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Stochastic leadtimes in a one-warehouse, N-retailer inventory system with the warehouse not carrying stock
European Journal of Operational Research 181 (2007) 1004–1013 www.elsevier.com/locate/ejor
Short Communication
Stochastic leadtimes in a one-warehouse, N-retailer inventory system with the warehouse not carrying stock Adriano O. Solis
a,*
, Charles P. Schmidt
b
a
b
The University of Texas at El Paso, Department of Information and Decision Sciences, 500 West University Avenue, El Paso, TX 79968-0544, United States The University of Alabama, Department of Information Systems, Statistics and Management Science, Tuscaloosa, AL 35487 Received 18 August 2005; accepted 21 July 2006 Available online 24 October 2006
Abstract This study extends upon a multi-echelon inventory model developed by Graves, introducing in the one-warehouse, Nretailer case—as Graves suggested—stochastic leadtimes between the warehouse and the retail sites in place of the original deterministic leadtimes. Effects of stochastic leadtimes on required base stock levels at the retail sites in the case where the warehouse carries no stock (e.g., serves as a cross-dock point) were investigated analytically. Two alternative treatments of stochastic leadtime distributions were considered. Using as a baseline Graves’ computational study under deterministic leadtimes, results of the current study suggest that it may be better to use the deterministic model with an accurately estimated mean leadtime than a stochastic model with a poorly estimated mean leadtime. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Inventory; Multi-echelon inventory system; Stochastic leadtimes; Fixed replenishment intervals
1. Introduction Graves (1996) introduced a multi-echelon inventory model assuming what he refers to as virtual allocation. Each site follows a base stock (or ‘‘order-up-to’’) policy. Focusing on a two-echelon system involving a central warehouse (CW) and N retailers, he specifies search procedures for the optimal combination of base stock levels at the CW and the retail sites minimizing the average system onhand inventory under each of two service level crite* Corresponding author. Tel.: +1 915 747 7757; fax: +1 915 747 5126. E-mail address: [email protected] (A.O. Solis).
ria: a probability of no stockout criterion and a fill rate criterion. He then undertakes a computational study involving 64 test cases for each criterion. In addition to making a number of other observations, Graves makes the assessment that the benefit of going from a policy with the CW not carrying stock to the optimal policy (where the CW base stock level is positive) in terms of the reduction in average system inventory—which Eppen and Schrage (1981) refer to as a depot effect—‘‘seems fairly small except when there are many retailers.’’ This observation suggests that in many instances a policy where the CW carries no stock (e.g., as in BarnesSchuster and Bassok, 1997; Mitra and Chatterjee, 2004) may as well be adopted. This case results in
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.07.008
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
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Table 1 Summary of test scenarios Scenario
Length of CW order cycle h1
External supplier leadtime s1
CW to retailer leadtime sj
No. of retail sites N
Retail site demand rate kj
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2
1
1
2
1
5
5
4
1
5
4
5
18 6 3 2 18 6 3 2 18 6 3 2 18 6 3 2
2 6 12 18 2 6 12 18 2 6 12 18 2 6 12 18
what Eppen and Schrage (1981) refer to as a joint ordering effect. It is consistent with the logistics technique known as cross-docking at distribution centers, a strategy made famous by Wal-Mart Stores, Inc. Under cross-docking, goods are shipped by suppliers to a distribution center, immediately broken down into smaller quantities and consolidated into truckloads of goods coming from various suppliers, and accordingly dispatched to retail stores without ever being held in inventory at the distribution center (Stalk et al., 1992; Simchi-Levi et al., 2003; Chopra and Meindl, 2004). In the current study, Graves’ model is extended by introducing stochastic leadtimes between the CW and the retail sites in place of the original deterministic leadtimes. It investigates analytically the effects of stochastic leadtimes on appropriate base stock levels at the retail sites in the case where the CW carries no stock. The search procedures pertaining to certain stochastic leadtime distributions are mathematically established, for each of the two service level criteria. Finally, a computational study (using the same 64 test cases Graves used for each service criterion) is undertaken, and the resulting average system inventory levels under stochastic leadtimes are compared with those under deterministic leadtimes. Graves’ model is relatively complex, and so are his mathematical results. Since the current study extends Graves’ model, we request the reader to refer to the original article (Graves, 1996). In his computational study, Graves used test scenarios all based on independent Poisson demand at the
retail sites with a single system demand rate k1 = 36. Identical retail sites are assumed, with the number N of retail sites being 2, 3, 6, or 18. Accordingly, the retail site demand rates kj are 18, 12, 6, or 2, respectively. The length of the retail site order cycle is fixed at hj = 1. Four different parameter combinations hh1, s1, sji, involving the length of the CW order cycle (h1), the leadtime (s1) from the external supplier to the CW, and the leadtime (sj) from the CW to the retail site, are tested. This resulted in 16 test scenarios, summarized in Table 1. For the probability of no stockout service criterion, four different levels of a were used: 0.80, 0.90, 0.95, and 0.975. Similarly, four different fill rate levels b were tested: 0.95, 0.98, 0.99, and 0.999. As a result, for each service criterion, a total of 64 test cases were utilized by Graves. We use the same test cases in our own computational studies. Section 2 discusses the introduction of stochastic leadtimes into Graves’ model and the results of our computational studies. Finally, we conclude the paper and propose some possible avenues for future research. 2. Stochastic leadtimes Graves (1996) suggested a number of possible extensions of his model, including stochastic leadtimes sj from the CW to the retail sites. The current study investigates effects of stochastic leadtimes sj from the CW to the retail sites when the CW carries no stock (B1 = 0)—in which case it takes the role simply of an order processing/coordinating
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center—under either a probability of no stockout criterion or a fill rate criterion. In the deterministic leadtime case, tr = s1 + h1 + sj is a constant and demand over the time interval (0, tr], Dj(0, tr), is a Poisson random variable with discrete density function (d.d.f.) f ðx; kj ½s1 þ h1 þ sj Þ 1 ¼ ½kj ðs1 þ h1 þ sj Þx expðkj ½s1 þ h1 þ sj Þ x! I f0;1;2;...g ðxÞ:
ð1Þ
