and deterministic lead times, Schneider [19,20] has obtained approximations for the reorder point when a servic level requirement is specified and the order quantity is predetermined. The purpose of this paper is to extend his results to a general class of (s, S) inventory systems in which the review is not necessarily periodic, and to
North-Holland European Journal of Operational Research 17 (1984) 175-190 0377-2217/84/$3.00 O 1984, Elsevier Science Publishers B.V. (North-Holland)
176
H.C. Tijms, H. Groenevelt/ Simple approximationsfor the reorderpoint
present simple approximations that can routinely be used in practice. By a direct approach which is simpler and more general than the approach in [19], we present a unifying treatment of both periodic review and continuous review (s,S) inventory systems. To our knowledge (cf. also [21]), we obtain for the first time tractable approximations for the continuous review (s,S) system taking account of undershoots of the reorder point, and thus improve upon the classical continuous review (s,Q) inventory system ignoring undershoots. Also, our approach covers in a natural way the case of stochastic lead times of replenishment orders provided the probability of orders crossing in time is negligible for the (s, S) policies of interest. In our analysis we assume backlogging of excess demand and focus on the service level defined as the fraction of demand satisfied directly from stock on hand. However, the analysis applies equally well to different service measures like the probability of no stockout during lead time. Also the analysis requires only slight modifications for the (s,S) inventory system in which excess demand is lost. This paper is organized as follows. In section 2 we give the derivation of the approximations for the reorder point. In Section 3 we discuss 2-moments approximations which are suited for routine use in practice. In particular, we consider 2-moments approximations by fitting normal respectively gamma distributions to the empirical demand distributions. Numerical experience with the approximations is given in section 4. We found that the normal approximations give excellent results for the required service level when the coefficient of variation of the demand in the lead time plus review time does not exceed 0.5; otherwise good 2-moments approximations may be obtained by fitting gamma distributions to the demand distributions. Also, we discuss the sensitivity of the reorder point to the higher moments of the demand distributions. Recommendations about the use of the approximations are given in Section 4.
2. The heuristic analysis We consider a stochastic inventory system in which the sequence of successive epochs at which demand events occur can be described by a renewal process, i.e. the successive interoccurrence times of demand events form a sequence of positive, independent and identically distributed random variables. Thus, letting N ( t ) -- the number of demand events in (0, t], the counting process { N(t), t >/0 } is a renewal process. The successive demands for a single commodity are nonnegative, independent random variables with common probability distribution function F with given mean P-1 and standard deviation o~. The successive demands are assumed to be independent of the process ( N(t)} generating the demand events. For ease we suppose that F has a probability density f. The state of the system is only reviewed just after each Rth occurrence of a demand event. Here the review parameter R is a prespecified positive integer. Assuming that excess demand is backlogged, the control of the system is based on the inventory position defined as the stock on hand minus backorders plus stock on order. The control rule is an ( s , S ) rule under which the inventory position is ordered up to the level S if at a review the inventory position is at or below s and no ordering is done otherwise. The lead time of each replenishment order is a nonnegative random variable L with given mean # ( L ) and standard deviation o(L). We assume that L is bounded and discrete-valued. The standard periodic review and continuous review inventory systems can be modelled as special cases within the above class of (s,S) inventory systems. Periodic review inventory system: Take a deterministic process N(t) with unit jumps at equidistant epochs t = 1,2 ..... i.e. the demand process is described by a sequence of (aggregated) demands occurring in periods t - 1,2 ..... The review of the inventory position is only at the beginning of each Rth period. Continuous reoiew inventory system: Take a Poisson process for the process { N(t)} and take R = 1, i.e. the successive times between demand epochs are independent random variables with a common exponential distribution and the inventory position is reviewed just after each demand transaction. In this paper it is assumed that the difference S-s indicating the order quantity is predetermined, e.g. by
H. C. Tijms, 14. Groenevelt / Simple approximationsfor the reorder point
177
the economic order quantity formula. We concentrate on the determination of the reorder point s under the following service level constraint: fraction of demand to be met directly from stock on hand >1 fl,
