Optimal replenishment for a periodic review inventory system with two supply modes

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Production, Manufacturing and Logistics

Optimal replenishment for a periodic review inventory

system with two supply modes

Chi Chiang

*

Department of Management Science, National Chiao Tung University, 1001 Ta Hsueh Road, 300 Hsinchu, Taiwan, ROC Received 1 October 1999;accepted 3 May 2002

Abstract

In this paper, we study periodic inventory systems with long review periods. We develop dynamic programming models for these systems in which regular orders as well as emergency orders can be placed periodically. We identify two cases depending on whether or not a ﬁxed cost for placing an emergency order is present. We show that if the emergency supply mode can be used, there exists a critical inventory level such that if the inventory position at a review epoch falls below this critical level, an emergency order is placed. We also develop simple procedures for computing the optimal policy parameters. In all cases, the optimal order-up-to level is obtained by solving a myopic cost function. Thus, the proposed ordering policies are easy to implement.

Keywords: Inventory;Periodic review;Dynamic programming;Ordering policy

1. Introduction

In this paper, we study a periodic review inventory system in which there are two resupply modes: namely a regular mode and an emergency mode. Orders placed via the emergency mode, compared to orders placed via the regular mode, have a shorter lead time but are subject to larger variable costs and/or ﬁxed order costs.

Many studies in the literature address this problem. See, e.g., Veinott (1966) and Whittmore and Saunders (1977) for periodic review models, and Moinzadeh and Nahmias (1988) and Moinzadeh and Schmidt (1991) for continuous review models. Earlier periodic review models, however, have focused on the situation in which supply lead times are a multiple of a review period. Such models could be regarded as an approximation of continuous review models, as the review periods can be modeled as small as, say, one working day. In contrast with earlier studies, we consider periodic inventory systems in which the review periods are long so that they are possibly larger than the supply lead times. For example, a retailer may

*_{Tel.: +886-3-5712121x57125;fax: +886-3-5713796.}

E-mail address:[email protected](C. Chiang).

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place regular replenishment orders biweekly or monthly while the supply lead time is of the order of one week. This class of problems is important since periodic systems with long review periods are commonly used to achieve economics in the inventory review as well as in the coordination and consolidation of orders for diﬀerent items. A common way to implement the coordination mechanism is through a periodic system in which on a given day the inventory status of all items from the same supplier are reviewed and the appropriate orders are placed.

We will develop dynamic programming models for the periodic system described above, and devise optimal ordering policies that minimize the expected discounted cost over an infinite horizon, which in-cludes item cost, inventory holding cost, and shortage cost. Note that Chiang and Gutierrez (1996) study the exact same periodic system as in this paper. They find that either only the regular supply mode is used, or there exists an indifference inventory level such that if the inventory position at a review epoch falls below this level, an emergency order is placed. This result is similar to the results obtained in this paper. However, Chiang and Gutierrez investigate only the case in which emergency orders have same variable costs (as the regular orders) but larger fixed order costs. Moreover, they assume the use of an order-up-to-R policy and allow the placement of a regular order or an emergency order (not both) at a periodic review epoch while minimizing the expected total cost per period. The ordering policy developed in their model thus is not an optimal policy. In contrast, this paper considers more general cases in which placing an emergency order incurs larger variable costs and/or fixed order costs. Furthermore, we develop optimal policies that allow for possible regular and emergency orders at a periodic review epoch while minimizing the expected discounted cost over an infinite horizon. See Table 1 for the differences between these two papers.

In addition, Chiang and Gutierrez (1998) and Chiang (2001) study a diﬀerent inventory system with long review periods. They assume that while regular orders are placed periodically (as in this paper), emergency orders can be placed continuously since the inventory status of items is known at any time. However, the inventory status of items may not be continuously updated in practice. For example, a retailer may review

Table 1

Diﬀerences among the models in Chiang and Gutierrez (1996), Chiang and Gutierrez (1998), and this paper

Chiang and Gutierrez (1996) This paper Chiang and Gutierrez (1998) Periodic systems considered Long review periods Long review periods Long review periods (also

called order cycles) within which there are a number of short periods (of length of one working day) How the inventory status

of items is updated?

On a periodic basis On a periodic basis On a continuous basis When emergency orders

are placed?

At periodic epochs At periodic epochs At the beginning of each short period

Cost of placing an emergency order

A positive ﬁxed cost only A positive ﬁxed cost and/or a larger variable cost

A larger variable cost only How inventory carrying

cost is charged?

Based on average inventory Based on average inventory Based on inventory at the end of each short period How shortage cost is

charged?

Charged at the time shortage is satisﬁed

Charged at the end of each short period as long as shortage exists Criterion used Minimize expected total cost

per period

Minimize expected discounted cost over an inﬁnite horizon

Minimize expected discounted cost over an inﬁnite horizon Is the ordering policy

developed optimal?

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inventory only before coordinating the orders for a group of items to a distribution center biweekly or monthly. Also, reviewing inventory and placing orders on a periodic basis is desirable in situations such as when vendors make routine visits to customers to take new orders (see, e.g., Chase and Aquilano, 1995). In addition, if the supplier of a line of items accepts orders only at particular times (see, e.g., Starr, 1996), there is no reason why we should review inventory frequently. Moreover, if items are time-consuming, difficult, or expensive to count, they should be reviewed only periodically to save costs. Consequently, for all such situations, if inventory on hand for an item is dangerously low at any time between two periodic reviews, an emergency order could be placed only later when a periodic review takes place. Finally, it is also possible that placing emergency orders at other than review epochs is significantly more expensive for the same reason that a periodic policy is usually followed in the first place: consolidation of orders to a single vendor. To summarize, while Chiang and GutierrezÕs (1998) model allows the placement of emergency orders virtually at any time between two periodic reviews, our model places emergency orders only at periodic epochs. See Table 1 for other differences between these two papers.

We will consider two cases depending on whether or not a ﬁxed cost for placing an emergency order is present. We show that under certain conditions, the emergency supply channel is never used, and that under other conditions, the emergency channel can be used. For the latter case, we develop optimal policies and identify the critical inventory level at a review epoch such that if the inventory position at a review epoch falls below this critical level, an emergency order is placed. We also will develop simple procedures for computing optimal policy parameters. In all cases, the optimal order-up-to level is obtained by solving a myopic cost function. Thus, the proposed ordering policies are easy to implement.

