Optimal control policy for a standing order inventory system

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Production, Manufacturing and Logistics

Optimal control policy for a standing order inventory system

Chi Chiang

Department of Management Science, National Chiao Tung University, Hsinchu, Taiwan, ROC Received 12 August 2005; accepted 2 August 2006

Available online 16 November 2006

Abstract

In this paper, we consider a standing order inventory system in which an order of fixed size arrives in each period. Since demand is stochastic, such a system must allow for procurement of extra units in the case of an emergency and sell-offs of excess inventory. Assuming the average-cost criterion, Rosenshine and Obee (Operations Research 24 (1976) 1143–1155) first studied such a system and devised a 4-parameter inventory control policy that is not generally optimal. The current paper uses dynamic programming to determine the optimal control policy for a standing order system, which consists of only two operational parameters: the dispose-down-to level and order-up-to level. Either the average-cost or discounted-cost criterion can be assumed in the proposed model. Also, both the backlogged and lost-sales problems are investigated in this paper. By using a convergence theorem, we stop the dynamic programming computation and obtain the two optimal parameters.

Keywords: Inventory; Emergency order; Dynamic programming; Storage constraint; Lost-sales; Standing order

1. Introduction

Inventory control systems in the literature are generally divided into two groups: continuous-review models and periodic-continuous-review models. The for-mer typically assumes a ﬁxed order size, while the latter usually predetermines the period length. Since demand is stochastic in the real world, the order interval for the former is thus variable, while the order quantity for the latter varies period by period. A standing order inventory system is a periodic-review one in which the order size is also ﬁxed. However, as Rosenshine and Obee [15] pointed

out, it must allow for procurement of extra units in the case of an emergency and sell-offs of excess inventory if necessary. Assuming that demand in different periods is independently and identically distributed and demand not satisfied at once is backlogged, Rosenshine and Obee hypothesized that the size of a standing order is greater than or equal to the mean demand of a period and devised a 4-parameter inventory control policy for such a system: the storage capacity, emergency order point, size of a standing order, and emergency order-up-to level (i.e., if inventory exceeds the storage capacity, the excess inventory is sold off, and if inventory falls below the order point, an emergency order is placed to raise inventory to the order-up-to level). Using a Markov chain approach, they determined the latter

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two operational parameters, given the former two. Consequently, the inventory policy they devised is not generally optimal.

In this paper, we use dynamic programming to derive the optimal control policy for a standing order system considered in[15]. We assume (as in[15]) that the emergency unit item cost is higher than the regu-lar unit cost, which in turn is greater than the unit sell-off revenue. Also, we assume that the size of standing orders is predetermined by the buyer. The optimal policy derived has only two operational parameters: the dispose-down-to level and order-up-to level. If inventory at a review epoch is lower than the order-up-to level, an emergency order is placed to raise inventory to this level, and if inven-tory at a review epoch is higher than the dispose-down-to level, inventory is sold off down to this level. A standing order inventory system has many attractive features compared to a base-stock peri-odic-review model [15]. The fixed cost for placing periodic orders is eliminated and lead-time does not exist. Also, suppliers are more likely to offer a certain form of price breaks or discounts for items delivered under a standing order. Moreover, a sup-plier does not suffer from the bullwhip effect if a standing order is negotiated with its buyer.

A standing order system bears resemblance to supply contracts with a ﬁxed periodic delivery. Sev-eral studies have recently been done on this area. Anupindi and Akella[1]investigate a ﬁnite-horizon periodic commitment model with a response time to adjustments in the order quantity. Henig et al.[10]

design a periodic inventory/transportation model where both downward and upward adjustments in the order quantity are permitted. Bassok et al. [3]

present a supply contract problem with periodic commitments and limited flexibility to change the purchase quantity. Ehrhardt[8]considers the prob-lem of selecting a fixed replenishment quantity to be delivered in each of n consecutive periods in the future. Janssen and de Kok[12]discuss a two-sup-plier periodic model where one suptwo-sup-plier delivers a fixed quantity while the amount delivered by the other is governed by an order-up-to policy. Urban

