An order expediting policy for continuous review systems with manufacturing lead-time

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Short Communication

An order expediting policy for continuous review systems with manufacturing

lead-time

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 8 January 2009 Accepted 12 August 2009 Available online 20 August 2009

Keywords: Supply mode Inventory Order expediting Order splitting Continuous review policy

a b s t r a c t

Expedited shipments are often seen in practice. When the inventory level of an item gets dangerously low after an order has been placed, material managers are often willing to expedite the order at extra ﬁxed and/or variable costs. This paper proposes a single-item continuous-review order expediting inventory policy, which can be considered as an extension of ordinary ðs; Q Þ models. Besides the two usual opera-tional parameters: reorder point s and order quantity Q, it consists of a third parameter called the expe-dite-up-to level R. If inventory falls below R at the end of the manufacturing lead-time, the buyer can request the upstream supplier to deliver part of an outstanding order via a fast transportation mode. The amount expedited will raise inventory to R, while the remaining order is delivered via a slow (reg-ular) supply mode. Simple procedures are developed to obtain optimal operational parameters. Compu-tational results show that the proposed policy can save large costs for a ﬁrm if service level is high, demand variability is large, the extra cost for expediting is small, or the manufacturing lead-time is long. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction

Expedited shipments are quite common in practice. When the inventory level of an item gets dangerously low after an order has been issued, material managers are often willing to expedite the order at extra ﬁxed and/or variable costs. By employing a fast transportation mode (e.g., by air), the buyer can have an outstand-ing order arrive earlier than planned. The use of the Internet has greatly enhanced the ability of a buyer to track the status of pur-chase orders. Through the online updating of orders or direct email communication (or simply phone calls), the buyer may request the upstream supplier to deliver an order or part of it via a fast supply mode if necessary.

Besides order expediting, a commonly suggested course of action when faced with an urgent situation is to place a faster or emergency order in addition to an already issued regular order (e.g.,Chiang, 2003). Although both options enable a ﬁrm to acquire materials in time and achieve the service level promised to customers, expedit-ing of existexpedit-ing orders does not create new orders that must be mon-itored or managed. AsPerona and Miragliotta (2004)emphasized, by restricting the placement of fast orders, a ﬁrm may reduce its operational complexity. However, to our knowledge, research on the issue of order expediting in inventory systems is limited.Allen and D’Esopo (1968)proposed an ðs; Q Þ policy with a third opera-tional parameter called the expediting level. They suggested that

when inventory drops to this level, an outstanding order is shipped after a short period.Bookbinder and Cakanyildirim (1999) consid-ered a deterministic-demand continuous review model where lead-time is made endogenous through an expediting factor. Also, Lawson and Porteus (2000) and Arslan et al. (2001)respectively investigated multi-echelon and make-to-order inventory systems with expediting, andBregman (2009)suggested a heuristic for solv-ing the dynamic probabilistic project expeditsolv-ing problem.

In this paper, we propose a new single-item inventory policy with order expediting. We consider continuous review systems where lead-time consists of two components: manufacturing lead-time and delivery lead-time. The former is the time needed for the supplier to manufacture the quantity ordered which includes the set-up time, job processing time and waiting time; the latter is the time needed to deliver the ﬁnished products. The former can be either constant or random depending on how con-gestion at the manufacturing facility affects the job waiting time; the latter is a deterministic interval, though it can take on two dif-ferent values corresponding respectively to the regular and fast transportation modes. Order expediting can occur due to the use of a fast mode and/or the effort to reduce the job waiting time. In this paper, we assume the manufacturing lead-time to be con-stant, and focus on the use of a fast mode to expedite supply. Note thatCakanyildirim et al. (2000)studied a deterministic-demand inventory model where lead-time is also made up of two periods: one for materials handling, waiting and set-up and the other for the manufacturing of a lot size. The time needed for delivery of a ﬁnished lot was not considered in their model.

*Tel.: +886 35712121 57166.

E-mail address:[email protected]

Contents lists available atScienceDirect

European Journal of Operational Research

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The proposed policy can be considered as an extension of ordin-ary ðs; Q Þ models. As an ðs; QÞ model operates, the proposed policy places an order for Q units whenever inventory drops to s; in addi-tion, it expedites part of an outstanding order if inventory falls be-low R at the end of the manufacturing lead-time. The amount expedited will raise inventory to R, which is thus called the expe-dite-up-to level, while the remaining order is delivered via a regular supply mode. Consequently, an outstanding order is split into two deliveries. The use of order splitting to reduce the shortage proba-bility and/or inventory costs has received much attention in the inventory literature (see, e.g.,Sculli and Wu, 1981; Kelle and Silver, 1990; Kelle and Miller, 2001; Lau and Zhao, 1993; Chiang and Ben-ton, 1994; Chiang and Chiang, 1996). Kiesmuller et al. (2005) investigated the use of both order splitting and expediting in a periodic review setting, and showed that an intelligent choice of supply modes could save considerable costs for a ﬁrm.