2. Assumptions and notation

Suppose that we now review inventory at a periodic epoch t with n periods remaining (until the end of the planning horizon) and want to decide whether to place a regular order and/or an emergency order. Denote the length of a period by T. In this paper, as in Chiang and Gutierrez (1996), we assume that T is not a decision variable and is exogenously determined. Denote the unit item cost of the regular and emergency supply modes by c1and c2respectively, where c16c2and the lead time of the two modes by s1

and s2 respectively, where s2<s1. Assume, as in Chiang and Gutierrez (1996), that s2< T and s3¼

s1 s2< T (i.e., there is no order cross-over). Assume also that there is a ﬁxed cost K for placing an

emergency order, which represents the extra expense of making a special arrangement with the supplier. The fixed cost of placing a regular order is assumed to be negligible or zero. One possible reason for this is that a regular order for an item is part of a joint order (due to the consolidation of orders for different items, as mentioned in Section 1) which consists of a mix of products shipped in the same truck, and the ordering cost (including the transportation cost) for a joint order is incurred every time a joint order is placed and thus is irrelevant to individual items. Other reasons include the routine administrative work (for placing a regular order) and the use of information technology such as EDI. As a result, we will focus on the fol-lowing two cases in our analysis: (i) K¼ 0 and (ii) K > 0. It is assumed in the first case that c1< c2. Finally,

there is an inventory holding cost at a rate of h per unit held per unit time and a shortage cost at p per unit, charged at the time shortage is satisﬁed.

In addition to the cost and operation parameters deﬁned above, we will refer to the following notation. ðxÞþ maxfx; 0g.

Df the ﬁrst derivative of the function f. dðxÞ 1 if x is positive and 0 if x is 0. a the discount factor.

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X1 demand during the upcoming time interval ½t; t þ s1Þ (i.e., X1 is the demand during the regular

supply lead time).

X2 demand during the upcoming time interval½t; t þ s2Þ (i.e., X2 is the demand during the emergency

supply lead time).

X3 demand during the time interval½t þ s2; tþ s1Þ (i.e., X3is the demand during the inter-arrival time

between the two supply modes). Since X1¼ X2þ X3, X1 is stochastically larger than X2.

X4 demand during the time interval½t þ s1; tþ T þ s2Þ.

fiðXiÞ the probability density function (p.d.f.) of Xi, i¼ 0, 1, 2.

GðH ; r; RÞ the expected total cost per period (excluding a possible ﬁxed ordering cost of K) given that the inventory position (i.e., inventory on hand plus inventory on order minus backorder) at the review epoch is H, the inventory position after a possible emergency order is r, and the inventory position after a possible emergency order and a possible regular order is R. See Fig. 1 for a realization of the inventory process that depicts the model considered in this paper.

CnðH Þ the expected discounted cost with n periods remaining given that the inventory position is H and an

optimal ordering policy is used at every review epoch. CnðH Þ satisﬁes the functional equation

CnðH Þ ¼ min

H 6 r 6 RfKdðr H Þ þ GðH ; r; RÞ þ aEX0Cn1ðR X0Þg ð1Þ

where C0ðH Þ 0.

Other assumptions made in our analysis include the following:

(1) Shortage (if any) should be satisﬁed as early as possible. This means that if the inventory position is negative at a review epoch, an emergency order should be placed immediately and its size should be suﬃcient to raise the inventory position to a non-negative level, i.e., r should be greater than or equal to zero. In the next section, however, r is allowed to be negative to get a complete picture of the function GðH ; r; RÞ.

(2) Demand is non-negative and independently distributed in disjoint time intervals. Demand X in a time interval of length s has mean ks, where k is the mean demand rate per unit time (e.g., EðX3Þ ¼ ks3Þ. Also, demand is assumed to follow the normal, Poisson, gamma, or geometric

distri-bution.

(3) R is assumed to be greater than zero. In addition, let R be the minimum order-up-to level satisfy-ing PrðX1> RÞ 6 uR, where uR is a maximum allowable probability that a regular order placed fails

to clear any backorders (u_R ¼ 0:01, for example). We assume throughout the analysis that R P R,

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i.e., a regular order when arriving will clear any backorders with probability almost 1. Note that the ordinary (R, T) model, by assuming ﬁrst that backorders are incurred only in very small quan-tities, also assumes that a regular order when arriving is almost always suﬃcient to meet any out-standing backorders (see, e.g., Hadley and Whitin, 1963). The assumption R P R is reasonable if the ratio h=p (which determines the optimal probability of a stockout in the Newsboy problem) is not too large.

(4) In computing the inventory holding cost within a time interval, it is assumed that the inventory on hand decreases with a constant rate. In addition, the exact holding cost expressions which involve a variable in the denominator will be approximated by a lower bound (as in Chiang and Gutierrez, 1996). This assumption is similar to the one in the ordinary (R, T) model in which the expected on-hand inventory is approximately equal to the expected net inventory. As these approximations underestimate the true holding cost, the optimal order-up-to levels derived may be somewhat a little large.

3. An expression for GðH; r; RÞ

In this section, we develop an explicit expression for GðH ; r; RÞ. Notice that at the current review epoch t, the expected cost incurred before time Tþ s2 has been determined by the decision made in the previous

review. Thus, GðH ; r; RÞ, which consists of item cost, inventory holding cost, and shortage cost, is derived for the upcoming time interval½t þ s2; tþ T þ s2Þ. The item cost is simply c2ðr H Þ þ c1ðR rÞ. If a regular

order is placed at the current epoch, inventory on hand will increase at time tþ s1Hence, in developing the

inventory holding or shortage cost expression, we need to consider two disjoint time intervals½t þ s2; tþ s1Þ

and½t þ s1; tþ T þ s2Þ separately.

