Periodic review inventory models with stochastic supplier's visit intervals

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Periodic review inventory models with stochastic supplier’s

visit intervals

Chi Chiang

Department of Management Science, National Chiao Tung University, Hsinchu, Taiwan, ROC

a r t i c l e

i n f o

Article history:

Received 30 September 2006 Accepted 20 March 2008 Available online 22 June 2008 Keywords: Inventory model Periodic review Supply chain Lost-sales Dynamic programming

a b s t r a c t

Periodic review inventory models are widely used in practice, especially for inventory systems in which many different items are purchased from the same supplier. However, all periodic review models have assumed a ﬁxed length of the review periods. In practice, it is possible that the review periods are of a variable length. Such periodic systems result mainly from supply uncertainties. For example, the supplier visits the downstream retailers and replenishes inventories for them, but does not always come in constant intervals. This may be because retailers are geographically dispersed in the supply chain, the supplier is in a relatively more powerful position, or the supplier simply does not have a reliable visit schedule. In such situations, the replenishment cycle length is random in nature. In this paper, we use dynamic programming to model such institutional contexts. We assume that the supplier’s visit intervals are independently and identically distributed. With a suitable transformation, the back-logged periodic review model derived becomes a standard discrete-time model. The computation shows that ignoring the variability of the supplier’s visit intervals can incur extremely large losses, especially if shortage is costly, demand variability is low, and/or the replenishment lead-time is short.

1. Introduction

Though the use of computer systems has made continuous review models more attractive, periodic re-view models are still applied in many situations (e.g., Prasad et al., 2005; Silver et al., 1998), especially for inventory systems in which the coordination of ordering and transportation for different items is important (which is especially true if these items are purchased from the same supplier). Also, as Porteus (1985) observes, continuous review systems that keep inventory records current, but order periodically, are equivalent to periodic review systems. Often, periodic systems have the review periods that are possibly longer than the supply lead-times.

One fundamental assumption about periodic systems is that the review periods are of a ﬁxed length. In practice, however, the review periods may be of a variable length. Such periodic systems result mainly from supply uncer-tainties. For example, many supermarkets have suppliers who come to visit regularly and replenish the inventory of various items (and even sell) for them. However, the supplier does not always come in constant (say, 10 days) intervals. Depending on her visit plans or work schedules and loads, she often arrives at a particular supermarket one or few hours (or days) early or late. The elapsed time between two consecutive visits varies basically.Ertogral and Rahim (2005)also observed institutional settings or constraints that are internal to the supply chain, in which the supplier is strategically dominant, in a relatively more powerful position, and/or the retailers are located in a geographically disadvantageous remote location, so that the supplier decides when to visit and replenish the retailers’ inventories. In general, for such situations, the Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/ijpe

Int. J. Production Economics

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replenishment epochs are not under retailers’ control; rather, they are under the supplier’s control. Hence, if the supplier arrives at a particular retailer in irregular intervals, the replenishment cycle length for that retailer is random in nature.

To our knowledge, the possibility of stochastic review periods or replenishment intervals has not been investi-gated in the inventory literature, though there are some works on inventory models with supply uncertainties (e.g.,Lee et al., 1997;Mohebbi, 2004;Ozekici and Parlar, 1999). It was studied only recently byErtogral and Rahim (2005)who derived the expected proﬁt per replenishment cycle by assuming independently and identically distrib-uted (i.i.d.) replenishment intervals, constant demand and zero replenishment lead-time.

In this paper, we use dynamic programming to model the supply chain situations where the supplier’s visit intervals (i.e., replenishment intervals) are random. We will assume that the supplier’s visit intervals are i.i.d., as in Ertogral and Rahim (2005). However, unlike Ertogral and Rahim, we will assume stochastic demand which is usually true in the real world; also, we will allow the replenishment lead-time to be positive (i.e., it may take a positive time to replenish inventories after the supplier arrives at a retailer and reviews his inventories). We will develop both the backlogged and lost-sales periodic review inventory models. With a suitable transformation, the backlogged model derived becomes a standard discrete-time model. Thus, an order-up-to policy is known to be optimal for the inﬁnite horizon problem. This is also true of the lost-sales problem with zero lead-time (due to a result from the inventory literature). For the lost-sales problem with positive lead-time, we suggest a simple heuristic policy inHadley and Whitin (1963).