We now consider the situation where sj is no longer preset and known. This is a departure from the original deterministic leadtime sj in Graves’ model, as in many others in the literature on multi-echelon inventory systems—and, particularly more so, from the zero leadtime assumed in some other models (e.g., McGavin et al., 1993; Nahmias and Smith, 1994). A zero leadtime may be plausible where regularly scheduled deliveries from the CW are made overnight while the retail sites are closed. Graves (1996) likewise assumes regularly scheduled shipments. He offers the motivation that, in a multi-item distribution system where each item occupies only a portion of a truckload, a fixed replenishment schedule allows consolidation of item shipments and, accordingly, transportation economies. This scenario may appear to allow a fixed, common positive leadtime for shipments between the CW and the retail site—where one truck services one retail site within the latter’s order cycle. But, while the time of the mth order occasion, pj(m), remains fixed, the time at which the mth order is received by site j, rj(m), may vary. Hence, the leadtime sj = rj(m) pj(m) may be treated as a random variable that varies according to a number of possible factors, for example, truck, road, and weather conditions, or even loading times for the consolidated, multi-item shipments. We assume the realized time of receipt rj(m) and, as a result, the leadtime sj do not depend on the demand process or the quantity ordered of the specific item under consideration. For Graves’ original results to continue to apply in the stochastic leadtime case, the condition that orders do not cross—that is, rj(m1) 6 rj(m) for any m—will need to hold. Our treatment here of sj is based on hj = 1 (i.e., a retail site order cycle length equal to one time unit), as used in all 16 test scenarios in Graves’ computational study, but may be extended to other possible
values of hj. We wish to ensure that the distribution of leadtimes sj satisfies the requirement that there is ‘‘no order crossing’’—specifically, that rj(m) < rj(m + 1) for any m. This requires the range of leadtimes (maximum leadtime less minimum leadtime) to be at most hj. In our treatment, therefore, we require this range of leadtimes to be at most 1. Such a requirement is less restrictive (with respect to the variability of leadtimes) when the mean leadtime is small relative to hj, and becomes more restrictive otherwise. Relative to a mean leadtime of 1, leadtimes between 0.5 and 1.5, with a range equal to 1, offer a fair amount of variability (up to 50% below or above the mean). On the other hand, relative to a mean leadtime of 5, leadtimes between 4.5 and 5.5, also with a range equal to 1, would offer less variability (only up to 10% below or above the mean). In his computational study, Graves uses the fixed leadtimes sj = 1 and sj = 5. Corresponding to Graves’ deterministic leadtime sj = 1, we may, for instance, consider leadtimes that vary within the interval between 0.5 and 1.5, or between 0.25 and 1.25, or between 0.8 and 1.8, among other such possibilities for which the range is 1 (equal to hj) and sj = 1 is included between the minimum and maximum leadtimes. [The range could be some other value
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
Mart’s stores are replenished twice a week, on average (Simchi-Levi et al., 2003). In our analysis, we shall use a leadtime interval of (0.5,1.5) arbitrarily, in place of the deterministic leadtime sj = 1. The analysis will essentially proceed in the same manner regardless of the final interval that actually applies. Similarly, the interval (4.5,5.5) is used in place of the deterministic leadtime sj = 5. Likewise, regardless of the original deterministic leadtime (be it sj = 1 or sj = 5), the analysis will proceed in the same manner. Within this interval of (0.5,1.5) or (4.5,5.5), we shall look into various stochastic leadtime distributions and their effects on required base stock levels. For example, corresponding to Graves’ fixed leadtime sj = 1, we see that the uniform distribution sj unif(0.5,1.5), having the simple probability density function (p.d.f.) gðsj Þ ¼ I ð0:5;1:5Þ ðsj Þ;
ð2Þ
would satisfy the no order crossing requirement. In this case, the mean of sj is equal to 1. A more general leadtime distribution within the interval (0.5,1.5) is one in which sj 0.5 beta(a, b), where the p.d.f. of sj is specified by
Z
1007
1:5
PrfDj ð0; tr Þ 6 Bj jsj ggðsj Þdsj ) Z 1:5 (X Bj ¼ f ðx; kj ½s1 þ h1 þ sj Þ gðsj Þdsj
0:5
0:5
¼
x¼0
Bj Z 1:5 X x¼0
f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj
P a;
0:5
ð4Þ which R 1:5 requires evaluation of a finite sum of integrals f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj , with x = 0, 1, . . . , 0:5 Bj. R 1:5 The integral 0:5 f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj is a mixture distribution [refer, for instance, to Johnson and Kotz (1985)] of Poisson random variables having the parameter kj(s1 + h1 + sj), with the distribution of sj as mixing distribution. In the present case, the leadtime sj—used in specifying the Poisson parameter kj(s1 + h1 + sj)—is itself a random variable defined over the interval (0.5,1.5) and having p.d.f. g(sj). If sj unif(0.5,1.5), with p.d.f. given by (2), Z 1:5 f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj 0:5
1 ðsj 0:5Þa1 gðsj Þ ¼ Bða; bÞ ð1:5 sj Þb1 I ð0:5;1:5Þ ðsj Þ;
¼ ð3Þ
with the so-called beta function Bða; bÞ ¼ CðaÞCðbÞ .A CðaþbÞ random variable with a beta(a, b) distribution varies a between 0 and 1, and has a mean of aþb . This distribution is skewed to the right or left depending upon a a whether aþb < 0:5 or aþb > 0:5. The mean of sj, a when sj 0.5 beta(a, b), is equal to aþb þ 0:5 – and can be anywhere on the interval (0.5,1.5). In this case, the distribution of leadtimes sj would be skewed to the right or left when the mean of sj is less than 1 or greater than 1, respectively. The latter case is more general than the former case—with sj 0.5 beta(1,1) being equivalent to sj unif(0.5,1.5), and (3) reducing to (2). 2.1. Probability of no stockout as service criterion
1 ½PrfPðkj ½s1 þ h1 þ 0:5Þ 6 xg kj PrfPðkj ½s1 þ h1 þ 1:5Þ 6 xg;
where P(g) denotes a Poisson random variable with parameter g. [For a uniform leadtime distribution in place of a fixed leadtime sj = 5—i.e., where sj unif(4.5,5.5), with p.d.f. g(sj) = I(4.5,5.5) (sj)—condition (4) and Eq. (5) are simply modified by replacing occurrences of the constants 0.5 and 1.5, respectively, with 4.5 and 5.5.] For the more general case where sj 0.5 beta(a, b), if a and b are integers, it can be shown, using a series of two binomial expansions that Z 1:5 f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj 0:5
¼
x X b1 X s¼0
Let us consider the probability of no stockout service criterion. Since the leadtime sj is stochastic with p.d.f. g(sj), Graves’ specification of the probability of not stocking out (for the deterministic leadtime case) translates into
ð5Þ
ð1Þq expðkj cÞ
q¼0
Cðq þ s þ aÞ 1 ½1 PrfPðkj Þ s! kjqþa < q þ s þ ag
ðckj Þxs Cða þ bÞ : ðx sÞ! q!ðb 1 qÞ!CðaÞ ð6Þ
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A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
where c = s1 + h1 + 0.5 and P(g) denotes a Poisson random variable with parameter g. [In the case R 5:5 where sj 4.5 beta(a, b), 4:5 f ðx;kj ½s1 þ h1 þ sj Þ gðsj Þdsj also reduces to the same double summation expression as in (6), with c = s1 + h1 + 4.5.] Given a policy where the CW carries no stock (B1 = 0), under the probability of no stockout service criterion the search for the optimal base stock level Bj at the retail sites, requires looking for the smallest Bj that satisfies condition (4)—with the use of either (5) or (6), as the case may be.