[17] describes a multi-period ‘‘recurrent’’ newsven-dor problem where changes in the order quantity result in an additional cost to the buyer. Moinzadeh and Nahmias [13] consider a continuous-review inventory model where fixed as well as variable costs are incurred for any upward adjustments to the fixed order quantity. Chiang [6] devises an order-splitting periodic model where n fixed-size

ship-ments (except the ﬁrst one) are delivered in future time points that are evenly separated. Recently, Cheung and Yuan[4]extend the model of Anupindi and Akella [1] to an inﬁnite-horizon one with no extra costs incurred for units ordered beyond the periodic quantity. See, e.g., Anupindi and Bassok

[2] and Tsay et al. [16] for other related research on supply contracts with periodic commitments. See also, e.g., Chiang and Gutierrez [7]for a peri-odic-review inventory model with emergency orders. Note that Henig et al.’s model [10] and Rosen-shine and Obee’s model[15]are similar in the sense that both allow for emergency orders at a review epoch. The diﬀerence between them is that when excess inventory seems to exist at a review epoch, the former [10] permits the supplier to deliver a quantity that is less than the periodic commitment (with no refunds given), while the latter [15] gives the buyer the option of disposing of excess inven-tory (upon receipt of a standing order). It is seen below that Henig et al.’s model is a special case of the basic dynamic program developed.

The rest of this paper is organized as follows. In Section 2, we develop a dynamic programming model for the standing order inventory system described above, which incorporates both the back-logging and lost-sales cases. In Section3, we present a method for computing the optimal dispose-down-to level and order-up-dispose-down-to level. In Section4, we con-clude this paper.

2. A dynamic programming model

Let n be the demand of a period and u(Æ) its prob-ability density function. Demand is assumed to be non-negative and independently distributed in dif-ferent periods. In addition, we use the following notation.

l average demand of a period R the standing order size C the unit item cost

Ce the unit cost via the emergency mode

Cs the unit revenue of excess inventory sold oﬀ

h the inventory cost per unit held per period p the shortage cost per unit per period in the

backlogging case

p the shortage cost per unit in the lost-sales case (p should be larger than its counterpart p, for it usually includes the sales price) L expected holding and shortage costs of a

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a the one-period discount factor, 0 < a 6 1 I net inventory (i.e., on-hand inventory

minus backorder) in the backlogging case or on-hand inventory in the lost-sales case, before the receipt of R at a review epoch fn(I) the expected discounted cost of

procure-ment, holding, shortage, and emergency ordering (minus sell-oﬀ revenue) with n periods remaining until the end of the plan-ning horizon, given I at a review epoch, the standing order R, and an optimal policy is used

(X)+ max{X, 0}.

We assume Cs< C < Ce. Thus, it is not

econom-ical to order a positive quantity via the emergency mode while disposing of some inventory in the same period. Also, immediate delivery (and negligible ﬁxed costs) for emergency orders is assumed, as in

[10] and [15]. In addition, p > Ce is assumed (for

the use of the emergency mode to be meaningful). Let t(Æ) be a transition function that represents the starting inventory of the next review period. fn(I)

satisﬁes the recursive equation fnðIÞ ¼ min

QPRfY ðQÞ þ CR þ LðI þ R þ QÞ

þ aEfn1ðtðI þ R þ Q nÞÞg; ð1Þ

where f0(I) 0, Q is the quantity ordered via the

emergency mode (if positive) or the quantity dis-posed of (if negative) at a review epoch, and Y(Q) = max{CeQ, CsQ} is the emergency operation

cost which is piecewise linear. Note that in the back-logging problem, t(X) = X and L(Æ) is given by