The computation shows that ignoring the variability of the supplier’s visit intervals can incur unnecessary large costs, especially if shortage is costly, the replenishment lead-time is short, and/or demand variability is not high. It is thus important for a retailer to incorporate this variability into inventory models when the supplier does not visit in constant intervals. It would be better if the retailer can have the supplier to visit in more regular intervals (i.e., the visit interval has a smaller variability) so that his cost can be reduced, as shown in the computation. This may not be an easy task, since the institutional constraints are perhaps difﬁcult to change in the short run (for example, the supplier is in a relatively more powerful position as described above). For such institutional contexts, we suggest that the retailer should somehow persuade the supplier to visit more regularly. The retailer should at least communicate with the supplier often so that she understands the consequence of the irregular visit intervals and hopefully, she will continue to improve on her visit schedule in terms of the stability/reliability in the future.

Of course, it is possible that the supplier completely ﬁxes the visit interval after the retailer’s persuasion. The supplier and the retailer may even cooperate closely in the supply chain, or form a strategic alliance in the long run. Then the supplier will also visit the retailer and replenish his inventories more often (not only more regularly) so as

to further reduce his costs, and in return, the retailer could negotiate a long-term supply contract or purchase other products from the supplier, for example. All of these are certainly a signiﬁcant change of status-quo, i.e., a break-through of the supply chain. Note that we are not saying that it is not good to have the replenishment epochs under the supplier’s control; it may be one of the most efﬁcient ways of operating the supply chain (in terms of replenish-ing the downstream retailers’ inventories), especially for the institutional settings described above. We simply say that cooperation between the supplier and the retailer could result in a win–win situation. If indeed the supplier no longer visits the retailer in irregular intervals, then the ordinary periodic review models found in textbooks can be used, i.e., the periodic review models derived in this paper need not be used.

2. Backlogged periodic review inventory models

We ﬁrst assume that all demand not ﬁlled immediately is backlogged. Let c denote the unit item cost and L the (deterministic) lead-time. Demand is stochastic with mean rate

m

per unit time, and is assumed to be non-negative and independently distributed in disjoint time intervals. Let T be the period length, i.e., the supplier’s visit interval. Successive T’s are assumed to be i.i.d. random variables. Let

f

( ) be the probability density function (pdf) of T and D the demand during T. Also let g( j

t

) be the conditional pdf of demand during a time interval of length

t

. Thus, g( jT) is the conditional pdf of D.

Let

a

be the discount rate, y the inventory position (i.e., inventory on hand minus backorder plus inventory on order) after an order is placed at a review epoch (i.e., upon the supplier’s visit), and H the expected one-period inventory holding and shortage cost (H is a function of y). Given time 0 at a review epoch, H is charged for the time interval [L, T+L). Denote Vn(x) as the expected discounted

cost with n periods remaining until the end of the planning horizon when the starting inventory position is x and an optimal ordering policy is used at every review epoch. Vn(x)

satisﬁes the functional equation: VnðxÞ ¼ min

xpyfe

aL_{½cy þ HðyÞ}

þET½eaTEDjT½Vn1ðy DÞg eaLcx, (1)

where the procurement cost c(y–x) is paid upon delivery. The above dynamic program is an inventory problem with discrete but random epochs. Let

b

¼ET[eaT].