ðBj kj ½s1 þ h1 þ sj Þgðsj Þdsj 4:5
¼ Bj kj ðs1 þ h1 þ
a þ 4:5Þ: aþb
ð10Þ
for sj unif(4.5,5.5), the integral R[Similarly, 5:5 ðB k ½s þ h1 þ sj Þgðsj Þdsj reduces to Bj j j 1 4:5 kj(s1 + h1 + 5).] Using (9), we may rewrite (8) in the form Z 1:5 E½fAj ðs1 þ h1 þ sj Þg gðsj Þdsj 0:5
Bj X x¼0
In investigating the fill rate service criterion, on the other hand, we continue to apply the heuristic that Graves uses, which approximates (for ‘‘reasonable’’ fill rates) the expected backorders over the CW order cycle, using the expected backorders at (the point in time just before) tr = s1 + h1 + sj. The requirement derived by Graves, applicable to the fill rate criterion in the deterministic leadtime case, transforms into
ðBj xÞ
Z
1:5
0:5
f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj
B j þ kj s 1 þ h1 þ
a þ 0:5 : aþb
ð11Þ
Given a policy where the CW carries no stock (B1 = 0), the search for the optimal base stock level Bj at the retail sites, under the fill rate service criterion, requires looking for the smallest Bj that satisfies condition (7)—with the use of (11), and either (5) or (6), as the case may be. 2.3. Our computational study
1:5
E½fAj ðs1 þ h1 þ sj Þg gðsj Þdsj 6 ð1 bÞkj h1 ;
0:5
ð7Þ where Aj(t) denotes the available inventory at j at time t. With B1 = 0, we obtain Z 1:5 E½fAj ðs1 þ h1 þ sj Þg gðsj Þdsj 0:5
¼
Z
1:5
(
0:5
Z
Bj X
) ½ðBj xÞf ðx; kj ½s1 þ h1 þ sj Þ gðsj Þdsj
x¼0 1:5
fBj kj ðs1 þ h1 þ sj Þggðsj Þdsj :
ð8Þ
0:5
In the general case where sj 0.5 beta(a, b), Z
5:5
¼
2.2. Fill rate as service criterion
Z
Z
1:5
ðBj kj ½s1 þ h1 þ sj Þgðsj Þdsj
0:5
¼ Bj kj ðs1 þ h1 þ
a þ 0:5Þ: aþb
ð9Þ
particular, for sj unif(0.5,1.5), the integral R[In 1:5 ðBj kj ½s1 þ h1 þ sj Þgðsj Þdsj reduces to Bj 0:5 kj(s1 + h1 + 1).] For sj 4.5 beta(a, b), we have
We computed the required base stock levels Bj for the probability of no stockout criterion—for each of the four values of a (0.80, 0.90, 0.95, and 0.975) and each of the 16 test scenarios—corresponding to a number of stochastic leadtime distributions. We used beta distributions with (i) a = b = 1 (i.e., a uniform distribution), (ii) a = 6 and b = 2, and (iii) a = 2 and b = 6. The means and standard deviations of the various stochastic leadtime distributions (associated with deterministic leadtimes sj = 1 and sj = 5, respectively) are summarized in Table 2. Programs were written in C to compute—for each service criterion—the retail site base stock levels Bj that would minimize thePaverage system N inventory as approximated by 1 Bj 0:5k1 h1 k1 s 1 . Table 3 presents the computed required retailer base stock levels for the 16 test scenarios when a = 0.95, and provides a summary of the changes in approximate average system inventory. This table shows that when sj 0.5 beta(1,1) or sj 4.5 beta(1,1), the mean increase in average system inventory across the 16 test scenarios is 1.0%. The corresponding mean increases are 0.6%, 0.8%, and 1.2% when a = 0.80, 0.90, and 0.975, respectively, and the
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
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Table 2 Stochastic leadtime distributions in computational study Leadtime distribution
Mean of beta(a, b)
Interval of sj
Mean of sj
Standard deviation of sj
sj 0.5 beta(2,6) sj 0.5 beta(1,1) sj 0.5 beta(6,2)
0.25 0.50 0.75
(0.5, 1.5) (0.5, 1.5) (0.5, 1.5)
0.75 1.00 1.25
0.144 0.289 0.144
sj 4.5 beta(2,6) sj 4.5 beta(1,1) sj 4.5 beta(6,2)
0.25 0.50 0.75
(4.5, 5.5) (4.5, 5.5) (4.5, 5.5)
4.75 5.00 5.25
0.144 0.289 0.144
overall mean increase across all 64 test cases is 0.9%. When sj 0.5 beta(6,2) or sj 4.5 beta(6,2), as seen in Table 3, the mean increase in average system inventory is 4.2% when a = 0.95. The corresponding mean increases when a = 0.80, 0.90, and 0.975 are 5.0%, 4.0%, and 4.3%, respectively. Across all 64 test cases, the overall mean increase is 4.4%. Table 3 further shows that, when sj 0.5 beta(2,6) or sj 4.5 beta(2,6), the mean decrease in average system inventory when a = 0.95 is 4.5%. The mean decreases are 3.9%, 4.5%, and 3.6% when a = 0.80, 0.90, and 0.975, respectively, and the overall mean decrease across all 64 test cases is 4.1%. Table 4, on the other hand, reports the required retailer base stock levels under the fill rate criterion with b = 0.99 for the 16 test scenarios, and provides a summary of the corresponding changes in estimated average system inventory. When sj 0.5 beta(1,1) or sj 4.5 beta(1,1), the mean increase in average system inventory is 1.0%. The corresponding mean increases when b = 0.95, 0.98, and 0.999 are 0.4%, 0.9%, and 1.5%, respectively, and the overall mean increase across all 64 test cases is 1.0%. For sj 0.5 beta(6,2) or sj 4.5 beta(6,2), Table 4 shows that the mean increase in average system inventory when b = 0.99 is 5.1%. When b = 0.95, 0.98, and 0.999, the corresponding mean increases are 3.8%, 4.5%, and 4.0%, respectively, and the overall mean increase is 4.4% for all 64 test cases. For sj 0.5 beta(2,6) or sj 4.5 beta(2,6), Table 4 reports a mean decrease of 3.7% in average system inventory when b = 0.99. Corresponding mean decreases are 5.0%, 4.0%, and 3.6% when b = 0.95, 0.98, and 0.999, respectively. The overall mean decrease across all 64 test cases is 4.0%. The assumption had been taken that stochastic leadtimes sj vary within some observed interval— such as (0.5,1.5) or (4.5,5.5)—containing the erstwhile deterministic leadtime value. In using the required retailer base stock level for the determinis-
tic leadtime case as a baseline for comparison, we noted that this baseline level would need to be adjusted in view of two possible factors: (i) a mean different from the deterministic leadtime value; and (ii) variability within the observed leadtime interval. When the leadtime distribution is skewed to the right or to the left within this interval—e.g., sj 0.5 beta(2,6) or sj 0.5 beta(6,2) within the interval (0.5,1.5)—a decrease or an increase in the baseline level of Bj arises in most test cases. However, when the mean of the distribution is equal to the deterministic leadtime value—e.g., sj 0.5 beta(1,1), which is equivalent to the uniform distribution sj unif(0.5,1.5)—and only the variability within the observed leadtime interval needs to be addressed, increases in required base stock levels apply in just a little more than half of the test cases. 2.4. Alternative treatment of leadtime distributions An alternative treatment of stochastic leadtimes would be one in which leadtimes sj vary according to some distribution, but with a mean equal to the value assumed in the deterministic leadtime case. Under such treatment, various distributions of sj would not vary within the same preset interval of observed leadtimes. Instead, the interval of leadtimes would differ depending upon the actual distribution, although the range of leadtimes (i.e., the difference between maximum and minimum leadtimes) is still limited to be at most hj in line with the no order crossing requirement. Under this alternative treatment, if the range of leadtimes is 1, then the distribution may take the leadtime a a form sj 1 aþb betaða;bÞ or sj 5 aþb
betaða;bÞ, depending upon whether the deterministic leadtime is 1 or 5, respectively. After conducting a computational study similar to that reported in Section 2.3, we were able to
1010
sj = 1 or sj = 5
sj 0.5 beta(1,1) or sj 4.5 beta(1,1)
sj 0.5 beta(6,2) or sj 4.5 beta(6,2)
sj 0.5 beta(2,6) or sj 4.5 beta(2,6)
N
Computed Bj
Average inventory
Computed Bj
Average inventory
D Average inventory (%)
Computed Bj
Average inventory
D Average inventory (%)
Computed Bj
Average inventory
D Average inventory (%)
18 6 3 2 18 6 3 2 18 6 3 2 18 6 3 2
13 32 60 86 23 60 112 164 28 73 138 202 37 99 190 278
162 120 108 100 342 288 264 256 270 204 180 170 432 360 336 322
13 33 61 89 23 60 113 166 28 73 139 204 37 100 190 280