LðX Þ ¼ Z Xþ 0 hðX nÞuðnÞ dn þ Z 1 Xþ pðn X ÞuðnÞ dn; ð2Þ

while in the lost-sales problem, t(X) = (X)+and

LðX Þ ¼ Z X 0 hðX nÞuðnÞ dn þ Z 1 X pðn X ÞuðnÞ dn: ð3Þ

It is assumed that Q PR, i.e., the quantity sold oﬀ at a review epoch is less than or equal to the standing order size (note that this is really not a restrictive assumption, as we shall see later that a stationary policy is optimal in the long run and

there is a maximum inventory level SU such that Q PR holds naturally). Notice that Rosenshine and Obee [15] use the undiscounted-cost (i.e., the average cost per period) criterion, while our model allows for both the undiscounted-cost and dis-counted-cost criteria (thus, the full unit emergency cost and unit sell-oﬀ revenue, rather than the mar-ginal cost or loss as in[15], should be used). Henig et al.’s model[10]is a special case of our model with Cs= 0. Also, both Henig et al. and Rosenshine and

Obee did not consider the lost-sales problem. Let Z = I + R + Q, i.e., the inventory level after a possible emergency order or disposal is made at a review epoch. We express model(1) by

fnðIÞ ¼ min

ZPIfY ðZ I RÞ þ CsRþ LðZÞ

þ aEfn1ðtðZ nÞÞg; ð4Þ

where the constant item cost (C Cs)R is excluded

for simplicity. Letting

GnðZÞ ¼ LðZÞ þ aEfn1ðtðZ nÞÞ; ð5Þ

we can write model(4)by fnðIÞ ¼ min ZPIfY ðZ I RÞ þ CsRþ GnðZÞg; ð6Þ which simpliﬁes to fnðIÞ ¼ min ZPIþRfCeZþ GnðZÞg CeðI þ RÞ þ CsR ð7Þ or fnðIÞ ¼ min I6Z6IþRfCsZþ GnðZÞg CsI; ð8Þ

depending on whether a possible emergency order or disposal is made at a review epoch.

Let Df and DDf be respectively the ﬁrst and sec-ond derivatives of the function f.

Lemma 1. fn(I) is convex.

Proof (By induction). f0(I) is convex (and Df0(I) P

Ce). Assume that fn1(I) is convex (and Dfn1(I) P

Cefor the lost-sales case). In the backlogging case,

fn(I) is convex since Y(Æ) is convex and the holding

and shortage costs in (1) are linear (apparently, these costs in L can be allowed to be not linear but convex). In the lost-sales case, it is seen from

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DGnðZÞ ¼ Z Z 0 huðnÞ dn Z 1 Z puðnÞ dn þ a Z Z 0 Dfn1ðZ nÞuðnÞ dn; DDGnðZÞ ¼ ðh þ pÞuðZÞ þ a Z Z 0 DDfn1ðZ nÞuðnÞ dn

þ aDfn1ð0ÞuðZÞ P ðh þ pÞuðZÞ

þ a Z Z

0

DDfn1ðZ nÞuðnÞ dn

aCeuðZÞ > 0;

since p > Ce. In addition, we see from(7) and (8)that

as Ce> Csand fn(I) is convex, Dfn(I) PCe. h

For inventory models with convex ordering costs, see, e.g., Porteus [14] for optimal policies. Here, we include a speciﬁc analysis of model (6)

with piecewise linear ordering costs (as in [10]). Let SLn minimize CeZ + Gn(Z) and SUnminimize

CsZ + Gn(Z). Since Cs< Ce, SLn is smaller than

SUn. It follows from (7) and (8) that the optimal

policy is to order the amount SLn I R at cost

Ce per unit if I + R 6 SLn, sell the amount

I + R SUn(respectively R) at price Cs per unit if

I + R P SUnP I (respectively if I P SUn), and do

nothing (i.e., neither order via the emergency mode, nor sell oﬀ inventory) if SLn6I + R 6 SUn (see

also Lemma 1 of[10]). In other words, Z¼ I; f_nðIÞ ¼ LðIÞ þ aEfn1ðtðI nÞÞ

if I P SUn; ð9Þ

Z¼ SUn; fnðIÞ ¼ CsðSUn IÞ þ LðSUnÞ

þ aEfn1ðtðSUn nÞÞ if Iþ R P SUnP I; ð10Þ Z¼ I þ R; f_nðIÞ ¼ CsRþ LðI þ RÞ þ aEfn1ðtðI þ R nÞÞ if SLn6Iþ R 6 SUn; ð11Þ Z¼ SLn; fnðIÞ ¼ CsRþ CeðSLn I RÞ þ LðSLnÞ þ aEfn1ðtðSLn nÞÞ if Iþ R 6 SLn: ð12Þ

We can see above that the optimal control policy for the ﬁnite-horizon model is governed by the two operational parameters: the emergency order-up-to level SLnand the dispose-down-to level SUn.