Using a standard approach in semi-Markov decision processes (e.g., Puterman, 1994, p. 542) and deﬁning

j

( )ET[eaTg( jT)]/

b

, i.e.,

b

j

( ) ¼ ET[eaTg( jT)] is the

discount density of D and

j

( ) is the ‘‘normalized’’ pdf of D, we can express ET[eaTEDjT[Vn1(y–D)]] by

ETeaTEDjT½Vn1ðy DÞ ¼ Z 1 0 eaT Z 1 0 Vn1ðy DÞgðDjTÞdD

f

ðTÞdT ¼ Z 1 0 Vn1ðy DÞ Z 1 0 eaT_gðDjTÞ

_f

_ðTÞdT dD

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¼

b

Z 1 0 Vn1ðy DÞ Z 1 0 eaT_gðDjTÞ

_f

_{ðTÞ dTÞ=}

_b

dD ¼

b

Z 1 0 Vn1ðy DÞ

j

ðDÞ dD ¼

b

ED½Vn1ðy DÞ, (2)

where the expectation EDis taken over the pdf

j

( ). After

the above transformation, Vn(x) is written as

VnðxÞ ¼ min

xpy e

aL_{½cy þ HðyÞ}

þ

b

ED½Vn1ðy DÞ

eaL_cx. ₍₃₎

The original problem in (1) is now a standard discrete-time model.

Next, we give an expression for H(y). Let h be the holding cost per unit held per unit time and p the shortage cost per unit. Assume that an order when arriving is almost always sufﬁcient to meet any outstanding back-orders (seeHadley and Whitin, 1963, p. 239, for a detailed discussion). Thus, backorders that occur during the time interval [L, L+T) can be computed for the interval [0, L+T) and the expected on-hand inventory immediately after the arrival of an order is yL

m

. If Dpy–L

m

, as the expected on-hand inventory just prior to the arrival of the next order is yL

m

D, the average holding cost over the interval [L, L+T) is hT(yL

m

0.5D). On the other hand, if D4yL

m

, the expected on-hand inventory falls to zero at some time during the interval [L, L+T). Assuming that the expected inventory decreases linearly with time (e.g., Hadley and Whitin, 1963, p. 238), the expected on-hand inventory falls to zero at time L+T(yL

m

)/D; since the average on-hand inventory over the interval [L, L+T(yL

m

)/ D) is 0.5(yL

m

), the average holding cost over this interval, which is also the average holding cost over the interval [L, L+T), is 0.5hT(yL

m

)2_{/D. As D appears in the}

deno-minator, this exact expression will complicate the subsequent analysis. We thus use a lower bound hT(yL

m

0.5D) for this expression, i.e., the same one as when DpyL

m

(seeChiang (2003)for a similar approach in the two-supply-mode setting). This approximation is similar to the one in the ordinary periodic model where the expected on-hand inventory approximately equals the expected net inventory (Hadley and Whitin, 1963, pp. 237–239). Hence, the average holding cost over the interval [L, L+T), after taking the expectation of D, is hT(yL

m

0.5T

m

). It follows that if T is constant,

HðyÞ ¼ hTðy L

m

0:5 T

m

Þ þ

Z 1 y

pð

z

yÞgð

z

jT þ LÞ dz (4)

(Hadley and Whitin, 1963, p. 240). For the present model in which T is a variable, H(y) is given by

HðyÞ ¼ ET½hTðy L

m

0:5 T

m

Þ

þ Z 1

y

pð

z

yÞg_ð

_z

_{jLÞ dz;} ₍₅₎

where g*( jL)ET[g( jT+L)]. Since H(y) is convex, Vn(x) in

(3) is a convex function (by induction and Proposition B-4 ofHeyman and Sobel, 1984). Hence, a stationary order-up-to policy (i.e., base-sorder-up-tock policy) is known order-up-to be optimal for the inﬁnite horizon problem. To obtain the optimal

order-up-to level y*, we minimize the following myopic function:

JðyÞ ¼ cyð1

b

Þ þHðyÞ (6) (e.g.,Veinott and Wagner, 1965, p. 527). Thus, we set the ﬁrst derivative of J(y) to zero:

cð1

b

Þ þhE½T Z 1 y pg ð

z

jLÞd

z

¼0 or Z 1 y gn ð

z

jLÞ d

z

¼ fcð1

b

Þ þhE½Tg=p (7) and solving for the optimal y*. J is basically the expected cost of the upcoming period. Notice that the constant scaling factor eaL

(discounted to the present time) and procurement cost eaL_c

_b

_E

D[D] are not included in (6) for

simplicity. As the ratio {c(1–

b

)+hE[T]}/p should be less than 1 (since the average backorder level is assumed to be small [Hadley and Whitin, 1963, p. 241]), y* is guaranteed to be obtained. Let y0_{be the optimal y found if T is ﬁxed.}

3. Lost-sales periodic review inventory models

Suppose now that demand not satisﬁed at once is lost. Assume that L is less than the minimum T (i.e., there is at most one order outstanding). Let D1be the demand during

the lead-time L and D2 the demand during the time

interval [L, T) (thus D ¼ D1+D2). Also, let z be the order

quantity placed at a review epoch and redeﬁne x as the starting on-hand inventory. Let ( )+_{max{ , 0}. Then,}

Vn(x) satisﬁes the recursive equation:

VnðxÞ ¼ min zX0 e aL_{ðcz þ E} D1½Hððx D1Þ þ þzÞÞ þET eaTED1;D2jT½Vn1ðððx D1Þ þ þz D2ÞþÞ (8) Consider ﬁrst the simplest case of L ¼ 0. Then (8) reduces to VnðxÞ ¼ min zX0 cz þ Hðx þ zÞ þETeaTEDjT½Vn1ððx þ z DÞþÞ ¼min

xpyfcy þ HðyÞ

þETeaTEDjT½Vn1ððy DÞþÞ cx. (9) If T is constant, (9) simpliﬁes to VnðxÞ ¼ min xpy cy þ HðyÞ þ e aT_E D½Vn1ððy DÞþÞ cx. (10) Veinott and Wagner (1965, p. 528)showed that the lost-sales model in (10) could be viewed as a backlog model in which a credit of eaT_{c is given to each unit of demand}

actually backlogged (an order-up-to policy is thus opti-mal). For the model in (9) where T is stochastic, we can transform (9) to (as in the backlogged model)

VnðxÞ ¼ min

xpyfcy þ HðyÞ þ

b

ED½Vn1ððy DÞ þ

Þg cx, (11) where the expectation ED is taken over the pdf

j

( ).

Hence, an order-up-to policy is optimal for the inﬁnite horizon problem and the optimal order-up-to level y* is

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obtained by minimizing JðyÞ ¼ cyð1

b

Þ þHðyÞ

b

c

Z 1 y

ðD yÞ

j

ðDÞ dD; (12)

where the last term (i.e., the difference between (6) and (12)) is the credit given to demand not satisﬁed and the exact one-period holding and shortage cost is given by

HðyÞ ¼ ET 0:5hT y þ Z y 0 ðy DÞgðDjTÞdD þ Z1 y pðD yÞgðDjTÞdD (13)

(which is convex in y). The holding cost of (13) is derived based on the average of beginning and ending inventories of a period. Also, the shortage penalty p has a different meaning in the lost-sales model and it usually includes the unit sales revenue. As J(y) is convex, y* is found by solving its ﬁrst-order condition, i.e.,

cð1

b

Þ þET 0:5hT 1 þ Z y o gðDjTÞdD Z 1 y pgðDjTÞdD þcET eaT Z 1 y gðDjTÞdD ¼0 (noting that

b

j

( ) ¼ ET[eaTg( jT)]), or cð1

b

Þ þET 0:5hTð1 þ 1 Z1 y gðDjTÞdDÞ Z 1 y pgðDjTÞdD þcET eaT Z 1 y gðDjTÞdD ¼0, i.e., ET ð0:5hT þ p eaTcÞ Z 1 y gðDjTÞdD ¼cð1