162 126 111 106 342 288 267 260 270 204 183 174 432 366 336 326
0.0 5.0 2.8 6.0 0.0 0.0 1.1 1.6 0.0 0.0 1.7 2.4 0.0 1.7 0.0 1.2
14 34 63 92 23 61 116 169 28 75 142 208 38 101 193 283
180 132 117 112 342 294 276 266 270 216 192 182 450 372 345 332
11.1 10.0 8.3 12.0 0.0 2.1 4.5 3.9 0.0 5.9 6.7 7.1 4.2 3.3 2.7 3.1
12 31 57 82 22 58 109 160 27 71 135 198 36 98 187 274
144 114 99 92 324 276 255 248 252 192 171 162 414 354 327 314
11.1 5.0 8.3 8.0 5.3 4.2 3.4 3.1 6.7 5.9 5.0 4.7 4.2 1.7 2.7 2.5
247.06
1.0
Scenario
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Mean
244.63
254.88
4.2
233.63
4.5
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
Table 3 Computed required retailer base stock levels under various leadtime distributions when a = 0.95
sj = 1 or sj = 5
sj 0.5 beta(1,1) or sj 4.5 beta(1,1)
N
Computed Bj
Average inventory
Computed Bj
Average inventory
D Average Computed inventory (%) Bj
Average inventory
D Average Computed inventory (%) Bj
Average inventory
D Average inventory (%)
18 6 3 2 18 6 3 2 18 6 3 2 18 6 3 2
14 33 59 84 25 61 112 163 28 72 134 196 38 98 186 272
180 126 105 96 378 294 264 254 270 198 168 158 450 354 324 310
14 33 61 87 25 62 114 165 28 72 135 197 38 99 186 273
180 126 111 102 378 300 270 258 270 198 171 160 450 360 324 312
0.0 0.0 5.7 6.3 0.0 2.0 2.3 1.6 0.0 0.0 1.8 1.3 0.0 1.7 0.0 0.6
198 138 117 108 396 306 276 264 288 204 180 168 468 366 333 320
10.0 9.5 11.4 12.5 4.8 4.1 4.5 3.9 6.7 3.0 7.1 6.3 4.0 3.4 2.8 3.2
180 114 96 88 360 282 258 246 270 186 159 148 432 348 315 300
0.0 9.5 8.6 8.3 4.8 4.1 2.3 3.1 0.0 6.1 5.4 6.3 4.0 1.7 2.8 3.2
248.13
1.0
236.38
3.7
Scenario
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Mean
245.56
sj 0.5 beta(6,2) or sj 4.5 beta(6,2)
15 35 63 90 26 63 116 168 29 73 138 201 39 100 189 277
258.13
5.1
sj 0.5 beta(2,6) or sj 4.5 beta(2,6)
14 31 56 80 24 59 110 159 28 70 131 191 37 97 183 267
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
Table 4 Computed required retailer base stock levels under various leadtime distributions when b = 0.99
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observe that changes in base stock levels from the deterministic case to the stochastic case, in many instances, either may not be required or may be fairly small if required, when mean leadtimes coincide with the deterministic leadtime value, and only the variability of leadtimes (as measured by standard deviation, for instance) is to be accounted for. It became apparent that the required changes in base stock levels under the earlier treatment of stochastic leadtimes (giving rise to variations in terms of both means and standard deviations) result mainly from the differences in means, except in the case where the uniform distribution applies. This observation suggests that it may be better to use the deterministic model with an accurately estimated mean leadtime than a stochastic model with a poorly estimated mean leadtime. 3. Conclusion and further research In this study, we extended Graves’ one-warehouse, N-retailer model by introducing, as Graves suggested, stochastic leadtimes sj between the CW and the retail sites in place of the original deterministic leadtimes. Effects of stochastic leadtimes on required base stock levels in the case where the CW carries no stock were investigated analytically. Mathematical models were derived for each of the two service criteria under investigation, along with computational forms for such models, to determine the appropriate retailer base stock level Bj. Two alternative treatments of stochastic leadtime distributions were considered. Under the first treatment, leadtimes are assumed to vary within some interval, with means of leadtime distributions being possibly different from the deterministic leadtime value. Adjustments to the required retailer base stock level (as computed for the deterministic leadtime case) address both the change in mean values and the variability of the stochastic leadtimes. Under the second treatment, the mean of the distribution is assumed to be equal to the deterministic leadtime value, so that an adjustment to the required level is brought about only by the variability of the stochastic leadtimes. Under the latter treatment, adjustments were not found to be necessary in many instances, or, if at all necessary, to result in fairly small increases in the base stock level and average system inventory. The latter observation would appear to suggest that it may be better to use the deterministic model with an accurately esti-
mated mean leadtime than a stochastic model with a poorly estimated mean leadtime. Where the CW carries stock (B1 > 0), a similar analytical investigation of the effects of stochastic leadtimes on the optimal base stock policy hB1, Bji does not appear possible. Extensive simulation studies are indicated in view of the nature of the search procedures for optimal levels of B1 and Bj. Admittedly, our assumption in this study that the range of stochastic leadtimes be at most hj, in order to ensure that orders do not cross, is restrictive. Less restrictive assumptions may be possible, for instance, by focusing on an order-arrival mechanism instead of the leadtime distribution, as Zipkin (1986) does. The expression tr = s1 + h1 + sj is crucial to the definition and analysis of uncovered demand in Graves’ model. We note that tr is a random variable that varies with the CW to retailer leadtime sj, when the latter is not deterministic but varies according to some leadtime distribution. Hence, the distribution of uncovered demand and, accordingly, the selection of a base stock policy that seeks to minimize average system inventory may be influenced by stochastic leadtimes sj. The same could be said about stochastic leadtimes s1 from the external supplier to the CW. Further, in investigating stochastic external leadtimes s1, it would be interesting to find out whether the model is more sensitive to randomness in s1 or to randomness in sj. References Barnes-Schuster, D., Bassok, Y., 1997. Direct shipping and the dynamic single-depot/multi-retailer inventory system. European Journal of Operational Research 101 (3), 509–518. Chopra, S., Meindl, P., 2004. Supply Chain Management: Strategy Planning and Operation. Prentice Hall, Upper Saddle River, NJ. David, F.R., 1999. Strategic Management: Concepts and Cases. Prentice Hall, Upper Saddle River, NJ. Eppen, G., Schrage, L., 1981. In: Schwarz, L.B. (Ed.), MultiLevel Production/Inventory Control Systems, Theory and Practice, TIMS Studies in the Management Sciences, Vol. 16. North Holland, Amsterdam, pp. 51–67. Graves, S.C., 1996. A multiechelon inventory model with fixed replenishment intervals. Management Science 42 (1), 1–18. Johnson, N.L., Kotz, S., 1985. Encyclopaedia of Statistical Sciences, Vol. 5. John Wiley and Sons, New York. McGavin, E.J., Schwarz, L.B., Ward, J.E., 1993. Two-interval inventory-allocation policies in a one-warehouse, N-identicalretailer distribution system. Management Science 39 (9), 1092–1107.
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013 Mitra, S., Chatterjee, A.K., 2004. Leveraging information in multi-echelon inventory systems. European Journal of Operational Research 152, 263–280. Nahmias, S., Smith, S.A., 1994. Optimizing inventory levels in a two-echelon retailer system with partial lost sales. Management Science 40 (5), 582–596. Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E., 2003. Designing and Managing the Supply Chain: Concepts Strategies and Case Studies. McGraw-Hill, New York, NY.
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Stalk, G., Evans, P., Shulman, L.E., 1992. Competing on capabilities: the new rule of corporate strategy. Harvard Business Review, 57–69, March–April 1992. Zipkin, P., 1986. Stochastic leadtimes in continuous-time inventory models. Naval Research Logistics Quarterly 33, 763–774.