(1)
where fl is a specified number. We note that the analysis below applies equally well to different service measures to be discussed in Remark 1. We give a unifying but heuristic treatment for the determination of the reorder point in the class of ( s , S ) inventory systems considered. In order to obtain implementable results, we have to compromise between mathematical and practical standpoints. Of course, each approximate step in the analysis should be based on sound principles and after all the approximate results should be validated. Before deriving the approximations, we introduce the following notation. Denote by F ~k~ and f~k~ the probability distribution function and the probability density of the total demand at k consecutive demand epochs, k >t 1. Let/z k and o k be the mean and standard deviation o f f tk). Note that lak = k/~ 1 and o k = olvrk-. Also, define the review time as the time elapsed between two consecutive reviews and note that the total demand in the review time has probability distribution function F t R). To do the approximate analysis, we make the following assumptions for the ( s , S ) rules achieving the required service level:
Assumptions. (i) the probability that replenishment orders cross in time or arrive simultaneously is negligible, (ii) S - s is sufficiently large compared with the average demand /t R in the review time (say, S - s >/1.5#n), (iii) the stock on hand just after arrival of a replenishment order is positive except for a negligible probability. We now first analyse the service level of a given ( s , S ) rule. Define a replenishment cycle as the time elapsed between two consecutive arrivals of replenishment orders. By a standard result in renewal theory (cf. Ross [18]), we have that the fraction of demand not satisfied directly from stock on hand equals the average amount that goes short per replenishment cycle divided by the average demand per replenishment cycle. Using Assumption (iii), the average amount that goes short per cycle is'approximately equal to the average shortage present just prior to the arrival of a replenishment order. Hence the fraction of demand not satisfied directly from stock on hand = (the average shortage, present just prior to the arrival of a replenishment order)/(the average demand per replenishment cycle).
(2)
To determine the ratio in (2), we tag one of the replenishment orders. Define the random variables Z = the undershoot of the reorder point s at the review at which the tagged replenishment order is placed, = the total demand in the lead time of the tagged replenishment order. Now, we can easily give the denominator of the ratio in (2). Since the average demand per replenishment cycle is equal to the average order size, we have the average demand per replenishment cycle = S - s + E Z .
(3)
We need a tractable approximation for the distribution of Z. Therefore, note that in renewal-theoretic terms Z is the excess variable associated with the random variable representing the review time demand. Recalling that the review time demand has probability distribution function FtR) and using Assumption (ii), we have by a standard result in renewal theory that (cf. [18]), P r { Z < ~ x ) = t a - R'
f0x/
1-F'R'(y)}dy,
x>~O,
(4)
H.C Tijms, H. Groenevelt / Simple approximations for the reorder point
178
independently of S - s. This approximate result requires that o~/#~ is not extremely small, i.e. the review time demand is not nearly deterministic. Also,
EZ= (ok + # ~ ) / 2 # , .
(5)
To determine the numerator of the ratio in (2), we first ,~bserve that the inventory position just prior to the placing of the tagged order equals s - Z. Also, by Assumption (i), any replenishment order placed earlier than the tagged order has been arrived when the tagged order comes in, while no order placed later than the tagged order arrives earlier than the tagged order. Hence the stock on hand just prior to the arrival of the tagged replenishment order = s - Z - 4. Denoting by
h(x)
the probability density of Z + ~, we have
the average shortage present just prior to the arrival of a replenishment order =ff°(x-s)h(x)dx. In Appendix 1, it is shown after some manipulations that
f~(x-s)h(x)dx=(2#R)-'(f°°(x-s)Z~l(x)dx-f~(x-s)2~(x)dx),
(6)
where (x) = the probability density of the total demand in the lead time plus review time, 4 (x) = the probability density of the total demand in the lead time. Now, by the relations (2)-(3) and (5)-(6), we have approximately determined the service level for a given (s, S) policy. Thus the reorder point s achieving the required service level fl (cf. (1)) can approximately be determined by solving the equation
where in the right side of (7) the difference S - s is given. Note that the relation (7) involves the complete demand densities 77 and 4. In the next section we give solution methods for (7) by using 2-moments approximations. Now we discuss a possible simplification of (7). If the service level/3 is close to 1 and the demand pattern is relatively smooth, it can heuristically be argued that the relation (7) may approximately be simplified to f ~ ( x - s )2r/(x)dx = (1 - / 3 )2~tR( S - s + ( ok + #~ )/2#R ).