Notic-ing in (4) that the total ordering cost Y(Z I R) + CsR is non-negative and L(Æ) is also

non-negative, we have

Theorem 1. If a < 1, then as n! 1, lim SLn= SL,

lim SUn= SU, and SL and SU minimize CeZ + G(Z)

and CsZ + G(Z), respectively, where

GðZÞ ¼ LðZÞ þ aEf ðtðZ nÞÞ; fðIÞ ¼ min

ZPIfY ðZ I RÞ þ CsRþ GðZÞg:

Proof. (It is basically the same as that of the second part of Theorem 1 of [10].) We verify that condi-tions (a)–(d) and (f) in Theorem 8–15 of Heyman and Sobel [11] hold here. Conditions (b), (c), and (d) of the theorem are immediate. For condition (a), consider the (non-optimal) base-stock policy and let B(I) denote its (inﬁnite-horizon discounted) expected costs when initial inventory is I. By the non-negativity of L (and Y(Z I R) + CsR), fn

is monotone increasing, and since fn(I) 6 B(I) for

every n, condition (a) is valid. Furthermore, for a given I we get from (1)that the optimal Q satisﬁes Y(Q) + CR 6 B(I) because L and f0 are

non-nega-tive. Thus, Q can be bounded from above and con-dition (f) is valid. h

Hence, a stationary policy (SL, SU) is optimal in the long run for the discounted-cost criterion. SU is then the maximum inventory level after a possible emergency order or disposal is made at a review epoch (if we ignore the ﬁrst possible review epochs when I > SU).

Rosenshine and Obee [15] considered a storage capacity IMAX such that if inventory at a review epoch exceeds IMAX, the excess inventory is sold oﬀ (see Federgruen and Zipkin[9]for a related peri-odic problem with limited production capacity). Suppose that our basic model has such a storage constraint, i.e.,

fnðIÞ ¼ min

I6Z6IMAXfY ðZ I RÞ þ CsRþ LðZÞ

þ aEfn1ðtðZ nÞÞg: ð13Þ

If the optimal SU obtained (by usingTheorem 2 be-low) without the constraint Z 6 IMAX is less than or equal to IMAX, the storage capacity will not constitute an eﬀective constraint. Otherwise, assume that SUn> IMAX but SLn< IMAX (if

IM-AX 6 SLn as well, IMAX is the only operational

parameter for fn(I) and the analysis is simpliﬁed

and thus omitted). Then,

Z ¼ IMAX; f_nðIÞ ¼ CsðIMAX IÞ þ LðIMAXÞ

þ aEfn1ðtðIMAX nÞÞ if Iþ R P IMAX;

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Z ¼ I þ R; f_nðIÞ ¼ CsRþ LðI þ RÞ þ aEfn1ðtðI þ R nÞÞ if SLn6Iþ R 6 IMAX; ð15Þ Z ¼ SLn; fnðIÞ ¼ CsRþ CeðSLn I RÞ þ LðSLnÞ þ aEfn1ðtðSLn nÞÞ if Iþ R 6 SLn: ð12Þ

Lemma 2. If IMAX < SUn, fn(I) is a convex

function.

Proof (By induction). f0(I) is convex. Assume that

fn1(I) is convex. Gn(Z) is convex (as shown in the

proof of Lemma 1). It follows from (8) that as SUn minimizes CsZ + Gn(Z) and IMAX < SUn,

Dfn(I) <Cs if I + R < IMAX; on the other hand,

we see from (14) that Dfn(I) =Cs if

I + R P IMAX. Also, by(7) and (12), Dfn(I) =Ce

if I + R 6 SLn and Dfn(I) P Ce if I + R P SLn.

Since Cs< Ce and fn(I) is convex for

SLn6I + R 6 IMAX by (15), it follows that fn(I)

is convex. h

Also, Theorem 1 holds here (without lim SUn= SU that minimizes CsZ + G(Z)).