Short Communication
Stochastic leadtimes in a one-warehouse, N-retailer inventory system with the warehouse not carrying stock Adriano O. Solis
a,*
, Charles P. Schmidt
b
a
b
The University of Texas at El Paso, Department of Information and Decision Sciences, 500 West University Avenue, El Paso, TX 79968-0544, United States The University of Alabama, Department of Information Systems, Statistics and Management Science, Tuscaloosa, AL 35487 Received 18 August 2005; accepted 21 July 2006 Available online 24 October 2006
Abstract This study extends upon a multi-echelon inventory model developed by Graves, introducing in the one-warehouse, Nretailer case—as Graves suggested—stochastic leadtimes between the warehouse and the retail sites in place of the original deterministic leadtimes. Effects of stochastic leadtimes on required base stock levels at the retail sites in the case where the warehouse carries no stock (e.g., serves as a cross-dock point) were investigated analytically. Two alternative treatments of stochastic leadtime distributions were considered. Using as a baseline Graves’ computational study under deterministic leadtimes, results of the current study suggest that it may be better to use the deterministic model with an accurately estimated mean leadtime than a stochastic model with a poorly estimated mean leadtime. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Inventory; Multi-echelon inventory system; Stochastic leadtimes; Fixed replenishment intervals
1. Introduction Graves (1996) introduced a multi-echelon inventory model assuming what he refers to as virtual allocation. Each site follows a base stock (or ‘‘order-up-to’’) policy. Focusing on a two-echelon system involving a central warehouse (CW) and N retailers, he specifies search procedures for the optimal combination of base stock levels at the CW and the retail sites minimizing the average system onhand inventory under each of two service level crite* Corresponding author. Tel.: +1 915 747 7757; fax: +1 915 747 5126. E-mail address: [email protected] (A.O. Solis).
ria: a probability of no stockout criterion and a fill rate criterion. He then undertakes a computational study involving 64 test cases for each criterion. In addition to making a number of other observations, Graves makes the assessment that the benefit of going from a policy with the CW not carrying stock to the optimal policy (where the CW base stock level is positive) in terms of the reduction in average system inventory—which Eppen and Schrage (1981) refer to as a depot effect—‘‘seems fairly small except when there are many retailers.’’ This observation suggests that in many instances a policy where the CW carries no stock (e.g., as in BarnesSchuster and Bassok, 1997; Mitra and Chatterjee, 2004) may as well be adopted. This case results in
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.07.008
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
1005
Table 1 Summary of test scenarios Scenario
Length of CW order cycle h1
External supplier leadtime s1
CW to retailer leadtime sj
No. of retail sites N
Retail site demand rate kj
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2
1
1
2
1
5
5
4
1
5
4
5
18 6 3 2 18 6 3 2 18 6 3 2 18 6 3 2
2 6 12 18 2 6 12 18 2 6 12 18 2 6 12 18
what Eppen and Schrage (1981) refer to as a joint ordering effect. It is consistent with the logistics technique known as cross-docking at distribution centers, a strategy made famous by Wal-Mart Stores, Inc. Under cross-docking, goods are shipped by suppliers to a distribution center, immediately broken down into smaller quantities and consolidated into truckloads of goods coming from various suppliers, and accordingly dispatched to retail stores without ever being held in inventory at the distribution center (Stalk et al., 1992; Simchi-Levi et al., 2003; Chopra and Meindl, 2004). In the current study, Graves’ model is extended by introducing stochastic leadtimes between the CW and the retail sites in place of the original deterministic leadtimes. It investigates analytically the effects of stochastic leadtimes on appropriate base stock levels at the retail sites in the case where the CW carries no stock. The search procedures pertaining to certain stochastic leadtime distributions are mathematically established, for each of the two service level criteria. Finally, a computational study (using the same 64 test cases Graves used for each service criterion) is undertaken, and the resulting average system inventory levels under stochastic leadtimes are compared with those under deterministic leadtimes. Graves’ model is relatively complex, and so are his mathematical results. Since the current study extends Graves’ model, we request the reader to refer to the original article (Graves, 1996). In his computational study, Graves used test scenarios all based on independent Poisson demand at the
retail sites with a single system demand rate k1 = 36. Identical retail sites are assumed, with the number N of retail sites being 2, 3, 6, or 18. Accordingly, the retail site demand rates kj are 18, 12, 6, or 2, respectively. The length of the retail site order cycle is fixed at hj = 1. Four different parameter combinations hh1, s1, sji, involving the length of the CW order cycle (h1), the leadtime (s1) from the external supplier to the CW, and the leadtime (sj) from the CW to the retail site, are tested. This resulted in 16 test scenarios, summarized in Table 1. For the probability of no stockout service criterion, four different levels of a were used: 0.80, 0.90, 0.95, and 0.975. Similarly, four different fill rate levels b were tested: 0.95, 0.98, 0.99, and 0.999. As a result, for each service criterion, a total of 64 test cases were utilized by Graves. We use the same test cases in our own computational studies. Section 2 discusses the introduction of stochastic leadtimes into Graves’ model and the results of our computational studies. Finally, we conclude the paper and propose some possible avenues for future research. 2. Stochastic leadtimes Graves (1996) suggested a number of possible extensions of his model, including stochastic leadtimes sj from the CW to the retail sites. The current study investigates effects of stochastic leadtimes sj from the CW to the retail sites when the CW carries no stock (B1 = 0)—in which case it takes the role simply of an order processing/coordinating
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center—under either a probability of no stockout criterion or a fill rate criterion. In the deterministic leadtime case, tr = s1 + h1 + sj is a constant and demand over the time interval (0, tr], Dj(0, tr), is a Poisson random variable with discrete density function (d.d.f.) f ðx; kj ½s1 þ h1 þ sj Þ 1 ¼ ½kj ðs1 þ h1 þ sj Þx expðkj ½s1 þ h1 þ sj Þ x! I f0;1;2;...g ðxÞ:
ð1Þ