(8)
It was pointed out to us by Dr. Helmut Schneider that in the periodic review model the simplified relation (8) arises for the service measure defined as the ratio of the average existing backlog and the average demand per unit time, see [20]. It will intuitively be clear that the latter service measure is very close to the B-service measure for high service levels and smooth demand. Clearly the reorder point determined by (8) is larger than or equal to the reorder point determined by (7) and so we may in general expect that the service level of the (s,S) rule induced by (8) is larger than or equal to the service level of the (s,S) rule induced by (7). Our numerical investigations show that in particular for irregular (erratic) demand the relation (8) is too crude and leads to erroneous results for the service level; this fact was not recognized in the studies [19,20] for the periodic review (s,S) system. Also, we found that for the continuous review system the second term in the left side of (7) plays in general a much more crucial role than in the periodic review system. Detailed recommendations about the use of the equations (7) and (8) for the reorder point are given in Section 4. For the periodic review (s,S) inventory system with deterministic lead times the relations (7) and (8) were already obtained in Schneider [19,20] by using the asymptotic results in Roberts [17]. The above
H. C Tijms, 14. Groenevelt / Simple approximations for the reorder point
179
approach is simpler and more insightful and it allows for a unifying treatment of a general class of (s,S) inventory systems with stoichastic lead times including periodic review and continuous review systems. We conclude this section with two remarks about different service measures and the (s, S) inventory system in which excess demand is lost.
Remark 1. Different service measures Another widely used service measure requires that the probability of no shortage just prior to the arrival of a replenishment order is at least P1; this criterion indicates the fraction of orders for which expedition might be necessary. By the same analysis as above we find that under this service measure the reorder point s can approximately be determined from
f f (x-s)7(x)dx- f f (x-s)4(x)dx= 1 - p,. The Pi-service measure has the drawback that it does not indicate how often a stockout occurs per unit time. Therefore consider the service measure requiring that the average number of stockouts per unit time is at most P0. This service measure can easily be related to the Pl-service measure. To do this, we note that the average length of a replenishment cycle follows from the renewal-theoretic result that the average demand per unit time equals the average demand per cycle divided by the average lenght of a cycle. The average demand per cycle is given by (3), while the average demand per unit time equals v#~ where v is the long-run average number of demand events per unit time. Thus under the service constraint that the average number of stockouts per unit time is at most P0, the reorder point s can approximately be determined from
ff
ff
+
I
Remark 2. The lost-sales (s,S) inventory system In this Remark we briefly discuss the (s,S) inventory system in which excess demand is lost, cf. also [3,15]. The above results need only slight modifications for the lost-sales case. An examination of the above analysis shows that the average lost demand per replenishment cycle is approximately given by (6). Also, the average demand per cycle is approximately equal to the average lost demand per cycle plus the average order size that is approximately given by the right side of (3). Thus, to achieve that the fraction of demand to be met from stock on hand is at least/3, the reorder point s can approximately be determined from the relation (7) in which 1 - 13 is replaced by (1 - fl)/fl. Clearly, for fl close to 1 the backlogging and lost-sales systems will not differ significantly.