The proposed policy minimizes the expected total cost per unit time, subject to a service level constraint. Both the no-short-age probability and fill rate service measures are used in this research. We develop simple procedures to compute optimal operational parameters. Computational results indicate that the proposed policy yields significant cost savings, especially if service level is high, demand variability is large, the extra (fixed and/or variable) costs for expediting are small, or the manufacturing lead-time is long.

Notice thatChiang (2002)devised a different order expediting policy, where when inventory falls to an expediting level at the end of manufacturing, an entire order is sent via a fast supply mode. Simple thought reveals that if expediting of orders entails large costs, it is economical to expedite only part of an order. Also, Chiang’s model used only the no-shortage probability service mea-sure. Moreover, expediting supply may not be always possible. Krishnamoorthy and Raju (1998a,b)introduced the use of ‘‘local purchase” of the item which runs out of stock. They considered three different types of local purchases: for the ﬁrst two cases, a re-tailer purchases from some other nearby shop for one or s units of the item when a demand arises during the stock out period; for the third case, the retailer makes a local purchase of S units and also cancels the outstanding order.

The rest of this paper proceeds as follows. In Section 2, we review ordinary ðs; QÞ models. In Section3, we propose a continu-ous-review order expediting policy. Section 4 presents some computational results. Section5concludes this research.

2. Review of (s, Q) models

In an ðs; Q Þ model, when the inventory position (i.e., inventory on hand + inventory on order backlog) drops to s, an order for Q units is placed and received after a lead-time L. The literature on ðs; Q Þ models is huge and can be divided into two groups: one with an exact cost formulation and the other with an approximate cost formulation. This paper will employ both of the exact and approximate cost expressions. Note that fuzzy theory can also be used to develop an ðs; QÞ model (e.g.,Handﬁeld et al., 2009).

Let A be the ﬁxed cost of ordering and/or set-up for the manu-facturing of an order, c be the unit item cost, r be the inventory holding cost rate per unit time, D be the average demand per unit time, YTbe the demand during a time interval of length T, and fTðÞ

be its probability density function. Demand not ﬁlled immediately is backlogged and assumed to be continuous for convenience of notation. Under constant lead-time but otherwise fairly general conditions, the exact average inventory level is

Iexact¼ ð1=Q Þ Z sþQ s Z y 0 ðy nÞfLðnÞdn dy ð1Þ

(e.g.,Zheng, 1992), where the inventory position in steady state is uniformly distributed on ðs; s þ Q. On the other hand, the approxi-mate average inventory level is

Ia

¼ s DL þ :5Q ð2Þ

(Hadley and Whitin, 1963, pp. 162–167); Ia_{is accurate when the}

ex-pected number of shortages per cycle is quite small, or the shortage probability is sufﬁciently low (e.g.,Lau and Lau, 2002). It is assumed that Q is large enough for the probability of more than one out-standing order, i.e., PrðYL>Q Þ, to be negligible or approximately

zero (Hadley and Whitin, 1963, pp. 162 and 201). This assumption is often used in the literature of continuous review models (e.g., Kelle and Silver, 1990; Bookbinder and Cakanyildirim, 1999). The average total cost per unit time is expressed by

TCðs; QÞ ¼ AD=Q þ rcI; ð3Þ

where I is given by(1)or(2). Notice that the shortage cost is not in-cluded, because it is rarely out-of-pocket and quantifying it is diffi-cult in practice. Instead, management often specifies a desired service level. Service level is usually defined as the probability of no shortage during an order cycle, denoted by

a

, or the fraction of demand ﬁlled directly from stock (i.e., the ﬁll rate), denoted by b (see, e.g.,Silver et al., 1998for other service measures). In the for-mer case, one minimizes(3)subject to

PrðYL>sÞ 6 1

a

: ð4Þ

In the latter case, one minimizes(3)subject to

Z 1

s

ðn sÞfLðnÞdn 6 ð1 bÞQ : ð5Þ

To ﬁnd the optimal solution s_{and Q}

, we observe that as s be-comes lower, I is smaller, whether I is given by(1)or(2). Thus if a

a

service level is applied, s_{is obtained by ﬁnding the smallest}

(inte-ger, for discrete demand) value of s satisfying(4), and Q is ob-tained by the EOQ formula

Q ¼ ð2DA=rcÞ0:5

ð6Þ if I is given by(2), or obtained by directly minimizing(3)if I is given by(1).