We ﬁrst evaluate the inventory holding cost for the time interval½t þ s2; tþ s1Þ. If r < X2 the on-hand

inventory is zero for any time during the interval ½t þ s2; tþ s1Þ. However, if r P X2 and the demand X3

during the time interval½t þ s2; tþ s1Þ is less than or equal to r X2, then the average inventory holding

cost over the interval½t þ s2; tþ s1Þ is hs3ðr X2 0:5X3Þ. On the other hand, if r P X2and X3> r X2the

average inventory holding cost is 0:5hs3ðr X2Þ 2

=X3. As the variable X3appears in the denominator, this

exact expression will substantially complicate the subsequent analysis. To overcome this problem, as mentioned in Section 2, we use a lower bound hs3ðr X2 0:5X3Þ for this expression, i.e., the same one

as when X36r X2. To summarize, if r P X2, the average inventory holding cost over the time

inter-val ½t þ s2; tþ s1Þ is hs3ðr X2 0:5X3Þ and when taking the expected value of X3, becomes hs3ðr

X2 0:5ks3Þ.

Similarly, if R P X1, the average inventory holding cost over the time interval ½t þ s1; tþ T þ s2 is

hðT s3Þ½R X1 0:5kðT s3Þ. However, as R P R is assumed, PrðX1> RÞ is approximately zero, i.e.,

PrðR P X1Þ approximately equals 1. Thus, the average holding cost over the interval ½t þ s1; tþ T þ s2Þ is

approximated by hðT s3Þ½R ks1 0:5kðT s3Þ.

Next, we develop the shortage cost expression of GðH ; r; RÞ. Since shortage cost is assumed to be charged at the time shortage is satisﬁed, we consider two upcoming epochs tþ s2 and tþ s1 when an emergency

order (if placed) and a regular order (if placed) arrive respectively. We ﬁrst evaluate the shortage cost incurred at the epoch tþ s2. There are two possible cases if shortage occurs: ﬁrst, the emergency order

will clear the backorders (i.e., if X26r), and second, the emergency order fails to clear the backorders

(i.e., if X2> r). In the former case, the amount of shortage is ðX2 H Þþ, while in the latter case, the

amount of shortage is the quantity ordered, i.e., r H . Similarly, if the regular order placed clears the backorders (i.e., if X16R), the amount of shortage isðX1 rÞþ;otherwise (i.e., if X1> R), it is the quantity

ordered (R r). Again, as R P R is assumed, PrðX1> RÞ is approximately zero. Thus, the latter case is not

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Combining the above analysis, we obtain GðH ; r; RÞ ¼ c2ðr H Þ þ c1ðR rÞ þ Z r 0 hs3ðr X2 0:5ks3Þf2ðX2Þ dX2 þ hðT s3Þ½R ks1 0:5kðT s3Þ þ p Z r H ðX2 H Þf2ðX2Þ dX2þ ðr H Þ Z 1 r f2ðX2Þ dX2þ Z 1 r ðX1 rÞf1ðX1Þ dX1 ¼ c2ðr H Þ þ c1ðR rÞ þ Z r 0 hs3ðr X2 0:5ks3Þf2ðX2Þ dX2 þ hðT s3Þ½R ks1 0:5kðT s3Þ þ p Z 1 H ðX2 H Þf2ðX2Þ dX2 Z 1 r ðX2 rÞf2ðX2Þ dX2þ Z 1 r ðX1 rÞf1ðX1Þ dX1 : ð2Þ Let G1ðH Þ ¼ c2Hþ p Z 1 H ðX2 H Þf2ðX2Þ dX2; ð3Þ G2ðrÞ ¼ ðc2 c1Þr þ Z r 0 hs3ðr X2 0:5ks3Þf2ðX2Þ dX2 þ p Z 1 r ðX1 rÞf1ðX1Þ dX1 Z 1 r ðX2 rÞf2ðX2Þ dX2 ; ð4Þ G3ðRÞ ¼ c1Rþ hðT s3Þ½R ks1 0:5kðT s3Þ: ð5Þ

Then GðH ; r; RÞ can be expressed as

GðH ; r; RÞ ¼ G1ðH Þ þ G2ðrÞ þ G3ðRÞ: ð6Þ

Notice that G1ðH Þ is easily seen to be convex. Also, G3ðRÞ in (5) is a linear (thus convex) function, since it is

derived based on the assumption R P R. We next analyze the functional form of G2ðrÞ. It follows from (4)

that for r 6 0, G2ðrÞ is equal to ðc2 c1Þr þ pðks1 ks2Þ, i.e., it is increasing on r (it is constant if c2¼ c1).

For r > 0, the ﬁrst and second derivatives of G2ðrÞ are given by

DG2ðrÞ ¼ ðc2 c1Þ þ hs3 Z r 0 f2ðX2Þ dX2 0:5ks3f2ðrÞ þ p Z 1 r f2ðX2Þ dX2 Z 1 r f1ðX1Þ dX1 ; ð7Þ DDG2ðrÞ ¼ hs3ff2ðrÞ 0:5ks3Df2ðrÞg þ pff1ðrÞ f2ðrÞg: ð8Þ

It can be seen from (7) that DG2ðrÞ ¼ ðc2 c1Þ þ hs3>0 as r becomes very large, and DG2ðrÞ ¼

ðc2 c1Þ 0:5hs3ks3f2ðrÞ as r approaches zero from the right. For the normal or Poisson demand, there is

at most one positive r, denoted by rI equating (8) to zero (see Appendix). It can be shown that this also

holds for the gamma or geometric demand. If rI does not exist, G2ðrÞ is convex on r > 0;otherwise if rI

exists, G2ðrÞ is concave for r 2 ð0; rIÞ and convex on r > rI (see Appendix for details). Consequently, it

follows that if c2> c1, then either G2ðrÞ is increasing on all r, or there exist ^rrand r , which are respectively

the (unique) local maximum and minimum of G2ðrÞ, such that G2ðrÞ is increasing on r < ^rrdecreasing on

r2 ð^rr; r Þ, and increasing on r > r _{(see Fig. 2). Note that ^}_rr_P_{0 and r} _>_{0. On the other hand if c} 2¼ c1,

then either G2ðrÞ is non-decreasing on all r, or it is constant on r 6 0, decreasing on r 2 ð0; r Þ, and

in-creasing on r > r _{(see Fig. 3). Notice that DG}

2ðrÞ approximately equals (c2 c1Þ þ hs3for r P R, since by

assumption, PrðX1> rÞ is approximately zero for r P R (and so is PrðX2> rÞ). This implies that if G2ðrÞ has

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4. The case where ﬁxed cost of ordering is zero

We ﬁrst analyze the case in which there is no ﬁxed cost for placing an emergency order, i.e., K¼ 0 (and thus c2> c1). We show that under a certain condition, the emergency supply channel is never used, and that

under other conditions, the emergency channel can be used. For the latter situation, we show that there exists a critical level such that if the inventory position at a review epoch is below this critical level, an emergency order is placed. We also suggest a simple myopic cost function to solve for the optimal policy parameters. Hence, the proposed inventory control policy is easy to implement.