3. Two-moments approximations for the reorder point In practice it will often happen that only the first two moments of the demand densities 7 and 4 are available instead of the complete distributions. In this case one may solve (7) or (8) by fitting suitable probability densities to the demand densities ~ and 4 by matching the first two moments. In practice the normal and gamma densities are often used to fit the empirical demand densities, cf. also [4,5]. In this section we discuss these 2-moments approximations which are suited for routine use in practice. Before doing this, we give formulae for the first two moments of the demand densities 7 and 4. Denote by #(7) and o(7) the mean and the standard deviation of the density ~1. Similarly, for #(4) and 0(4). Routine calculations involving conditional expectations (cf. [18]) yield # ( 7 ) = (R+EN(L)}I~1,
o2(7)=Ro~+EN(L)o?+o2(N(L))#~,
where N(L) denotes the number of demand events in the lead time L of a replenishment order. The formulae for #(/~) and 0(4) follow by putting R -- 0 in the above formulae. In general we can invoke from
H.C Tijms, H. Groenevelt/ Simpleapproximationsfor the reorderpoint
180
renewal theory approximate expressions for EN(L) and o 2(N(L)) based on the first three moments of the interoccurrence times of the demand events and the first two moments of the lead time, e.g. cf. [23, p. 133]. We omit these general expressions, but we only give the formulae for the most important cases of periodic review and continuous review. In these cases we can easily give exact expressions for EN(L) and a 2(N(L)), since the process ( N ( t ) ) is deterministic for periodic review and is a Poisson process for continuous review. It is no restriction to assume that the lead time L is integer valued for the case of periodic review. We have for the periodic review model
= (R +
o2(,7) = Ro? + ELo? +
whereas for the continuous review model
o2(~)=o~+XELo~+(X2oZ(L)+XEL)#~,
# ( ~ ) = (1 + X E L ) , , ,
with ~, is the occurrence rate of demand transactions. In both models we have ~t(~j)= # ( r l ) - Rp,1 and o2(~) = o2(-q)- Ro~ (note R = 1 for the continuous review model). We now turn to 2-moments approximations for the reorder point. First, we discuss the simplified normal approximation which is obtained by solving the simplified equation (8) in which a normal density is fitted to the demand density rl(x) by matching the first two moments. Clearly, a normal fit can only be used when the coefficient of variation o(~)/~t(~l) is small (say, o(~)/~t(~)~< ½), since negative demand values cannot occur. Let
q ,,( x ,) = - - ~1e -
,'/2
and
~(x)= fx
q~(u)du
(9)
be the probability density and the probability distribution function of the standard normal distribution. Also, define
I(x)= f~(uWe note that
x)ep(u)du
and
J(x)= f~°(u-x)2ep(u)du.
(10)
(cf. [12]),
I(x)=¢(x)-x{1-*(x)}
and
J(x)=(l
+ x2){1-*(x)}-x¢(x).
(11)
Now, if we fit a normal density to the demand density ~ ( x ) by matching the first two moments and use the representation s =#(~) +ko(~),
(12)
then the simplified equation (8) reduces to
o2(
)JIk) =y0,
(13)
where Y0 = (1 - fl)2txR { S - s + (oa2 + #2a ) / 2 # R }.
(14)
The easiest way to compute the safety factor k and so the reorder point s is to approximate the inverse function k = J - 1 ( y o) by a rational function. From [20], we have
k=
a o + alw + a2w 2 + a3w 3 +,(w), bo + blw + b2w2 + b3w3
(15)
where for the case of Y0 ~< 0.5 2
w=(ln(1/y6)) a 3 --- 0,
1/2
b0= 1,
,
a 0 = -0.4188413, b a = 0.2134080,
a I = - 0.2554696, b 2 = 0.04439934,
a 2 ----0.5189103, b 3 = - 0.002639787,
H.C Tijms, H. Groenevelt / Simple approximationsfor the reorder point
181
while for the case of Y0 > 0.5, w -- Y0,
a 0 -- 1.125946,
al = - 1.319002,
b 0 = 1,
b 1 = 2.836738,
b 2 = 0.6559378,
a 2 = - 1.809643, b3 =
a 3 = - 0.1165009,
0.008220435.