b

Þ þhE½T. (14) If L is positive (deterministic), the lost-sales model given by (8) is difficult to solve. If T is fixed, see, e.g.,Hadley and Whitin (1963, p. 285),Morton (1971), andZipkin (2000, pp. 411–413) (note thatChiang (2006)recently develops optimal ordering policies for the case of LpT). For the present model in (8) where T is variable, heuristic approaches need to be used. We suggest that one uses the order-up-to policy in Hadley and Whitin (1963, pp. 240–242). Using our notation, the expected undis-counted cost of the upcoming time interval [L, T+L), if T is fixed, is expressed by HðyÞ ¼ hTðy

m

L 0:5T

m

Þ þ ð0:5hT þ p cÞ Z 1 y ð

z

yÞgð

z

jT þ LÞd

z

. (15) As a note, there is an error in expressions (5–11) of Hadley and Whitin. The integral should be multiplied by a factor of 0.5. Since the base-stock policy orders ﬁlled demands just as it does in a backlogged model (except that now lost sales are not counted), the average cycle stock (i.e., order quantity) over the interval [L, T+L) is given by

0:5 T

m

Z1 y ð

z

yÞgð

z

jT þ LÞd

z

(16)

instead of 0.5T

m

. Note that (16) is an approximation, since it ignores the effects of lost sales that can occur between

the time an order is placed and the time it arrives. Adding (16) to the safety stock, given by (5–10) of Hadley and Whitin, would yield the holding cost component of (15). For the present model in which T is a variable, the expected undiscounted cost of the upcoming time interval [L, T+L) is written by HðyÞ ¼ ET½hTðy

m

L 0:5T

m

Þ þET ð0:5hT þ p cÞ Z 1 y ð

z

yÞgð

z

jT þ LÞd

z

. (17)

The optimal order-up-to level y* is then obtained by minimizing (17), i.e., setting the ﬁrst derivative of (17) to zero: hE½T ET ð0:5hT þ p cÞ Z 1 y gð

z

jT þ LÞd

z

¼0, or ET ð0:5hT þ p cÞ Z 1 y gð

z

jT þ LÞd

z

¼hE½T (18)

and solving for the optimal y*. Let y0 _{be the optimal level}

found from (14) or (18) when T is ﬁxed.

4. Computational results

We investigate the effect of a variable T on the expected cost, if a retailer fails to incorporate it when developing inventory policies. In the following experiments, we assume that demand is normal with mean

mt

and variance

s

2

_t

_{for a time interval of length}

_t

_{. The common}

data used are

m

¼10/day (unit time is one day), h ¼ $0.1,

a

¼0, and E[T] ¼ 10 days. Also, T is either triangularly or uniformly (discretely) distributed. In the former case, Pr(T ¼ 8) ¼ Pr(T ¼ 12) ¼ 1/9, Pr(T ¼ 9) ¼ Pr(T ¼ 11) ¼ 2/9, and Pr(T ¼ 10) ¼ 1/3; in the latter case, Pr(T ¼ 8) ¼Pr(T ¼ 9) ¼ Pr(T ¼ 10) ¼ Pr(T ¼ 11) ¼ Pr(T ¼ 12) ¼ 1/5. Thus, T has a larger variability if it is uniformly distributed.

First consider the backlogged model with L ¼ 0.Table 1 reports computational results as p and

s

2

are varied. Three observations can be made fromTable 1. First, ignoring T’s variability, i.e., using y0 _{of the ordinary periodic model}

when in fact T is stochastic, can incur unnecessary large costs, especially if T is uniformly distributed (i.e., has a larger variability). Second, it appears that as the unit shortage cost p is higher, a ﬁrm incurs larger losses. This result can be seen by comparing (5) to (4). T’s variability affects J(y) only through its shortage cost component. As p is higher, T’s variability has a larger effect on J(y). Third (and probably most importantly), as

s

is smaller, T’s variability has a larger impact on J(y). In the extreme case where

s

¼0 (i.e., deterministic demand), ignoring T’s variability can increase the cost by more than 300 percent! This is possibly because the variability of demand during T plus L includes T’s variability and demand variability; the introduction of T’s variability into an inventory model increases the overall variability of demand during T plus L more signiﬁcantly when demand variability is smaller.