We ﬁrst state in Theorem 1 the conditions under which the emergency supply channel is never used and can be used respectively.

Theorem 1. Suppose that K ¼ 0. If G2ðrÞ is increasing on all r, the emergency supply channel will never be

used; otherwise if G2ðrÞ is increasing on r < ^rr, decreasing on r2 ð^rr; r Þ, and increasing on r > r , the

emer-gency channel can be used, i.e., if H < r _{, an emergency order is placed.}

To verify Theorem 1, we see from (1) and (6) that CnðH Þ is expressed as

CnðH Þ ¼ min

H 6 r 6 RfG2ðrÞ þ G3ðRÞ þ aEX0Cn1ðR X0Þg þ G1ðH Þ: ð9Þ

Fig. 2. G2ðrÞ with c2> c1.

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If G2ðrÞ is increasing on all r, then (9) reduces to

CnðH Þ ¼ min

H 6 RfG3ðRÞ þ aEX0Cn1ðR X0Þg þ G1ðH Þ þ G2ðH Þ; ð10Þ

i.e., r¼ H . This implies that the emergency supply channel will never be used, and thus the ordinary order-up-to-R policy is optimal. It can be seen from (7) that if p is very small (in comparison to c2 c1), DG2ðrÞ

may be positive for all r > 0.

Thus, we assume for the remaining of this section that G2ðrÞ has the shape as depicted in Fig. 2. We show

in the following that if the inventory position at a review epoch falls below r _{, an emergency order is placed.}

Notice that (9) can be written as CnðH Þ ¼ min

H 6 Rf minH 6 r 6 RfG2ðrÞg þ G3ðRÞ þ aEX0Cn1ðR X0Þg þ G1ðH Þ: ð11Þ

Since we assume that R P R, minH 6 r 6 RG2ðrÞ reduces to minH 6 rG2ðrÞ due to r < R. Hence, (11) simpliﬁes

to

CnðH Þ ¼ min

H 6 RfG3ðRÞ þ aEX0Cn1ðR X0Þg þ G1ðH Þ þ minH 6 rG2ðrÞ: ð12Þ

As we can see from Fig. 2, there exists r0such that G2ðr0Þ ¼ G2ðr Þ. It follows that minH 6 rG2ðrÞ ¼ G2ðH Þ

for H 6 r0, minH 6 rG2ðrÞ ¼ G2ðr Þ for H 2 ðr0; r Þ, and minH 6 rG2ðrÞ ¼ G2ðH Þ for H P r . In summary,

minH 6 rG2ðrÞ is a non-decreasing function of H. Deﬁne for each review epoch JnðRÞ as

JnðRÞ ¼ G3ðRÞ þ aEX0Cn1ðR X0Þ: ð13Þ

Denote by RnðH Þ the value of R minimizing JnðRÞ for R P H (note that RnðH Þ P R as R P R is required) and

denote by R

nthe value of R minimizing JnðRÞ for all R P R (i.e., R nis the global minimum). If RnðH Þ or R nis

not an unique minimum, the smallest such value is chosen. In general, (12) is expressed as CnðH Þ ¼ JnðRnðH ÞÞ þ G1ðH Þ þ min

H 6 rG2ðrÞ ð14Þ

(note that JnðRnðH ÞÞ is also a non-decreasing function of H). In particular, for H 6 R n,

CnðH Þ ¼ JnðR nÞ þ G1ðH Þ þ min

H 6 rG2ðrÞ: ð15Þ

The optimal ordering policy at a review epoch with n periods remaining can be described in three diﬀerent cases: (i) if H 2 ðr0; r Þ, order amounts r H and Rn r at unit cost c2 and c1respectively, (ii) if H 6 R n

but H 62 ðr0; r Þ, order an amount R _n H at unit cost c1, and (iii) if H > R _n, order an amount RnðH Þ H at

unit cost c1. Notice that the reason of not placing an emergency order for H 6 r0is because the incremental

item costðc2 c1Þðr H Þ is too large and shortages which have occurred can still be satisﬁed by a regular

order. However, we assume that if shortages occur, they should be satisﬁed as early as possible. Hence, as long as H < r _{an emergency order is always placed to bring the inventory position to r} _{. As a note, the}

probability of H 6 r0 is virtually zero as r0 is usually a large negative number (this is shown in the

com-putation).

We observe in practice that in most cases there exists a minimum divisible quantity and demand occurs in a multiple of this quantity. Since demand in a review period is non-negative and bounded, if follows that the state space for H is finite. Note that even if there does not exist a minimum divisible quantity and demand can occur in any finite non-negative real amount, the state space must be discretized when im-plemented on a digital computer. Moreover, the action space is also finite since in practice the order quantity is also bounded and orders will be placed in a multiple of a divisible quantity. As the dynamic program defined by (9) is a Markov decision process, it follows (Blackwell, 1962) that there exists an

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optimal policy that is stationary if the planning horizon is inﬁnite. Hence, the sequencefR

ng can be

re-placed by a single number R _{for the inﬁnite horizon problem. Consequently, r} _{and R} _{constitute the}

optimal policy parameters for the inﬁnite horizon model. Thus, if H < r _{, order amounts r} _{H and}

R _r _{at unit cost c}

2and c1respectively;otherwise if H 2 ½r ; R , order an amount R H at unit cost c1.