We now consider the situation where sj is no longer preset and known. This is a departure from the original deterministic leadtime sj in Graves’ model, as in many others in the literature on multi-echelon inventory systems—and, particularly more so, from the zero leadtime assumed in some other models (e.g., McGavin et al., 1993; Nahmias and Smith, 1994). A zero leadtime may be plausible where regularly scheduled deliveries from the CW are made overnight while the retail sites are closed. Graves (1996) likewise assumes regularly scheduled shipments. He offers the motivation that, in a multi-item distribution system where each item occupies only a portion of a truckload, a fixed replenishment schedule allows consolidation of item shipments and, accordingly, transportation economies. This scenario may appear to allow a fixed, common positive leadtime for shipments between the CW and the retail site—where one truck services one retail site within the latter’s order cycle. But, while the time of the mth order occasion, pj(m), remains fixed, the time at which the mth order is received by site j, rj(m), may vary. Hence, the leadtime sj = rj(m) pj(m) may be treated as a random variable that varies according to a number of possible factors, for example, truck, road, and weather conditions, or even loading times for the consolidated, multi-item shipments. We assume the realized time of receipt rj(m) and, as a result, the leadtime sj do not depend on the demand process or the quantity ordered of the specific item under consideration. For Graves’ original results to continue to apply in the stochastic leadtime case, the condition that orders do not cross—that is, rj(m1) 6 rj(m) for any m—will need to hold. Our treatment here of sj is based on hj = 1 (i.e., a retail site order cycle length equal to one time unit), as used in all 16 test scenarios in Graves’ computational study, but may be extended to other possible
values of hj. We wish to ensure that the distribution of leadtimes sj satisfies the requirement that there is ‘‘no order crossing’’—specifically, that rj(m) < rj(m + 1) for any m. This requires the range of leadtimes (maximum leadtime less minimum leadtime) to be at most hj. In our treatment, therefore, we require this range of leadtimes to be at most 1. Such a requirement is less restrictive (with respect to the variability of leadtimes) when the mean leadtime is small relative to hj, and becomes more restrictive otherwise. Relative to a mean leadtime of 1, leadtimes between 0.5 and 1.5, with a range equal to 1, offer a fair amount of variability (up to 50% below or above the mean). On the other hand, relative to a mean leadtime of 5, leadtimes between 4.5 and 5.5, also with a range equal to 1, would offer less variability (only up to 10% below or above the mean). In his computational study, Graves uses the fixed leadtimes sj = 1 and sj = 5. Corresponding to Graves’ deterministic leadtime sj = 1, we may, for instance, consider leadtimes that vary within the interval between 0.5 and 1.5, or between 0.25 and 1.25, or between 0.8 and 1.8, among other such possibilities for which the range is 1 (equal to hj) and sj = 1 is included between the minimum and maximum leadtimes. [The range could be some other value
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
Mart’s stores are replenished twice a week, on average (Simchi-Levi et al., 2003). In our analysis, we shall use a leadtime interval of (0.5,1.5) arbitrarily, in place of the deterministic leadtime sj = 1. The analysis will essentially proceed in the same manner regardless of the final interval that actually applies. Similarly, the interval (4.5,5.5) is used in place of the deterministic leadtime sj = 5. Likewise, regardless of the original deterministic leadtime (be it sj = 1 or sj = 5), the analysis will proceed in the same manner. Within this interval of (0.5,1.5) or (4.5,5.5), we shall look into various stochastic leadtime distributions and their effects on required base stock levels. For example, corresponding to Graves’ fixed leadtime sj = 1, we see that the uniform distribution sj unif(0.5,1.5), having the simple probability density function (p.d.f.) gðsj Þ ¼ I ð0:5;1:5Þ ðsj Þ;
ð2Þ
would satisfy the no order crossing requirement. In this case, the mean of sj is equal to 1. A more general leadtime distribution within the interval (0.5,1.5) is one in which sj 0.5 beta(a, b), where the p.d.f. of sj is specified by
Z
1007
1:5
PrfDj ð0; tr Þ 6 Bj jsj ggðsj Þdsj ) Z 1:5 (X Bj ¼ f ðx; kj ½s1 þ h1 þ sj Þ gðsj Þdsj
0:5
0:5
¼
x¼0
Bj Z 1:5 X x¼0
f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj
P a;
0:5
ð4Þ which R 1:5 requires evaluation of a finite sum of integrals f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj , with x = 0, 1, . . . , 0:5 Bj. R 1:5 The integral 0:5 f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj is a mixture distribution [refer, for instance, to Johnson and Kotz (1985)] of Poisson random variables having the parameter kj(s1 + h1 + sj), with the distribution of sj as mixing distribution. In the present case, the leadtime sj—used in specifying the Poisson parameter kj(s1 + h1 + sj)—is itself a random variable defined over the interval (0.5,1.5) and having p.d.f. g(sj). If sj unif(0.5,1.5), with p.d.f. given by (2), Z 1:5 f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj 0:5
1 ðsj 0:5Þa1 gðsj Þ ¼ Bða; bÞ ð1:5 sj Þb1 I ð0:5;1:5Þ ðsj Þ;
¼ ð3Þ
with the so-called beta function Bða; bÞ ¼ CðaÞCðbÞ .A CðaþbÞ random variable with a beta(a, b) distribution varies a between 0 and 1, and has a mean of aþb . This distribution is skewed to the right or left depending upon a a whether aþb < 0:5 or aþb > 0:5. The mean of sj, a when sj 0.5 beta(a, b), is equal to aþb þ 0:5 – and can be anywhere on the interval (0.5,1.5). In this case, the distribution of leadtimes sj would be skewed to the right or left when the mean of sj is less than 1 or greater than 1, respectively. The latter case is more general than the former case—with sj 0.5 beta(1,1) being equivalent to sj unif(0.5,1.5), and (3) reducing to (2). 2.1. Probability of no stockout as service criterion
1 ½PrfPðkj ½s1 þ h1 þ 0:5Þ 6 xg kj PrfPðkj ½s1 þ h1 þ 1:5Þ 6 xg;
where P(g) denotes a Poisson random variable with parameter g. [For a uniform leadtime distribution in place of a fixed leadtime sj = 5—i.e., where sj unif(4.5,5.5), with p.d.f. g(sj) = I(4.5,5.5) (sj)—condition (4) and Eq. (5) are simply modified by replacing occurrences of the constants 0.5 and 1.5, respectively, with 4.5 and 5.5.] For the more general case where sj 0.5 beta(a, b), if a and b are integers, it can be shown, using a series of two binomial expansions that Z 1:5 f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj 0:5
¼
x X b1 X s¼0
Let us consider the probability of no stockout service criterion. Since the leadtime sj is stochastic with p.d.f. g(sj), Graves’ specification of the probability of not stocking out (for the deterministic leadtime case) translates into
ð5Þ
ð1Þq expðkj cÞ
q¼0
Cðq þ s þ aÞ 1 ½1 PrfPðkj Þ s! kjqþa < q þ s þ ag
ðckj Þxs Cða þ bÞ : ðx sÞ! q!ðb 1 qÞ!CðaÞ ð6Þ
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where c = s1 + h1 + 0.5 and P(g) denotes a Poisson random variable with parameter g. [In the case R 5:5 where sj 4.5 beta(a, b), 4:5 f ðx;kj ½s1 þ h1 þ sj Þ gðsj Þdsj also reduces to the same double summation expression as in (6), with c = s1 + h1 + 4.5.] Given a policy where the CW carries no stock (B1 = 0), under the probability of no stockout service criterion the search for the optimal base stock level Bj at the retail sites, requires looking for the smallest Bj that satisfies condition (4)—with the use of either (5) or (6), as the case may be.