Here maxl{(w)l < 2.3 x 10 -4 for - 4 ~< k < 4. The computation of the simplified normal approximation by (12)-(15) is very easy and can routinely be done in practice. Next we discuss the computation of the normal approximation which is obtained by solving the relation (7) in which normal densities are fitted to the demand densities 71(x) and ~(x) by matching the first two moments. Therefore we first note that the equation (7) has a unique solution since the left side of (7) is decreasing in s and approaches 0 as s ~ ~ . Denoting by M ( s ) the left side of (7), we easily derive for the normal fit
while the derivative of M ( s ) is given by
M'(s)
=
-2o(7)1(s-
#(*/)
s
We can easily evaluate the function M ( s ) and its derivative by using the relation (11) and so we can effectively solve the equation (7) by the standard N e w t o n - R a p h s o n method. Since the function M ( s ) is convexly decreasing this iteration method converges for each starting point; as suggested in [19] a good starting point is the value s o determined by (12)-(15). Thus the normal approximation can be computed by the following algorithm:
Step O. Compute s o by (12)-(15). Let i ,= 0. Step 1. Given the current value s , compute the new value si÷ ~ from (with Y0 defined by (14)), s'+l '= si
(M(si) -Y0) M'(s,) "
using the relation (11) for the evaluation of the functions l ( x ) and J(x). The standard normal distribution function ~ ( x ) required in (11) could be calculated by using the approximation formula ~ ( x ) = 1 O(X)~.5=lCktk-st-C.(X) for x>~0 and O ( x ) = 1 - O ( - x ) for x~<0, where I{(x)[ < 7 . 5 x 10 -8, t = 1/(1 + ax), a = 0.2316419, c 1 = 0.319381530, c: = -0.356563782, c 3 = 1.781477937, c 4 = - 1.821255978 and c 5 = 1.330274429, cf. [1]. Step 2. If Is,+~ - si] ~< 0.1, then stop, otherwise i '-= i + 1 and go to Step 1. This algorithm converges very rapidly and in all cases tested it required at most 4 iterations. Also, this computational procedure for the normal approximation is easy to implement and suited for routine use. The normal demand distribution is typically adequate for representing the demand of fast-moving items with relatively smooth demand patterns. If the coefficient of variation of the demand in the lead time plus review time is larger than 0.5 (say), it becomes inadequate to fit normal demand distributions since negative demand cannot occur. The difficulty of negative demand is avoided when using g a m m a demand distributions. In m a n y practical inventory applications a g a m m a distribution fits well the empirical demand distribution, cf. also [5]. Also the g a m m a distribution is quite tractable for numerical calculations. For g a m m a densities the left side of (7) can be written in terms of incomplete g a m m a functions for which easily implementable numerical procedures are available. The incomplete g a m m a function is defined by
1 £
P( a,z ) = F( a---") e 't ~ 'at,
a > O,
where the term F ( a ) in the denominator is the complete g a m m a function, cf. [1,5]. The incomplete g a m m a
H. C Tijms, 14. Groenevelt / Simple approximations for the reorder point
182
function can easily be calculated from a series expansion whose positive terms can recursively be computed, see [1; 5 Appendix 2] for details. An excellent Fortran program for the incomplete gamma function is stated in [11]. If gamma densities are fitted to the demand densities 7(x) and ~(x) by matching the first two moments, we easily derive that the left side of (7) is given by M(s).= o 2 ( 7 ) ( a ( 7 ) + 1){1 - P(a(7) + 2,b(7)s)} - 2s~(7){1 - P(a(11) + 1,b(r/)s)} + s2{1 - P(a(7),b(7)s } - o2(l~)(a(l~) + 1)(1 - P(a(~) + 2,b(~j)s) } +2s~(~){1-P(a(~)+l,b(/~)s}-sZ{1-P(a(~),b(~)s)},
s>O,
where a(7) = 1~t2(7)/o2(7) and b(7) = ~(7)/o:(7). Similarly, for a(~j) and b(~). For s ~
= o2(7)
+
- o2(
) -
_ 2s {
-
}•
Moreover, the derivative of M(s) is given by
M'(s) = - 2 # ( 7 ) { 1 - P(a(7) + 1,b(~)s} + 2s{1 - P(a(~l),b(7)s)) +2#(~)(1-P(a(~)+l,b(l~)s}-2s(1-P(a(~),O(~)s)),
s>0.
From a numerical standpoint it is important to note that for the computation of M(s) and M'(s) we need only two evaluations of the incomplete gamma function in view of the recurrence relation (cf. [1]), + 1,z)
=
+ 1),
where F(a ÷ 1) = aF(a). Also, for the gamma fit equation (7) can effectively be solved by the standard Newton-Raphson method taking s o determined by (12)-(15) as starting point. The numerical procedure for the gamma approximation is less simple than the one for the normal approximation, but it is still tractable enough for practical purposes. None the less we give in Appendix 2 a simple approximate method to solve (7) for gamma distributed demand. We conclude this section by remarking that other popular demand distributions (cf. also [9]) for which (7) can effectively be solved by Newton-Raphson include the lognormal and the Weibull distributions. However, noting the role of the tail probabilities in (7), it should be pointed out that for larger values of the coefficients of variation of the demand densities 7(x) and ~(x) (i.e. for more erratic demand), the choice of the distributions to describe the empirical demand distributions gets more important, in particular when the required service level is close to 1.