Notice that H will never go above R _{as demand is non-negative.}

Next, we propose a simple model to compute the optimal order-up-to level R _{. (r} _{can be obtained by}

decreasing r from R until G2ðrÞ starts increasing.) Denote by CðH ; RÞ the expected cost per period for an

inventory position of H if r _{and an order-up-to level R are used for an inﬁnite horizon. Then, it follows}

that CðH ; RÞ ¼ G1ðH Þ þ G2ðr Þ þ G3ðRÞ for H < r ; CðH ; RÞ ¼ G1ðH Þ þ G2ðH Þ þ G3ðRÞ for H P r : Deﬁne FðH Þ as FðH Þ G1ðH Þ þ G2ðr Þ for H < r ; FðH Þ G1ðH Þ þ G2ðH Þ for H P r :

Then CðH ; RÞ ¼ F ðH Þ þ G3ðRÞ, i.e., the expected cost per period decomposes into a function of H (the

state) and a function of R (the action). Also, since the inventory position at a review epoch will never go above R, the order-up-to level R is always feasible at a review epoch. Moreover, the state H at a review epoch depends only on the action R at the previous review epoch and does not depend on the state at the previous epoch. As a result, all the conditions of Theorem 1 in Sobel (1981) are satisﬁed. It follows that minimizing a myopic cost function given by

JðRÞ ¼ G3ðRÞ þ aEX0FðR X0Þ ð16Þ

will minimize the expected discounted cost over an inﬁnite horizon. To minimize JðRÞ, we note that J ðRÞ is a convex function as FðH Þ and G3ðRÞ are convex. Thus, the optimum R can be obtained by, for example,

increasing R from R until JðRÞ starts increasing. In summary, we use the following method to obtain the optimal policy parameter r _{and R} _.

Step 1. Decrease r from R (or a large value of r) until r¼ r _{where DG}

2ðr Þ ¼ 0. As a note, if DG2ðrÞ never

becomes negative as r decreases, the emergency channel is never used.

Step 2. Increase R from R (or decrease R from a large value of R) until R¼ R _{(and JðRÞ starts increasing)}

where JðR _{Þ is the minimum of J ðRÞ.}

Note that if we let C0ðH Þ F ðH Þ, the expected discounted cost over an inﬁnite horizon is equal to

FðH Þ þ J ðR _{Þ=ð1 aÞ for any H 6 R} _{which depends on the initial state H. Hence, in Section 6, we will use}

JðRÞ only for computing the savings obtained from employing two supply modes. Incidentally, if only the regular supply mode is used, the optimal order-up-to level is obtained by solving also (16) except that FðH Þ G1ðH Þ þ G2ðH Þ for all H 6 R.

5. The case where ﬁxed cost of ordering is positive

In this section, we analyze the case in which there is a positive ﬁxed cost for placing an emergency order, i.e., K > 0. We show that a similar optimal policy exists, i.e., if the emergency supply channel can be used, there exists a critical level such that if the inventory position at a review epoch falls below this critical level, an emergency order is placed. We also propose a myopic cost function to solve for the optimal policy parameters.

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We ﬁrst state the condition under which the emergency supply channel is never used, no matter whether c2¼ c1 or not.

Lemma 1. Suppose that K > 0. If G2ðrÞ is non-decreasing on r, the emergency supply channel will never be

used.

To prove Lemma 1, we see from (1) and (6) that CnðH Þ is expressed by

CnðH Þ ¼ min H 6 r 6 RfKdðr H Þ þ G2ðrÞ þ G3ðRÞ þ aEX0Cn1ðR X0Þg þ G1ðH Þ ¼ min H 6 rfKdðr H Þ þ G2ðrÞ þ minr 6 RfG3ðRÞ þ aEX0Cn1ðR X0Þgg þ G1ðH Þ: ð17Þ Deﬁne JnðRÞ as JnðRÞ ¼ G3ðRÞ þ aEX0Cn1ðR X0Þ ð18Þ

and let FnðrÞ ¼ minr 6 RfJnðRÞg. Then CnðH Þ is expressed by

CnðH Þ ¼ min

H 6 rfKdðr H Þ þ G2ðrÞ þ FnðrÞg þ G1ðH Þ: ð19Þ

Denote by RnðrÞ the (smallest) value of R minimizing JnðRÞ for R P r (as before, RnðrÞ P R since R P R is

required) and R _nthe (smallest) value of R minimizing JnðRÞ for all R P R. In general, FnðrÞ ¼ JnðRnðrÞÞ. In

particular, for r 6 R

n, FnðrÞ ¼ JnðR nÞ. FnðrÞ is a non-decreasing function of r. Thus, if G2ðrÞ is also

non-decreasing on r, it follows from (19) that the emergency supply channel will never be used. Notice that this condition is similar to the condition for employing only the regular channel in Theorem 1.

We assume in the subsequent analysis that G2ðrÞ has the shape as depicted in Fig. 2 or Fig. 3. We ﬁrst

assume that c2> c1. We state in Theorem 2 the conditions under which the emergency supply channel is

never used and can be used respectively.

Theorem 2. Suppose that K > 0 and c2> c1. Suppose also that G2ðrÞ is increasing on r < ^rr, decreasing

on r2 ð^rr; r _{Þ, and increasing on r > r} _{. If K P G}

2ð^rrÞ G2ðr Þ, the emergency channel will never be used.

Otherwise if K < G2ð^rrÞ G2ðr Þ, the emergency channel can be used, i.e., there exists a critical level which is

less than r _{(i.e., the critical level when K}_{¼ 0, other things being equal), such that if H falls below this level, an}

emergency order is placed.

To show Theorem 2, we observe that as FnðrÞ is non-decreasing on r, if K P G2ð^rrÞ G2ðr Þ in Fig. 2, then

r¼ H , i.e., the emergency mode will never be used at any review epoch. We assume that K < G2ð^rrÞ G2ðr Þ. Deﬁne rU¼ inffr > ^rrjG2ðrÞ 6 K þ G2ðr Þg and rL ¼ supfr < ^rrjG2ðrÞ 6 K þ G2ðr Þg. As rU

and rL are independent of n, they are the same for all review epochs. Since r < Rand FnðrÞ is constant for

r < R, it follows that the optimal policy at a review epoch with n periods remaining is (i) if H 2 ðrL; rUÞ,

order amounts r _{H and R}

n r at unit cost c2 (plus a ﬁxed cost of K) and c1 respectively, i.e.,

CnðH Þ ¼ K þ G2ðr Þ þ Fnðr Þ þ G1ðH Þ ¼ K þ G2ðr Þ þ JnðR nÞ þ G1ðH Þ; ð20Þ

(ii) if H 6 R

n but H 62 ðrL; rUÞ, order an amount R n H at unit cost c1, i.e.,

CnðH Þ ¼ G2ðH Þ þ FnðH Þ þ G1ðH Þ ¼ G2ðH Þ þ JnðR nÞ þ G1ðH Þ; ð21Þ

and ﬁnally (iii) if H > R

n, order an amount RnðH Þ H at unit cost c1, i.e.,

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The reason of not placing an emergency order in case of H 6 rL is the large incremental item cost and/or