ðBj kj ½s1 þ h1 þ sj Þgðsj Þdsj 4:5
¼ Bj kj ðs1 þ h1 þ
a þ 4:5Þ: aþb
ð10Þ
for sj unif(4.5,5.5), the integral R[Similarly, 5:5 ðB k ½s þ h1 þ sj Þgðsj Þdsj reduces to Bj j j 1 4:5 kj(s1 + h1 + 5).] Using (9), we may rewrite (8) in the form Z 1:5 E½fAj ðs1 þ h1 þ sj Þg gðsj Þdsj 0:5
Bj X x¼0
In investigating the fill rate service criterion, on the other hand, we continue to apply the heuristic that Graves uses, which approximates (for ‘‘reasonable’’ fill rates) the expected backorders over the CW order cycle, using the expected backorders at (the point in time just before) tr = s1 + h1 + sj. The requirement derived by Graves, applicable to the fill rate criterion in the deterministic leadtime case, transforms into
ðBj xÞ
Z
1:5
0:5
f ðx; kj ½s1 þ h1 þ sj Þgðsj Þdsj
B j þ kj s 1 þ h1 þ
a þ 0:5 : aþb
ð11Þ
Given a policy where the CW carries no stock (B1 = 0), the search for the optimal base stock level Bj at the retail sites, under the fill rate service criterion, requires looking for the smallest Bj that satisfies condition (7)—with the use of (11), and either (5) or (6), as the case may be. 2.3. Our computational study
1:5
E½fAj ðs1 þ h1 þ sj Þg gðsj Þdsj 6 ð1 bÞkj h1 ;
0:5
ð7Þ where Aj(t) denotes the available inventory at j at time t. With B1 = 0, we obtain Z 1:5 E½fAj ðs1 þ h1 þ sj Þg gðsj Þdsj 0:5
¼
Z
1:5
(
0:5
Z
Bj X
) ½ðBj xÞf ðx; kj ½s1 þ h1 þ sj Þ gðsj Þdsj
x¼0 1:5
fBj kj ðs1 þ h1 þ sj Þggðsj Þdsj :
ð8Þ
0:5
In the general case where sj 0.5 beta(a, b), Z
5:5
¼
2.2. Fill rate as service criterion
Z
Z
1:5
ðBj kj ½s1 þ h1 þ sj Þgðsj Þdsj
0:5
¼ Bj kj ðs1 þ h1 þ
a þ 0:5Þ: aþb
ð9Þ
particular, for sj unif(0.5,1.5), the integral R[In 1:5 ðBj kj ½s1 þ h1 þ sj Þgðsj Þdsj reduces to Bj 0:5 kj(s1 + h1 + 1).] For sj 4.5 beta(a, b), we have
We computed the required base stock levels Bj for the probability of no stockout criterion—for each of the four values of a (0.80, 0.90, 0.95, and 0.975) and each of the 16 test scenarios—corresponding to a number of stochastic leadtime distributions. We used beta distributions with (i) a = b = 1 (i.e., a uniform distribution), (ii) a = 6 and b = 2, and (iii) a = 2 and b = 6. The means and standard deviations of the various stochastic leadtime distributions (associated with deterministic leadtimes sj = 1 and sj = 5, respectively) are summarized in Table 2. Programs were written in C to compute—for each service criterion—the retail site base stock levels Bj that would minimize thePaverage system N inventory as approximated by 1 Bj 0:5k1 h1 k1 s 1 . Table 3 presents the computed required retailer base stock levels for the 16 test scenarios when a = 0.95, and provides a summary of the changes in approximate average system inventory. This table shows that when sj 0.5 beta(1,1) or sj 4.5 beta(1,1), the mean increase in average system inventory across the 16 test scenarios is 1.0%. The corresponding mean increases are 0.6%, 0.8%, and 1.2% when a = 0.80, 0.90, and 0.975, respectively, and the
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
1009
Table 2 Stochastic leadtime distributions in computational study Leadtime distribution
Mean of beta(a, b)
Interval of sj
Mean of sj
Standard deviation of sj
sj 0.5 beta(2,6) sj 0.5 beta(1,1) sj 0.5 beta(6,2)
0.25 0.50 0.75
(0.5, 1.5) (0.5, 1.5) (0.5, 1.5)
0.75 1.00 1.25
0.144 0.289 0.144
sj 4.5 beta(2,6) sj 4.5 beta(1,1) sj 4.5 beta(6,2)
0.25 0.50 0.75
(4.5, 5.5) (4.5, 5.5) (4.5, 5.5)
4.75 5.00 5.25
0.144 0.289 0.144
overall mean increase across all 64 test cases is 0.9%. When sj 0.5 beta(6,2) or sj 4.5 beta(6,2), as seen in Table 3, the mean increase in average system inventory is 4.2% when a = 0.95. The corresponding mean increases when a = 0.80, 0.90, and 0.975 are 5.0%, 4.0%, and 4.3%, respectively. Across all 64 test cases, the overall mean increase is 4.4%. Table 3 further shows that, when sj 0.5 beta(2,6) or sj 4.5 beta(2,6), the mean decrease in average system inventory when a = 0.95 is 4.5%. The mean decreases are 3.9%, 4.5%, and 3.6% when a = 0.80, 0.90, and 0.975, respectively, and the overall mean decrease across all 64 test cases is 4.1%. Table 4, on the other hand, reports the required retailer base stock levels under the fill rate criterion with b = 0.99 for the 16 test scenarios, and provides a summary of the corresponding changes in estimated average system inventory. When sj 0.5 beta(1,1) or sj 4.5 beta(1,1), the mean increase in average system inventory is 1.0%. The corresponding mean increases when b = 0.95, 0.98, and 0.999 are 0.4%, 0.9%, and 1.5%, respectively, and the overall mean increase across all 64 test cases is 1.0%. For sj 0.5 beta(6,2) or sj 4.5 beta(6,2), Table 4 shows that the mean increase in average system inventory when b = 0.99 is 5.1%. When b = 0.95, 0.98, and 0.999, the corresponding mean increases are 3.8%, 4.5%, and 4.0%, respectively, and the overall mean increase is 4.4% for all 64 test cases. For sj 0.5 beta(2,6) or sj 4.5 beta(2,6), Table 4 reports a mean decrease of 3.7% in average system inventory when b = 0.99. Corresponding mean decreases are 5.0%, 4.0%, and 3.6% when b = 0.95, 0.98, and 0.999, respectively. The overall mean decrease across all 64 test cases is 4.0%. The assumption had been taken that stochastic leadtimes sj vary within some observed interval— such as (0.5,1.5) or (4.5,5.5)—containing the erstwhile deterministic leadtime value. In using the required retailer base stock level for the determinis-
tic leadtime case as a baseline for comparison, we noted that this baseline level would need to be adjusted in view of two possible factors: (i) a mean different from the deterministic leadtime value; and (ii) variability within the observed leadtime interval. When the leadtime distribution is skewed to the right or to the left within this interval—e.g., sj 0.5 beta(2,6) or sj 0.5 beta(6,2) within the interval (0.5,1.5)—a decrease or an increase in the baseline level of Bj arises in most test cases. However, when the mean of the distribution is equal to the deterministic leadtime value—e.g., sj 0.5 beta(1,1), which is equivalent to the uniform distribution sj unif(0.5,1.5)—and only the variability within the observed leadtime interval needs to be addressed, increases in required base stock levels apply in just a little more than half of the test cases. 2.4. Alternative treatment of leadtime distributions An alternative treatment of stochastic leadtimes would be one in which leadtimes sj vary according to some distribution, but with a mean equal to the value assumed in the deterministic leadtime case. Under such treatment, various distributions of sj would not vary within the same preset interval of observed leadtimes. Instead, the interval of leadtimes would differ depending upon the actual distribution, although the range of leadtimes (i.e., the difference between maximum and minimum leadtimes) is still limited to be at most hj in line with the no order crossing requirement. Under this alternative treatment, if the range of leadtimes is 1, then the distribution may take the leadtime a a form sj 1 aþb betaða;bÞ or sj 5 aþb
betaða;bÞ, depending upon whether the deterministic leadtime is 1 or 5, respectively. After conducting a computational study similar to that reported in Section 2.3, we were able to
1010
sj = 1 or sj = 5
sj 0.5 beta(1,1) or sj 4.5 beta(1,1)
sj 0.5 beta(6,2) or sj 4.5 beta(6,2)
sj 0.5 beta(2,6) or sj 4.5 beta(2,6)
N
Computed Bj
Average inventory
Computed Bj
Average inventory
D Average inventory (%)
Computed Bj
Average inventory
D Average inventory (%)
Computed Bj
Average inventory
D Average inventory (%)
18 6 3 2 18 6 3 2 18 6 3 2 18 6 3 2
13 32 60 86 23 60 112 164 28 73 138 202 37 99 190 278
162 120 108 100 342 288 264 256 270 204 180 170 432 360 336 322
13 33 61 89 23 60 113 166 28 73 139 204 37 100 190 280
162 126 111 106 342 288 267 260 270 204 183 174 432 366 336 326