If a b service level is applied, a Lagrangian formulation including (3) and (5)could be used and by solving the ﬁrst-order condition two equations involving s and Q are formed (e.g., Chiang and Chiang, 1996; Platt et al., 1997). However, the usual method of iter-ative substitution for ﬁnding s_{and Q}_{does not guarantee}

conver-gence of solutions. In this paper, we suggest that one ﬁnds the smallest value of s satisfying(5)for a given Q, i.e., s is set as a func-tion of Q, and then minimizes(3)over Q (e.g., using a Fortran pro-gram) whether I is given by (1) or (2) (see, e.g., Tijms and Groenevelt, 1984for a similar approach that determines s given Q in service-constrained inventory systems).

3. An order expediting inventory control policy

Assume that the lead-time L consists of two periods: the sup-plier’s manufacturing lead-time M and delivery lead-time N. While M is constant, N can be shortened to G (i.e., G < NÞ if a fast supply mode (e.g., by air) is used. Assume without loss of generality that the supplier arranges the order’s delivery at the end of M. (If the sup-plier has to arrange delivery at a time earlier than the end of M, the remaining manufacturing lead-time becomes part of G or N.) If inventory is dangerously low at the end of M, the buyer can request the upstream supplier to deliver part of an order via a fast mode. A natural question arises: how much of the outstanding order is expe-dited? The amount expedited cannot be too large in order to save procurement costs, while it cannot be too small because the

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on-hand inventory at the end of M plus the amount expedited are expected to satisfy backorders (if any) and meet demand during the upcoming time interval N. Thus, it seems that there is an inven-tory level (which is less than sÞ, denoted by R, such that if inveninven-tory falls to H < R at the end of M, the amount expedited equals R H while the amount not expedited is Q R þ H (seeFig. 1). We call R the expedite-up-to level.

Hence, we propose a new continuous review policy that con-sists of three decision variables: s; Q , and R, or equivalently,D;Q , and R whereD s R > 0. Let A0 be the ﬁxed cost of expediting and c0_{be the incremental (out-of-pocket) unit expediting cost. We}

next derive the average total cost per unit time for the proposed policy. Notice that the probability of expediting is PrðYM>DÞ and

the average amount expedited is

QE ZDþQ D ðf

D

ÞfMðfÞdf þ Q Z 1 DþQ fMðfÞdf; ð7Þ

which is a function ofDand Q. Since PrðYL>Q Þ is assumed to be

negligible, PrðYMPDþ Q Þ is approximately zero due to M < L

andD>0 and thus

QE

Z1

D

ðf

D

ÞfMðfÞdf; ð70Þ

which depends onDonly. Also,Chiang and Chiang (1996)showed that if an order is split into two deliveries in a continuous review system, the average cycle stock is reduced by the proportion of Q shipped in the second delivery times the average demand during the inter-arrival time between the two deliveries (see also Lau and Zhao, 1993in a two-supplier setting). Here, part of Q may be expedited (the expected proportion of Q expedited is QE=Q Þ as op-posed to delayed in Chiang and Chiang and the time between the expedited and normal shipments is N G. Thus, the average cycle stock is increased by ðQE=Q ÞDðN GÞ. It follows from(3)that the average total cost per unit time is written by

TCð

D

;R; Q Þ ¼ AD=Q þ ðA0_{D=Q ÞPrðY}

M>

D

Þ þ c0_DQE =Q þ rc½I þ ðQE=Q ÞDðN GÞ ¼ ðD=Q ÞfA þ A0PrðYM>

D

Þ þ ½c0_{þ rcðN GÞQ}E g þ rcI; ð8Þ

where I is given by(1)or(2)with s ¼Dþ R. Notice that the inven-tory position in steady state is still uniformly distributed on ðs; s þ Q if using(1)for I, since it remains unchanged with order expediting. Let

A fA þ A0PrðYM>

D

Þ þ ½c0þ rcðN GÞQEg: ð9Þ

Then,(8)can be written by the following expression, which is of the same form as(3)

TCð

D

;R; Q Þ ¼ AD=Q þ rcI ð10Þ

If service level is deﬁned as the probability of no shortage dur-ing an order cycle, one minimizes(10)subject to

Z D 0 Z 1 DþRf fNðnÞdn fMðfÞdf þ Z DþR D Z 1 DþRf fGðnÞdn fMðfÞdf þ Z 1 DþR fMðfÞdf þ Z 1 D fMðfÞdf Z1 R fNðnÞdn 6 1

a

: ð11Þ

The ﬁrst term of(11)is for the situation where expediting does not occur, while the second and third terms are the shortage probability before the expedited shipment arrives and the last term is the shortage probability before the remaining order arrives.