ﬁxed ordering cost. Since we assume that shortages (if any) should be satisﬁed as early as possible, we require that as long as H < rU, an emergency order is always placed to bring the inventory position to r

(note again that the probability of H 6 rL is approximately zero as rL is usually negative). Notice (by the

deﬁnition of rU) that the critical level rUis a decreasing function of K as long as K < G2ð^rrÞ G2ðr Þ. If we

use the same reasoning as for the case of K¼ 0, we see that there also exists an optimal policy that is stationary if the planning horizon is inﬁnite. Hence, the sequencefR

ng can be replaced by a single number

R _{for the inﬁnite horizon problem. Consequently, r}

U, r , and R constitute the optimal policy parameters

for the inﬁnite horizon model. Thus, if H < rU, the optimal policy is to order amounts r H and R r at

unit cost c2 (plus a ﬁxed cost of K) and c1 respectively;otherwise if H2 ½rU; R , the optimal policy is to

order an amount R _{H at unit cost c} 1.

As in the case of K¼ 0, we suggest a simple model to compute the optimal order-up-to level R _{. Denote}

by ^CCðH ; RÞ the expected cost per period for an inventory position of H if rU, r , and an order-up-to level R

are used for an inﬁnite horizon. It follows that ^ C CðH ; RÞ ¼ K þ G1ðH Þ þ G2ðr Þ þ G3ðRÞ for H < rU; ^ C CðH ; RÞ ¼ G1ðH Þ þ G2ðH Þ þ G3ðRÞ for H P rU: Deﬁne ^FFðH Þ as ^ F FðH Þ K þ G1ðH Þ þ G2ðr Þ for H < rU; ^ F FðH Þ G1ðH Þ þ G2ðH Þ for H P rU:

Then ^CCðH ; RÞ ¼ ^FFðH Þ þ G3ðRÞ. By verifying the conditions of Theorem 1 in Sobel (1981) as we did for the

case of K¼ 0, it follows that minimizing a myopic cost function given by ^

JJðRÞ ¼ G3ðRÞ þ aEX0FF^ðR X0Þ ð23Þ

attains optimality for the inﬁnite horizon model.

To minimize ^JJðRÞ, we observe that ^JJðRÞ is not a convex function as ^FFðH Þ is not. However, we see that D ^FFðH Þ < DF ðH Þ for H 2 ðrU; r Þ and D ^FFðH Þ ¼ DF ðH Þ for H < rU or H P r . This implies that

D ^JJðRÞ 6 DJ ðRÞ. If R _{is the optimal order-up-to level that minimizes J}_{ðRÞ, then DJ ðR} _{Þ ¼ 0 if R} _{6¼ R (the}

following theorem is also true if R _{¼ RÞ, which indicates D ^}_{JJðRÞ 6 DJ ðRÞ < 0 for R < R} _{(due to the}

convexity of JðRÞ). This completes the proof the following theorem.

Theorem 3. The optimal order-up-to level minimizing ^JJðRÞ is greater than or equal to the optimal order-up-to level minimizing JðRÞ.

To obtain the optimum R _{that minimizes ^}_{JJðRÞ, Theorem 3 gives a lower bound on R} _{while a large value of}

R serves as an upper bound. Thus, we can perform a simple search on the values of R between these two bounds and compare ^JJðRÞ. In summary, we use the following method to obtain the optimal policy pa-rameters rU, r and R .

Step 1. Obtain r _{by using the same procedure as in the cost of K}_{¼ 0.}

Step 2. Decrease r from r _{until r}_{¼ r}

U where G2ðrUÞ 6 K þ G2ðr Þ and G2ðrU 1Þ > K þ G2ðr Þ. As a

note, if there does not exist rU, indicating K P G2ð^rrÞ G2ðr Þ, the emergency channel is never

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Step 3. Set on R _{a lower bound (e.g., R or the value of R minimizing JðRÞ) as well as an upper}

bound. Search exhaustively on the values of R between these two bounds and select the R minimiz-ing ^JJðRÞ.

Likewise, we can derive the optimal ordering policy for the case of c2¼ c1. As we see from Fig. 3, if

KP G2ð0Þ G2ðr Þ, the emergency supply channel will never be used. We assume that K < G2ð0Þ G2ðr Þ.

Deﬁne rB¼ inffr > 0jG2ðrÞ 6 K þ G2ðr Þg. Then the optimal policy for the inﬁnite horizon model is (i)

if H < rB, order amounts r H and R r at unit cost c2(plus a ﬁxed cost of K) and c1respectively, and

(ii) otherwise if H 2 ½rB; R , order an amount R H at unit cost c1. To obtain the optimal order-up-to

level R _{in this case, we also minimize (23) except that ^}_F_F_{ðH Þ K þ G}

1ðH Þ þ G2ðr Þ for H < rBand ^FFðH Þ

G1ðH Þ þ G2ðH Þ for H P rB. A lower bound on R is obtained by pretending K¼ 0 and solving the resulting

convex function of (23) for the optimal R. A similar solution procedure (to the one for the case of c2> c1)

also can be employed.

6. Computational results

In this section, we present some computational results for the optimal ordering policies we develop in Sections 4 and 5.