0.0 5.0 2.8 6.0 0.0 0.0 1.1 1.6 0.0 0.0 1.7 2.4 0.0 1.7 0.0 1.2
14 34 63 92 23 61 116 169 28 75 142 208 38 101 193 283
180 132 117 112 342 294 276 266 270 216 192 182 450 372 345 332
11.1 10.0 8.3 12.0 0.0 2.1 4.5 3.9 0.0 5.9 6.7 7.1 4.2 3.3 2.7 3.1
12 31 57 82 22 58 109 160 27 71 135 198 36 98 187 274
144 114 99 92 324 276 255 248 252 192 171 162 414 354 327 314
11.1 5.0 8.3 8.0 5.3 4.2 3.4 3.1 6.7 5.9 5.0 4.7 4.2 1.7 2.7 2.5
247.06
1.0
Scenario
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Mean
244.63
254.88
4.2
233.63
4.5
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
Table 3 Computed required retailer base stock levels under various leadtime distributions when a = 0.95
sj = 1 or sj = 5
sj 0.5 beta(1,1) or sj 4.5 beta(1,1)
N
Computed Bj
Average inventory
Computed Bj
Average inventory
D Average Computed inventory (%) Bj
Average inventory
D Average Computed inventory (%) Bj
Average inventory
D Average inventory (%)
18 6 3 2 18 6 3 2 18 6 3 2 18 6 3 2
14 33 59 84 25 61 112 163 28 72 134 196 38 98 186 272
180 126 105 96 378 294 264 254 270 198 168 158 450 354 324 310
14 33 61 87 25 62 114 165 28 72 135 197 38 99 186 273
180 126 111 102 378 300 270 258 270 198 171 160 450 360 324 312
0.0 0.0 5.7 6.3 0.0 2.0 2.3 1.6 0.0 0.0 1.8 1.3 0.0 1.7 0.0 0.6
198 138 117 108 396 306 276 264 288 204 180 168 468 366 333 320
10.0 9.5 11.4 12.5 4.8 4.1 4.5 3.9 6.7 3.0 7.1 6.3 4.0 3.4 2.8 3.2
180 114 96 88 360 282 258 246 270 186 159 148 432 348 315 300
0.0 9.5 8.6 8.3 4.8 4.1 2.3 3.1 0.0 6.1 5.4 6.3 4.0 1.7 2.8 3.2
248.13
1.0
236.38
3.7
Scenario
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Mean
245.56
sj 0.5 beta(6,2) or sj 4.5 beta(6,2)
15 35 63 90 26 63 116 168 29 73 138 201 39 100 189 277
258.13
5.1
sj 0.5 beta(2,6) or sj 4.5 beta(2,6)
14 31 56 80 24 59 110 159 28 70 131 191 37 97 183 267
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013
Table 4 Computed required retailer base stock levels under various leadtime distributions when b = 0.99
1011
1012
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observe that changes in base stock levels from the deterministic case to the stochastic case, in many instances, either may not be required or may be fairly small if required, when mean leadtimes coincide with the deterministic leadtime value, and only the variability of leadtimes (as measured by standard deviation, for instance) is to be accounted for. It became apparent that the required changes in base stock levels under the earlier treatment of stochastic leadtimes (giving rise to variations in terms of both means and standard deviations) result mainly from the differences in means, except in the case where the uniform distribution applies. This observation suggests that it may be better to use the deterministic model with an accurately estimated mean leadtime than a stochastic model with a poorly estimated mean leadtime. 3. Conclusion and further research In this study, we extended Graves’ one-warehouse, N-retailer model by introducing, as Graves suggested, stochastic leadtimes sj between the CW and the retail sites in place of the original deterministic leadtimes. Effects of stochastic leadtimes on required base stock levels in the case where the CW carries no stock were investigated analytically. Mathematical models were derived for each of the two service criteria under investigation, along with computational forms for such models, to determine the appropriate retailer base stock level Bj. Two alternative treatments of stochastic leadtime distributions were considered. Under the first treatment, leadtimes are assumed to vary within some interval, with means of leadtime distributions being possibly different from the deterministic leadtime value. Adjustments to the required retailer base stock level (as computed for the deterministic leadtime case) address both the change in mean values and the variability of the stochastic leadtimes. Under the second treatment, the mean of the distribution is assumed to be equal to the deterministic leadtime value, so that an adjustment to the required level is brought about only by the variability of the stochastic leadtimes. Under the latter treatment, adjustments were not found to be necessary in many instances, or, if at all necessary, to result in fairly small increases in the base stock level and average system inventory. The latter observation would appear to suggest that it may be better to use the deterministic model with an accurately esti-
mated mean leadtime than a stochastic model with a poorly estimated mean leadtime. Where the CW carries stock (B1 > 0), a similar analytical investigation of the effects of stochastic leadtimes on the optimal base stock policy hB1, Bji does not appear possible. Extensive simulation studies are indicated in view of the nature of the search procedures for optimal levels of B1 and Bj. Admittedly, our assumption in this study that the range of stochastic leadtimes be at most hj, in order to ensure that orders do not cross, is restrictive. Less restrictive assumptions may be possible, for instance, by focusing on an order-arrival mechanism instead of the leadtime distribution, as Zipkin (1986) does. The expression tr = s1 + h1 + sj is crucial to the definition and analysis of uncovered demand in Graves’ model. We note that tr is a random variable that varies with the CW to retailer leadtime sj, when the latter is not deterministic but varies according to some leadtime distribution. Hence, the distribution of uncovered demand and, accordingly, the selection of a base stock policy that seeks to minimize average system inventory may be influenced by stochastic leadtimes sj. The same could be said about stochastic leadtimes s1 from the external supplier to the CW. Further, in investigating stochastic external leadtimes s1, it would be interesting to find out whether the model is more sensitive to randomness in s1 or to randomness in sj. References Barnes-Schuster, D., Bassok, Y., 1997. Direct shipping and the dynamic single-depot/multi-retailer inventory system. European Journal of Operational Research 101 (3), 509–518. Chopra, S., Meindl, P., 2004. Supply Chain Management: Strategy Planning and Operation. Prentice Hall, Upper Saddle River, NJ. David, F.R., 1999. Strategic Management: Concepts and Cases. Prentice Hall, Upper Saddle River, NJ. Eppen, G., Schrage, L., 1981. In: Schwarz, L.B. (Ed.), MultiLevel Production/Inventory Control Systems, Theory and Practice, TIMS Studies in the Management Sciences, Vol. 16. North Holland, Amsterdam, pp. 51–67. Graves, S.C., 1996. A multiechelon inventory model with fixed replenishment intervals. Management Science 42 (1), 1–18. Johnson, N.L., Kotz, S., 1985. Encyclopaedia of Statistical Sciences, Vol. 5. John Wiley and Sons, New York. McGavin, E.J., Schwarz, L.B., Ward, J.E., 1993. Two-interval inventory-allocation policies in a one-warehouse, N-identicalretailer distribution system. Management Science 39 (9), 1092–1107.
A.O. Solis, C.P. Schmidt / European Journal of Operational Research 181 (2007) 1004–1013 Mitra, S., Chatterjee, A.K., 2004. Leveraging information in multi-echelon inventory systems. European Journal of Operational Research 152, 263–280. Nahmias, S., Smith, S.A., 1994. Optimizing inventory levels in a two-echelon retailer system with partial lost sales. Management Science 40 (5), 582–596. Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E., 2003. Designing and Managing the Supply Chain: Concepts Strategies and Case Studies. McGraw-Hill, New York, NY.
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Stalk, G., Evans, P., Shulman, L.E., 1992. Competing on capabilities: the new rule of corporate strategy. Harvard Business Review, 57–69, March–April 1992. Zipkin, P., 1986. Stochastic leadtimes in continuous-time inventory models. Naval Research Logistics Quarterly 33, 763–774.