3. Computing SL and SU

Theorem 1does not reveal how to obtain SL and SU. Conjecturing that an (SL, SU) policy continues to be optimal over an inﬁnite horizon for the aver-age-cost criterion, Henig et al. used the Markov chain approach for computing SL and SU. Here, we conjecture as well that an (SL, SU) policy is opti-mal for our more general model if the long-run aver-age cost is to be minimized, and suggest using

Theorem 2 for computing SL and SU under either the average-cost or discounted-cost criterion.

Theorem 2. If there exists some n such that

(a) SUn= SUn1,

(b) Dfn(I) = Dfn1(I) for I 6 SUn, then SUi= SUn

and SLi= SLnfor i P n + 1.

Proof. If SUn= SUn1 and Dfn(I) = Dfn1(I) for

I 6 SUn, it follows from (5) that DGn+1(Z) =

DGn(Z) for I 6 SUn. As SUn minimizes CsZ +

Gn(Z), it also minimizes CsZ + Gn+1(Z), i.e.,

SUn+1= SUn. Also, due to SLn< SUn, SLn

mini-mizes CeZ + Gn+1(Z) as well, i.e., SLn+1= SLn. In

addition, by expressing fn+1(I) as in(9)–(12), it can

be easily seen that Dfn+1(I) = Dfn(I) for I 6 SUn+1.

Hence, the argument continues and SUi= SUn

and SLi= SLnfor all i P n + 1. h

As we see fromTheorem 2, if conditions (a) and (b) are satisﬁed, the sequences {SLi} and {SUi}

con-verge respectively to SL = SLnand SU = SUn and

thus the dynamic programming computation can be stopped (note that if a storage constraint is included and eﬀective,Theorem 2involves only con-dition (b) with SUnreplaced by IMAX). See Chiang

and Gutierrez[7]and Chiang[5]for a similar theo-rem that is applied to a backorder model in the two-supply-mode setting and a lost-sales model in the replenishment-cycle environment, respectively. SL and SU are then optimal operational parameters for the infinite-horizon model. Condition (a) is expected to be satisfied more quickly than condition (b), which is true of the following computation. The reason is that in most cases in practice there exists a minimum divisible quantity and demand occurs in a multiple of this quantity. Since demand in a period is non-negative and bounded, it follows that the state space for I is finite. Note that even if demand can occur in any finite non-negative amount, the state space must be discretized when implemented on a digital computer. Moreover, the space for SUn is also finite, since the order quantity is also

bounded in practice and orders will be placed in a multiple of the above divisible quantity.

As for condition (b), since Dfn(I) can be any real

number, to facilitate the computation, we use the following approximation: the ﬁrst derivatives of two consecutive cost functions could be regarded as equal when

Max

I6SUn

jDfnðIÞ Dfn1ðIÞj 6 e: ð16Þ

If e = 0.02, (16) was satisfied for all the 203 prob-lems in Tables 1–4(the average number of periods required is about 90). If e = 0.01,(16)was satisfied for all but five problems; if e = 0.005 instead, (16)

failed to be met for 22 problems. For these problems not solved for the inﬁnite horizon, the dynamic pro-gramming computation stopped in a period for which SLn and SUn are apparently incorrect (the

computation was aborted).

To illustrate, consider the base case: C = $100, Ce= $110, Cs= $90, l = 5 (with Poisson demand),

R = 5, a = 1, h = $1, p = $20. After solving, we ﬁnd that SL = 7 and SU = 16. In addition, we vary the

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value of Ce, Cs, and p in the base case to

investi-gate the eﬀect of these input parameters on the opti-mal control policy. Table 1 reports computational results for 27 problems. As we see, SL is non-increasing in Ce and SU is non-decreasing in Ce,

implying that emergency operations on both ends (whether purchases or disposals) are used less and less as Ce increases. Also, SU is non-increasing in

Cs and SL is non-decreasing in Cs, indicating that

emergency operations on both ends are used more frequently as Cs increases. In addition, as p

increases, both SL and SU tend to increase to avoid running out of goods (i.e., there would be

greater use of emergency purchases and lesser use of disposals).