We first assume that the fixed ordering cost K is zero. We investigate the effect of demand variance, unit shortage cost, emergency unit cost, and carrying cost/unit/unit time on the performance of the proposed model, as compared to that of the regular-mode-only model. Table 2 gives the sensitivity results for the first

Table 2

Computational results: sensitivity to demand variance and unit shortage cost

Parameters Regular-mode-only model Proposed model Savings (%) r2 _p _R _JðR _Þ _r 0 r R JðR Þ 10,000 $1.25 310 2701.7 a 2.5 416 2781.0 25 136 415 2775.5 0.20 5 489 2844.4 200 195 458 2821.5 0.81 10 545 2897.0 658 234 491 2855.3 1.44 40 629 2980.6 3591 290 542 2909.6 2.38 70 657 3008.7 6568 309 559 2928.3 2.67 100 673 3025.4 9554 320 568 2939.4 2.84 2500 1.25 343 2671.1 a 2.5 404 2717.2 42 144 387 2709.8 0.27 5 444 2751.8 265 173 406 2729.5 0.81 10 474 2779.8 744 192 422 2746.9 1.18 40 519 2825.2 3711 220 447 2775.2 1.77 70 533 2841.0 6699 230 457 2785.1 1.97 100 542 2850.5 9692 235 462 2791.1 2.08 1250 1.25 350 2662.5 a 2.5 402 2698.4 53 146 381 2689.8 0.31 5 431 2723.2 284 166 395 2705.3 0.66 10 452 2743.1 770 180 405 2717.7 0.93 40 484 2775.2 3746 200 423 2737.7 1.35 70 494 2786.4 6738 206 429 2744.6 1.50 100 501 2793.1 9733 210 433 2748.8 1.59 Data: k¼ 250:0, T ¼ 1:0, s1¼ 0:6, s2¼ 0:2, K ¼ $0, h ¼ $1.0/unit/unit time, c1¼ $10, c2¼ $11, a ¼ 0:98.

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two parameters and Table 3 gives the results for the last two parameters. The average CPU time of ob-taining the optimal order-up-to level for the proposed model is 0.33 seconds on an IBM 3081. Demand is assumed to be normal with mean ks and variance r2_s_{for a time interval of length s.}

It is clear from Table 2 that as the unit shortage cost p increases (other things being equal), the per-centage savings of the proposed model in comparison to the regular-mode-only model increases, i.e., the proposed model becomes more attractive. The same phenomenon is also observed by the changes in the demand variance r2_{. Other things being equal, the proposed model performs better as r}2_{increases. This is}

because a small amount of safety stock may be enough to avoid stockouts if demand variability is low. On the other hand, the use of a faster supply mode may also be needed in addition to carrying a relatively large amount of safety stock if demand variability is high. These ﬁndings agree with the ones in Chiang and Gutierrez (1996). Notice that r0is negative in our computation and PrðH 6 r0Þ is virtually zero.

Next, we see from Table 3 that other things being equal, the percentage savings of the proposed model increases as the carrying cost h per unit and unit time increases. Intuitively, if h is high, it is not advan-tageous to carry a large amount of inventory for a long period of time and thus the use of the emergency mode may be needed. In addition, we observe from Table 3 that as the emergency unit cost c2increases

(other things being equal), the proposed model becomes less attractive. Accompanying this is the result that as c2increases, the optimum R of the proposed model increases while the optimum r decreases, reﬂecting

the fact that the probability of using the emergency mode, i.e., PrðX0> R r Þ, becomes smaller. It can be

seen from (7) that as c2increases beyond a certain value (other things being equal), DG2ðrÞ P 0, i.e., only

the regular mode is used.

We next assume that there is a positive ﬁxed ordering cost K. Intuitively, if K > 0 (other things being equal), the expected cost over an inﬁnite horizon will be larger than when K¼ 0. For example, if we compare the savings of the case of s1¼ 0:6 and c2¼ $11 in Table 4 to the savings of the same case in Table

3, the savings is down from 1.77% to below 1.57%;and the larger the K (other things being equal), the smaller the savings of the proposed model. It is of no doubt that if the incremental ordering cost and/or unit cost is larger, the emergency mode is of less value to employ. Note that rL is negative in the computation

and PrðH 6 rLÞ is virtually zero.

Finally, for the case of c1¼ c2, we compare the performance of the proposed model and Chiang and

GutierrezÕs model. To facilitate the comparison, we compute the expected cost per period for the proposed model (using the optimal policy parameters rB, r and R obtained for minimizing the expected discounted

cost over an inﬁnite horizon) as follows:

EX0CCðH ; RÞ ¼ E^ X0FF^ðR X0Þ þ G3ðRÞ: ð24Þ

Table 3

Computational results: sensitivity to emergency unit cost and carrying cost/unit/unit time

Parameters Regular-mode-only model Proposed model Savings (%) c2 h R JðR Þ r0 r R JðR Þ $10.5 $0.5 533 2700.8 7697 232 448 2664.0 1.36 1.0 519 2825.2 7645 228 427 2753.8 2.53 2.0 501 3057.8 7545 221 407 2918.3 4.56 11.0 0.5 533 2700.8 3735 223 478 2677.4 0.87 1.0 519 2825.2 3711 220 447 2775.2 1.77 2.0 501 3057.8 3664 216 418 2947.4 3.61 12.0 0.5 533 2700.8 1759 212 502 2687.3 0.50 1.0 519 2825.2 1748 210 477 2795.5 1.05 2.0 501 3057.8 1726 207 442 2986.0 2.35 Data: k¼ 250:0, r2_{¼ 2500, T ¼ 1:0, s} 1¼ 0:6, s2¼ 0:2, K ¼ $0, c1¼ $10, p ¼ $40/unit, a ¼ 0:98.

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Expression (24) excludes the item cost (by subtracting c1k), since Chiang and GutierrezÕs model does not

include it (as it is constant). As we see from Table 5, the proposed model yields smaller costs than Chiang and GutierrezÕs model, given that all parameters are the same.