4. Numerical results
In this section we discuss both for periodic review and continuous review (s,S) inventory systems with backlogging of excess demand the performance of the approximations for the B-service level defined as the fraction of demand to be met directly from stock on hand. We concentrate on the following approximations for the reorder point: (i) the simplified normal approximation (snor.) resulting from (8) in which a normal density is fitted to 7(x) by matching the first two moments, (ii) the normal approximation (norm.) resulting from (7) in which normal densities are fitted to 7(x) and ~(x) by matching the first two moments, (iii) the gamma approximation (gamma) resulting from (7) in which gamma densities are fitted to 7(x) and ~(x) by matching the first two moments, (iv) the true approximation (true) resulting from (7) in which the actual demand densities 7(x) and/j(x) are used. Also, denote by C~ = o(7)/E7 and C~ = o(~)/E~ the coefficients of variation of the demand densities 7(x) and ~j(x) respectively. Extensive numerical investigations reveal that in many practical inventory applica-
H.C Tijms, H. Groenevelt / Simple approximations for the reorder point
183
tions the following 2-moments approximations for the reorder point yield an excellent performance for the /3-service level.
Recommendationsfor 2-moments approximationsfor the reorderpoint (A) Periodic review (s,S) inventory system. For the case of Cn < 0.5, use the simplified normal approximation provided both fl >/0.90 and C~ - C, >/0.01, otherwise use the normal approximation. For the case of Cn > 0.5, use the gamma approximation. (B) Continuous review (s,S) inventory system. For the case of Cn ~<0.5, use the simplified normal approximation provided both fl >/0.97 and C~ - Cn >/0.01, otherwise use the normal approximation. For the case of Cn > 0.5, use the gamma approximation. We wish to make two side-notes to these recommendations. First, the approximations are not warranted when the ordering frequency #~1(S - s) is less than 1½ and when the probability of orders crossing in time is not small (~< ~ , say). The effect of possible cross-overs of orders diminishes if fl gets closer to 1. Secondly, the reorder point becomes increasingly sensitive to more than the first 2 moments of the demand densities ~(x) and ~(x) when the coefficients of variation C, and C~ get larger and when the service level fl gets very close to 1. This phenomenon for erratic demand was also observed in [2]; the insensitivity conjectured in [14] is typical for rather smooth demand patterns provided the service level is not very close to 1. Thus for erratic demand some care should be exercised in using 2-moments approximations, in particular when the service level fl is close to 1. To illustrate this, we include in the tables 2 and 3 the lognormal approximation (logn.) resulting from (7) in which lognormal densities are fitted to ~(x) and ~(x); the fact that the lognormal distribution has a longer tail then the gamma distribution is typically reflected in the reorder point. Clearly, for erratic demand and for very high service levels it is always safer to use the true approximation whenever possible. Nevertheless in many practical inventory applications with erratic demand, gamma distributions give an excellent fit to the empirical demand distributions (cf. also [5]) and the associated 2-moments approximation show a good performance for the service level. Also, for most items in inventory control the demand "patterns are relatively smooth so that for those items one can in general safely use 2-moments approximations based on normal demand densities. In the test examples below the probability density f(x) of the demand transaction sizes is given by a negative binomial distribution or a Poisson distribution. These discrete distributions have as possible values the nonnegative integers. The negative binomial distribution is completely specified by t~l and otz and requires o~/#1 > 1, while the Poisson distribution is completely specified by #1 and has the property o~/~1 = 1. Also, we consider the following two distributions for the lead time L of a replenishment order, (a) P r { L = 2 } = l (b) P r { L = I } = ¼ ,
(EL=2, o2(L)=O), Pr{L=2}=½, Pr{L=3}=¼
(EL= 2 , 0 2 ( L ) =½).