In Table 2, we consider the lost-sales case and design the experiment such that p is equal to p + largest Ce in Table 1, and observe similar results

regarding how SL and SU will change due to an increase in the value of Ce, Cs, or p. Notice that if

SL 6 R, emergency orders are never placed. This is found in six problems ofTable 2where the diﬀer-ence between p and Ce is small. In addition, we

recall that the ordinary zero-time-lag lost-sales peri-odic problem could be viewed as a backorder model in which a credit of aC is given to each unit of

Table 1

Computation of the optimal operational parameters for a backlogged standing order system

Input parameters Operational parameters With a storage constraint p Cs C SL SU SL SU $2 0 110 3 23 4 20 150 5 26 5 20 200 7 30 8 20 50 110 1 18 Same 150 3 22 3 20 200 5 26 5 20 90 110 2 12 Same 150 1 18 Same 200 3 23 3 20 20 0 110 5 28 5 20 150 5 32 5 20 200 5 36 4 20 50 110 6 22 6 20 150 6 27 5 20 200 5 31 5 20 90 110 7 16 Same 150 6 22 6 20 200 5 28 5 20 200 0 110 9 31 9 20 150 9 35 8 20 200 9 39 8 20 50 110 9 25 9 20 150 9 30 9 20 200 9 34 8 20 90 110 10 18 Same 150 9 25 9 20 200 9 31 9 20

Data: l = 5 (with Poisson demand), R = 5, a = 1, h = $1, IMAX = 20.

Table 2

Computation of the optimal operational parameters for a lost-sales standing order system

Input parameters Operational parameters With a storage constraint p Cs Ce SL SU SL SU $202 0 110 8 30 8 20 150 7 33 7 20 200 2 34 2 20 50 110 8 24 8 20 150 7 28 7 20 200 2 30 2 20 90 110 9 17 Same 150 8 23 7 20 200 2 26 2 20 220 0 110 8 30 8 20 150 7 33 7 20 200 5 36 4 20 50 110 9 24 8 20 150 8 28 7 20 200 5 31 5 20 90 110 9 18 Same 150 8 24 8 20 200 5 28 5 20 400 0 110 9 31 9 20 150 9 35 9 20 200 9 39 8 20 50 110 10 25 10 20 150 9 30 9 20 200 9 34 8 20 90 110 10 19 Same 150 10 25 10 20 200 9 31 9 20

Data: l = 5 (with Poisson demand), R = 5, a = 1, h = $1, IMAX = 20.

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demand actually backlogged [18]. Here, if the lost-sales standing order model yields an optimal SL that is greater than or equal to R, it could also be viewed as a backorder model where a credit of aCeis given to each unit of demand actually

back-logged, i.e., L(Æ) is given by LðX Þ ¼ Z Xþ 0 hðX nÞuðnÞ dn þ Z 1 Xþðp aCeÞðn X ÞuðnÞ dn: ð17Þ

There are nine problems inTable 2where p aCeis

equal to p inTable 1. Three problems do not yield

SL that is greater than or equal to R and the other six have the same SL and SU as inTable 1.

Suppose that we add a storage constraint IMAX = 20 into the problems in Tables 1 and 2. The revised SU and SL are reported in the last two columns ofTables 1 and 2. As we see, when a storage constraint is included and eﬀective, SL may decrease. This is because if there is a storage capacity IMAX which is below SU, the buyer is more likely to have to sell oﬀ goods, and is thus more averse to spending money on an emergency order, thus lowering SL.