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Next, consider the backlogged model with a positive L (which equals, for example, 6 days). ComparingTable 2 to 1, we see that except for the case of deterministic demand, a positive L dilutes the effect of a variable T on J(y). In fact, as L is larger (other things being equal), ignoring T’s variability incurs smaller losses (more experiments are available from the author). This result can be explained by the same reason above: as the demand during T plus L becomes more volatile, the introduction of T’s variability into an inventory model has a less signiﬁcant effect on the cost.

In Tables 3 and 4, we consider the lost-sales models with zero and positive lead-times, respectively. The unit cost c is assumed to be $100 and p is varied such that (p–c) is the same as p in the backlogged models. As we see from these two tables, similar results to those inTables 1 and 2 are observed, though the percentage loss of ignoring T ’s variability seems to be a little higher in general.

Notice that if T is ﬁxed (equal to 10 days) rather than stochastic, the cost per period is signiﬁcantly reduced (by using the ordinary periodic review models found in text-books). For example, if

s

¼0, the optimal order-up-to level is 100 and the cost per period is only $50, as compared to $69.3 in the above tables. This illustrates the importance of a ﬁxed visit schedule on the part of the supplier. As a note, the cost per period is only $12.5 if T is ﬁxed and cut down to 5 days (and

s

¼0). This certainly signals a message: the retailer should somehow have the supplier to come more often and regularly, as mentioned in Section 1.

5. Conclusions

In this paper, we considered periodic inventory models with stochastic supplier’s visit intervals. We assumed that

Table 1

Effect of variable T on the expected cost (backlogged model with L ¼ 0)

sp y0 _{(A) T is triangularly distributed (B) T is uniformly distributed}

y* J(y*) J(y0₎ _{% Off} _y* _J(y*) _J(y0₎ _{% Off}

0 10 100 120 69.3 93.8 35.4 120 69.0 109.0 56.0 20 100 120 69.3 138.2 99.4 120 69.0 169.0 144.9 40 100 120 69.3 227.1 227.7 120 69.0 289.0 318.8 2 10 109 118 72.5 78.5 8.3 121 74.8 87.1 16.4 20 111 122 75.9 91.4 20.4 125 77.9 106.4 36.6 40 113 128 79.3 109.1 37.6 128 80.5 134.2 66.7 4 10 117 123 80.1 81.7 2.0 125 82.7 86.5 4.6 20 121 129 85.5 90.1 5.4 132 88.1 97.3 10.4 40 125 134 90.2 98.5 9.2 137 92.8 108.8 17.2 8 10 133 136 99.4 99.7 0.3 138 101.6 102.3 0.7 20 142 147 108.5 109.5 0.9 149 111.0 113.1 1.9 40 150 156 116.7 118.4 1.5 159 119.5 123.2 3.1 Table 2

Effect of variable T on expected cost (backlogged model with L ¼ 6)

0 10 160 180 69.3 93.8 35.4 180 69.0 109.0 56.0 20 160 180 69.3 138.2 99.4 180 69.0 169.0 144.9 40 160 180 69.3 227.1 227.7 180 69.0 289.0 318.8 2 10 171 179 74.0 78.3 5.8 182 76.5 85.5 11.8 20 174 183 77.9 87.6 12.5 186 80.1 99.0 23.5 40 176 187 81.2 101.8 25.4 190 83.1 120.6 45.1 4 10 181 186 84.5 85.5 1.2 188 86.9 89.3 2.8 20 187 193 90.7 93.0 2.5 196 93.2 98.0 5.2 40 192 199 96.2 100.2 4.2 202 98.7 107.0 8.4 8 10 202 204 109.9 110.0 0.1 205 111.6 111.8 0.2 20 213 217 120.8 121.1 0.3 219 122.8 123.7 0.7 40 223 228 130.5 131.3 0.6 230 132.8 134.6 1.4 Table 3