Although our primary goal is to test the service level of the approximate (s,S) rules, we assume in the examples a fixed setup cost K = 36 and linear holding cost h = 1. Thus in our test examples we predetermined S - s as the nearest integer to ~ (periodic review) and ~/2KXt~l/h (continuous review) respectively, where ~, is the occurrence rate of demand transactions. Also, for the 2-moments approximations the reorder point s was rounded to the nearest integer. For the true approximation, we used a discretized version of equation (7) and determined the reorder point as the largest integer s for which
(k-s)2,7(k) - E k~s
kss
with Y0 given by (14). To test the quality of the approximations we computed by an analytical method the actual value/3 (s,S) (say) of the service level of the approximate (s,S) rule, using the original (negative binomial or Poisson) distribution for the demand transaction sizes. We shall not discuss here this analytical method but refer to [8] for details. Also, for testing purposes, we have applied a Lagrangian method which
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searches for (but not necessarily finds) an (s, S)rule minimizing the average holding and setup costs within the class of (s,S) rules satisfying the t-service level constraint. In this Lagrangian method a fixed penalty cost ~r per unit shortage is introduced and is varied until its smallest value is found for which the associated average cost optimal (s,S) rule still satisfies the service level constraint, cf. [8] for details. Incidentally, it may be interesting to note that our numerical experiments support the empirical result that the average holding and setup costs achieved by choosing S - s equal to the economic order quantity are often quite close to the minimal average costs; e.g. in our test examples the relative difference percentage in costs of the true (s,S) rule and the Lagrangian (s,S) rule is typically between 0 and 4% and is occasionally as large as 10%. We note that the Lagrangian method and the analytical method to compute the exact service level are both exact up to neglecting the probability of orders crossing in time when the lead times are stochastic. In the test examples with stochastic lead times the actual values fl(s,S) for the service level have been validated by computer simulation. In the tables we give the various approximate (s,S) rules and the Lagrangian (s, S) rule (Lagr.) and the associated actual values fl(s, S) of the service level. The Tables 1 and 2 deal with the periodic review (s,S) inventory system, where in all examples we have chosen the review period R = 1. Table 3 deals with the continuous review (s,S) inventory system. It is indicated in the tables how the different parameters are varied.
Appendix
1. The d e r i v a t i o n of t h e b a s i c relation (6)
In this appendix we derive the relation (6). We first introduce some notation. For each t >/0, let
p~( t ) = Pr{ N( t ) = n }, n = 0,1 .... (withp0(0)
= 1,
by convention),
i.e. p~(t) is the probability that n demand events occur in (0,t]. Also, put for abbreviation ge(x) =
x>_.0.
Then, by (4), the probability density of the undershoot variable Z is approximately given by ge(x). Using the independence of the random variables Z and ~ and conditioning on the lead time L of the tagged order and on the number of demand events during this lead time, we find Pr{Z+~
Y~.Pr{Z + ~<~xIL=I}Pr{ L = I } 1
= Y~.Pr{L=I} Y'. P r { Z + ~ < ~ x l L = l , N ( I ) = n } p . ( I ) I
n>~O
= ]~Pr{L=/} I
[:o Po(I)
ge(y)dy+
E p.(I) n~l
::
Ft")(x-y)ge(Y)dy
The probability density h(x) of Z + ~j follows after differentiation (use thatF ~") (0)= O, n >/1),
h ( x ) = Y'.Pr{L=I I
'[
po(l)ge(x)+
Using the change of variable u = x - y
Y'.p.(l n~l
':0
f'")(x-y)g~(y)dy
]
, x>O.
in the latter integral and using the convolution formula
f ' " + ' ) ( x ) = foXf'")(u)f'm)(x-u)du,
n,m>: l,
]
.
H.C. Tijms, H. Groenevelt / Simple approximations for the reorderpoint
188
we find after differentiation
h'(x)
= • P r { L = l} [--po(I)g-Rlf(R)(X) 1 [
+ Y'. p,(l)ft")(x)ge(O) n>>.l
The first term between brackets in the last relation gives the probability density ~ ( x ) of the lead time plus review time demand, while the second term gives the probability density 4(x) of the lead time demand for x > 0 (note that 4(0)= E/Pr{ L -- l}Y..~op.(l)F(")(O), where F (°) (0) = 1 and F (') (0) -- 0, n >/1). Hence we can write
h'(x)= -g;~'[,7(x)-4(x)],
x>0.