Assume now that a = 0.999 (other input parame-ters being equal). We solve the same problems in

Table 1 and observe similar results, as shown in

Table 3

Input parameters Operational parameters With a storage constraint p Cs Ce SL SU SL SU $2 0 110 4 22 4 20 150 6 25 6 20 200 8 28 9 20 50 110 1 17 Same 150 4 21 4 20 200 6 25 6 20 90 110 2 12 Same 150 2 17 Same 200 5 21 5 20 20 0 110 5 28 5 20 150 5 31 5 20 200 5 35 4 20 50 110 6 22 6 20 150 5 26 5 20 200 5 30 5 20 90 110 7 15 Same 150 6 21 6 20 200 5 26 5 20 200 0 110 9 31 9 20 150 9 34 8 20 200 9 38 8 20 50 110 9 24 9 20 150 9 29 9 20 200 9 33 8 20 90 110 10 18 Same 150 9 24 9 20 200 9 29 9 20

Data: l = 5 (with Poisson demand), R = 5, a = 0.999, h = $1, IMAX = 20.

Table 4

Input parameters R = 6 R = 5 R = 4 p Cs Ce SL SU SL SU SL SU $2 0 110 33 7 4 22 5 111 150 40 7 6 25 5 146 200 47 7 8 28 5 188 50 110 22 7 1 17 5 62 150 33 7 4 21 5 97 200 42 7 6 25 5 140 90 110 6 7 2 12 5 24 150 22 7 2 17 5 60 200 35 7 5 21 5 102 20 0 110 1 13 5 28 8 113 150 2 13 5 31 8 149 200 4 13 5 35 8 191 50 110 3 12 6 22 8 65 150 1 13 5 26 8 100 200 2 13 5 30 8 143 90 110 5 11 7 15 8 27 150 3 12 6 21 8 63 200 0 12 5 26 8 105 200 0 110 7 16 9 31 11 116 150 6 17 9 34 11 151 200 6 17 9 38 11 194 50 110 8 15 9 24 11 67 150 7 16 9 29 11 103 200 6 17 9 33 11 145 90 110 9 14 10 18 11 30 150 8 15 9 24 11 65 200 7 16 9 29 11 108

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Table 3. If we compare results in these two tables, SL and SU inTable 3are less than or equal to their respective counterparts inTable 1. This is possibly due to the fact that shortage becomes less costly if the discounted-cost criterion is used.

Moreover, we vary R for the 27 problems in

Table 3. As we see from Table 4, as R is larger, the system is enabled to operate with a smaller amount of inventory, i.e., both SL and SU tend to decrease. This is because as R is larger, the amount of inventory bought at the cheaper C (as opposed to Ce) increases, thus increasing the willingness of the

system to dispose of inventory more easily (i.e., decreasing SU) as well as wait for the next shipment rather than placing an emergency order (i.e., decreasing SL). In addition, if R < l and Ceis large,

SU could be very high, indicating that the system probably will never dispose of inventory, and if R > l and p is small, SL could be very low, implying that emergency orders are probably never placed.

4. Conclusion

In this paper, we propose a dynamic program-ming model for the standing order inventory system where a ﬁxed quantity is delivered to the buyer in each period. The proposed basic model incorporates both the backlogged and lost-sales cases (note that the model can actually handle the partial backlog-ging case by writing t(X) = (X)+ b(X)+

where b is the fraction of excess demand backlogged, and expressing L appropriately). It also can include a possible storage constraint. Also, Henig et al.’s model is a special case of the basic model with the unit sell-oﬀ revenue equal to zero.

Since demand is stochastic in the real world, a standing order system must allow for sell-oﬀs and emergency orders. It is shown that the optimal con-trol policy is governed by the two operational parameters: the dispose-down-to level and order-up-to level, and these two parameters can be computed by using a convergence theorem. Compu-tational results show that as the emergency unit item cost increases or as the unit sell-oﬀ revenue decreases, the optimal dispose-down-to level may increase while the optimal order-up-to level may decrease.

Notice that we assume throughout the whole paper that the fixed cost for sell-offs and emergency orders is zero or negligible. It is possible that the fixed cost for sell-offs and/or emergency orders is not negligible. This provides a future research

direc-tion. Also, it is assumed that the size of standing orders is not a decision variable of the basic model, i.e., the issue of the optimal standing order size is not examined in this paper. It seems that the optimal standing order size depends on the unit item cost, the emergency unit item cost, the unit sell-oﬀ reve-nue, and other cost parameters. This provides another research direction.

Acknowledgement

The author would like to thank the two anony-mous referees for their constructive comments and suggestions.

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