Table 4

Computational results: optimal ordering policy when ﬁxed ordering cost is positive

Parameters Regular-mode-only model Proposed model Savings (%) s1 c2 K R JðR Þ rL rU=rB r R JJðR^ Þ 0.4 $10 $10 461 2804.9 – 158 181 408 2747.4 2.05 40 461 2804.9 – 144 181 413 2759.6 1.62 160 461 2804.9 – 123 181 434 2783.2 0.77 640 461 2804.9 – 93 181 455 2800.9 0.14 11 10 461 2804.9 1799 147 159 421 2767.8 1.32 40 461 2804.9 1769 136 159 427 2775.2 1.06 160 461 2804.9 1649 118 159 441 2789.4 0.55 640 461 2804.9 1169 90 159 457 2801.9 0.11 0.6 $10 $10 519 2825.2 – 219 240 413 2729.9 3.37 40 519 2825.2 – 204 240 418 2753.8 2.53 160 519 2825.2 – 181 240 484 2803.3 0.78 640 519 2825.2 – 147 240 513 2821.5 0.13 11 10 519 2825.2 3701 206 220 453 2780.9 1.57 40 519 2825.2 3671 195 220 469 2793.0 1.14 160 519 2825.2 3551 175 220 497 2810.8 0.51 640 519 2825.2 3071 143 220 514 2822.3 0.10 0.8 $10 $10 576 2844.5 – 277 297 419 2732.9 3.92 40 576 2844.5 – 261 297 420 2761.6 2.91 160 576 2844.5 – 237 297 548 2827.7 0.59 640 576 2844.5 – 201 297 571 2841.9 0.09 11 10 576 2844.5 5591 264 278 517 2809.5 1.23 40 576 2844.5 5561 252 278 535 2819.5 0.88 160 576 2844.5 5441 231 278 559 2833.6 0.38 640 576 2844.5 4961 196 278 573 2842.6 0.07 Data: k¼ 250:0, r2_{¼ 2500, T ¼ 1:0, s}

2¼ 0:2, h ¼ $1.0/unit/unit time, c1¼ $10, p ¼ $40/unit, a ¼ 0:98.

Table 5

Comparison of computational results obtained in Chiang and Gutierrez (1996) and this paper

Parameters Chiang and GutierrezÕs model Proposed model Savings (%) s1 p Ia R Cost per period rB r R Cost per period 0.4 $10 99 385 189.3 110 146 360 178.5 5.71 40 120 403 206.0 128 158 384 194.6 5.53 70 126 410 211.7 134 162 392 200.5 5.29 0.6 10 141 448 199.7 163 198 356 173.7 13.02 40 169 471 220.7 185 214 381 191.3 13.32 70 178 478 227.7 192 218 390 197.5 13.26 0.8 10 183 507 206.9 214 249 360 176.8 14.55 40 218 534 231.3 240 269 384 194.7 15.82 70 228 542 239.4 248 275 392 201.3 15.91 Data: k¼ 250:0, r2_{¼ 1250, T ¼ 1:0, s} 2¼ 0:2, h ¼ $1.0/unit/unit time, K ¼ $40. a

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7. Concluding remarks

In this paper, we develop dynamic programming models for an inventory system where regular orders as well as emergency orders can be placed periodically. We identify two important cases depending on whether or not a ﬁxed cost for placing an emergency order is present. We show that if the emergency supply channel can be used, there exists a critical inventory level such that if the in-ventory position at a review epoch falls below this level, an emergency order is placed. We also develop simple procedures for computing the optimal policy parameters. In all cases, the optimal order-up-to level is obtained by solving a myopic cost function. Thus, the proposed ordering policies are easy to implement.

Notice that we assume throughout the entire analysis that the cost of placing a regular order is zero. If there is also a positive ﬁxed cost for placing a regular order, the ordering policies developed in this paper are no longer optimal. The reason is that the policies developed will place at least a regular order as long as the demand during the preceding period is not zero. However, it is obvious that we do not order at all if the cost of placing a regular order is positive and the demand during the preceding period is very small (which is theoretically possible as demand is stochastic). For example, if the de-mand during the preceding period is only a few units compared to a mean dede-mand of several hundred, then ordering these few units at a review epoch is apparently not economical. This is just like the ordinary case of using only the regular supply mode. The ordering-up-to-R policy is optimal with zero ordering costs. On the other hand, the ordering-up-to-R policy is no longer optimal with positive ordering costs. There exists an additional parameter that triggers the placement of an order (it is well known that the (s, S) type policy is then optimal).

How to employ the two supply modes when a ﬁxed cost of placing a regular order is also present provides a future research direction. A preliminary investigation shows that the form of the optimal dering policy is rather complex. Nevertheless, since the review periods studied in this paper are long, or-dering always up to R _{(after a possible emergency order is placed) as described in Section 5 should be a}

nearly optimal policy if the ﬁxed cost of placing a regular order is not large (see, e.g., Hax and Candea, 1984), which is especially true nowadays when information technology such as EDI has increasingly gained use in industry.

Appendix

We show that for normal or Poisson demand, there is at most one positive r, denoted by rI, equating (8)

to zero.

For normal demand (with mean ks and variance r2_s_{for a time interval of length s), (8) reduces to}

DDG2ðrÞ ¼ hs3f2ðrÞf1:0 ðks3ðks2 rÞ=2r2s2Þg þ pff1ðrÞ f2ðrÞg ¼ pf2ðrÞ ðhs3=pÞ ðhs3ks3ðks2 rÞ=2r2s2pÞ þ ðf1ðrÞ=f2ðrÞÞ 1:0 ¼ pf2ðrÞ ðhs3=pÞ n ðhs3ks3ðks2 rÞ=2r2s2pÞ þ ðt1=t2Þðs2=s1Þ0:5expf0:5s3ðr2 k2s1s2Þ=r2s1s2g 1:0 o ðA:1Þ where t1 and t2 are normalizing constants (greater than 1) as demand is non-negative. As the

expres-sion within the big parentheses increases on r, there exists at most one positive r that can equate (A.1) to zero.

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For Poisson demand, (8) (interpreted as the second diﬀerence) is expressed by hs3P2ðrÞf1:0 ðs3ðks2 rÞ=2s2Þg þ pfP1ðrÞ P2ðrÞg

¼ pP2ðrÞ ðhsf 3=pÞ ðhs3s3ðks2 rÞ=2s2pÞ þ ðP1ðrÞ=P2ðrÞÞ 1:0g

¼ pP2ðrÞ ðhsf 3=pÞ ðhs3s3ðks2 rÞ=2s2pÞ þ ðs1=s2Þrexpf ks3g 1:0g ðA:2Þ

where P1ðÞ (resp. P2ðÞ) is the Poisson density function for demand during the time interval ½kT ; kT þ s1Þ

(resp.½kT ; kT þ s2Þ). As before, the expression within the big parentheses increases on r. Thus, there exists at

most one positive r that can equate (A.2) to zero.

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