Finally we obtain
f;(x s)h(x)d --
=
--
)2=
__ 1
f; ( X - - s ) 2 h . ' ( x ) d x
Here it is assumed that the demand d e n s i t y f ( x ) has a finite third moment so that lira x ~ ~x:'h ( x ) = 0. This completes the verification of (6).
E(Z
+ 4): < ¢¢ and so
Appendix 2. An approximate method for gamma distributed demand This appendix gives for gamma distributed demand an approximate method to solve the equation (7) avoiding the evaluation of incomplete gamma integrals. We can always approximate the gamma density by a simpler probability density having the same first two or three moments, cf. also [6]. Consider a gamma density with mean m and coefficient of variation C. Distinguish between 0 < C ~< 1 / v ~ and C > 1/vr2.
Case 1, 0 < C ~< 1 / v ~ . Then 1/k <~C: < 1/(k - 1) for a unique integer k >/2, and the following mixture of E k_l and E L densities with the same scale parameters has the same first two moments as the gamma density, xk-2
xk-1
k(x)=pb~-'--e-b"+(1-p)bk--e
-h', x>~O,
where
-11--[ k C : _ ( k ( l + C : ) _ k 2 C : ) , / 2 ]
k-p.,
P= 1+C 2
Case 2, C > 1 / v ~ . Then the following probability density gamma density, k(x)=pble-b''+(1-p)b2e -h2',
x>~0,
k(x)
has the same first
three moments
as the
189
H.C. Tijms, H. Groenevelt / Simple approximations for the reorder point Table A.1 The reorder points s G and SME (~: = 8 and L = 2) o]/~
5
8
10
15
25
40
SG,
SME
$G'
$ME
SG'
SME
$G'
SME
SG'
SME
SG"
$ME
0.90 0.95 0.99
26, 32, 44,
26 32 44
30, 38, 55,
30 38 55
34, 43, 63,
33 42 61
42, 54, 81,
41 54 82
57, 75, 117,
58 76 118
81, 108, 170,
82 108 169
(P)
0.90 0.95 0.99
31, 39, 58,
30 38 55
35, 45, 68,
35 45 68
38, 50, 75,
38 50 76
46, 60, 92,
46 61 95
61. 81, 126,
61 82 129
84, 112, 177,
85 113 178
(C)
where 2+ b12• = m -
2 { C. . .2. .- ½ } 1/2 I+C 2 and
m
=
p
b:(b2m-1) b2_bl
Note that b 1 > b2 > 0. Also, 0 ~
/1 and p < 0 otherwise. Further p ~ ~ as C 2 ~ ½. The above densities k(x) have the same unimodal shape as the gamma density. Now, consider the equation (7). By approximating the gamma densities .~(x) and ~(x) by the corresponding densities k(x), we can easily solve the resulting equation (7) by the Newton-Raphson iteration method where no incomplete gamma integrals have to be evaluated. For, we only have to compute cumulative Poisson probabilities, since for any integerj >/1 and a > 0,
f~ ~(x - s)2a: 1(j)x:-' ~_.
1) Qi+:(s) - ---~2sj Qj(s) +s2Qj l(s), e-aXdx = j ( j +a-----S---
(A.1)
where Qi(s)= E i~=o e-,,,~(as)~/k.W, while the derivative of the left side of (A.1) is given by 2sQ:_ l ( s ) (2 j/a) Q: (s). Note that the Poisson probability q/(s) = e-"S(as ) J/j ! can recursively be computed by using
%(s) = (as/j)qi_ l(s). Excellent results are obtained by the approximate method which solves (7) by approximating the gamma demand densities 7/(x) and ~(x) by the mixtures of Erlang (exponential) densities. In Table A.1 we give for a number of examples from Tables 2 and 3 the reorder points s c and SME obtained by solving the equation (7) with the gamma densities and the mixtures of Erlang densities respectively. We denote by ( P ) and (C) the respective examples for periodic and continuous review.
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H.C. TOms, H. Groenevelt / Simple approximationsfor the reorderpoint
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