Effect of variable T on expected cost (lost-sales model with L ¼ 0)

0 110 100 120 69.3 96.3 39.0 120 69.0 112.5 63.0 120 100 120 69.3 140.8 103.2 120 69.0 172.5 150.0 140 100 120 69.3 239.7 245.9 120 69.0 292.5 323.9 2 110 109 118 72.8 79.7 9.5 122 75.2 88.8 18.1 120 111 122 76.1 92.3 21.3 125 78.0 107.8 38.2 140 113 128 79.3 109.8 38.4 128 80.5 135.3 68.1 4 110 117 124 80.6 82.6 2.5 126 83.2 88.8 6.7 120 121 129 85.7 90.6 5.7 133 88.3 98.1 11.1 140 125 135 90.3 98.9 9.5 138 92.8 109.3 17.8 8 110 133 137 100.1 100.4 0.3 139 102.4 103.1 0.7 120 142 147 108.8 110.0 1.1 149 111.4 113.7 2.1 140 150 157 116.9 118.7 1.5 159 119.6 123.5 3.3 Table 4

Effect of variable T on expected cost (lost-sales model with L ¼ 6)

0 110 160 180 69.3 96.3 39.0 180 69.0 112.5 63.0 120 160 180 69.3 140.8 103.2 180 69.0 172.5 150.0 140 160 180 69.3 239.7 245.9 180 69.0 292.5 323.9 2 110 171 179 74.4 79.4 6.7 182 76.8 87.0 13.3 120 174 184 78.1 88.3 13.1 187 80.3 100.1 24.7 140 176 188 81.3 102.4 26.0 190 83.2 121.4 45.9 4 110 181 186 85.0 86.4 1.6 189 87.5 90.4 3.3 120 187 193 90.9 93.4 2.8 196 93.5 98.7 5.6 140 192 199 96.3 100.4 4.3 202 98.8 107.4 8.7 8 110 202 204 110.8 111.0 0.2 206 112.5 113.0 0.4 120 213 217 121.1 121.6 0.4 219 123.2 124.3 0.9 140 223 228 130.7 131.6 0.7 230 133.0 134.9 1.4

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the supplier’s visit intervals were independently and identically distributed. With a suitable transformation, the backlogged dynamic programming model derived became a standard discrete-time model. In addition, we suggested a simple order-up-to policy for the lost-sales periodic problem with positive lead-time. The periodic review policies developed in this paper were thus easy to implement.

The computation showed that ignoring the variability of the supplier’s visit intervals could incur large losses if shortage was costly. It also showed that a retailer was more vulnerable to this variability if the replenishment lead-time was short and/or demand variability was small. This was because the introduction of T’s variability into an inventory model increased the overall variability of demand during T plus L more signiﬁcantly when lead-time was shorter and/or demand variability was smaller. In the extreme case where demand was deterministic, ignoring T’s variability could increase the retailer’s cost by more than 300 percent!

Moreover, the computation showed that a retailer could avoid some losses by reducing the variability of the supplier’s visit intervals (e.g., from a uniform to triangular distribution). This might not be an easy task, because the replenishment epochs were under the supplier’s control and such institutional constraints were perhaps difﬁcult to change in the short run. However, the retailer could discuss this issue and explain its effect on his cost with the suppler. The retailer should at least communicate with the supplier often so that she would come to visit and replenish inventories more punctually (or even in con-stant intervals) in the future.

References

Chiang, C., 2003. Optimal replenishment for a periodic review inventory system with two supply modes. European Journal of Operational Research 149, 229–244.

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