To ﬁnd the optimal combination ofD, Q , and R, we ﬁrst state the following lemma.

Lemma 1. For a certain value of D, as R decreases, the shortage probability during an order cycle, i.e., the left-hand side of (11), increases.

G

Amount expedited

Inventory level

M

N

M

Q

Amount not expedited

s

R

Time

N

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Proof. As R decreases, the four terms, except the second one, in the left-hand side of(11)increase. But the inner integral of the second term is less than one; hence, as R decreases, the second and third terms combined increase. h

The above lemma intuitively makes sense. Also, we observe that as R becomes lower, the total cost in(10)is smaller. Thus, for a cer-tainD, we ﬁnd the smallest (integer, for discrete demand) value of R satisfying(11)and Q can be found from(10)by the equation

Q ¼ ð2DA=rcÞ0:5 ð12Þ

if I is given by(2), or obtained by directly minimizing(10)if I is gi-ven by(1), as in the ordinary no-expediting model. Then, one can perform a simple search on (integer) values ofDin order to deter-mine the optimal solution.

On the other hand, if service is deﬁned as the ﬁll rate, one min-imizes(10)subject to Z D 0 Z 1 DþRf ½n ð

D

þ R fÞfNðnÞdn fMðfÞdf þ Z DþR D Z 1 DþRf ½n ð

D

þ R fÞfGðnÞdn fMðfÞdf þ Z 1 DþR ½f ð

D

þ RÞ þ DGfMðfÞdf þ Z1 D fMðfÞdf Z 1 R ðn RÞfNðnÞdn 6_{ð1 bÞQ :} _ð13Þ

The ﬁrst term in the left-hand side of(13)is the average backorder if expediting does not occur. The second and third terms are the average amount backlogged before the expedited shipment arrives, while the last term is the expected backorder before the remaining order arrives (notice the same range of integrals in(11) and (13)).

To ﬁnd the optimal combination ofD;Q , and R, we suggest that one treats(10)as a function ofD, as in the case of

a

service levels, and follows the solution procedure described above for the ordin-ary ðs; QÞ models. In other words, for a certainD, one ﬁnds the smallest value of R satisfying(13)for a given Q, due to the follow-ing lemma, and then minimizes(10)over Q whether I is given by (1)or(2).

Lemma 2. GivenDand Q, as R decreases, the fraction of demand not ﬁlled directly from stock, i.e., the left-hand side of(13)divided by Q, increases.

Proof. Omitted (similar to that ofLemma 1). h

Lemma 2echoesLemma 1and makes intuitive sense as well. Again, one carries out a simple search on values of D to obtain the optimal solution.

4. Computational results

Consider the base case: unit time = 1 year = 250 (working) days, D ¼ 500 units=year, A ¼ $16, A0

¼ $4, r ¼ 25%=year, c ¼ $4:0=unit, c0_{¼ $0:5=unit,} _{L ¼ :1 years,} _{M ¼ :08 years,} _{N ¼ :02 years,}

G ¼ :004 years,

a

(or b) = 99.9%. Demand is Poisson with mean DT for a period of length T. Note that the average inventory should be added by one-half if demand is Poisson and one uses Ia _{for I}

(Hadley and Whitin, 1963, pp. 186–187).

Consider ﬁrst employingChiang’s model (2002)that expedited a whole outstanding order if necessary and used only the

a

service level; the cost saving is only 0.80%. If the proposed policy is used, we ﬁnd thatD¼ 49; R ¼ 18; Q ¼ 128, and the savings is 2.92%. Noticing PrðYMPDþ Q Þ in(7)for the base case (which is

approx-imately zero), we see that it is very unlikely that one has a whole order shipped via the fast mode. This result seems to be observed throughout the computation. If a b service level is applied, then D¼ 49; R ¼ 11; Q ¼ 130, and the savings is 0.98%. More results are shown in Tables 1–3. It is clear that the proposed policy is superior to Chiang’s model. Also, the higher the service level, the larger the percentage savings given by the proposed policy. This re-sult is expected, for high service levels justify the expediting of part of an outstanding order.

Notice that we used both Iexact_{and I}a _{to compute the optimal}

solutions and their costs. The optimal operational parameters found are the same for all problems solved inTables 1–8, which is probably because we have set service at high levels, so that it is worthwhile to expedite part of an outstanding order. For a given optimal solution, the expected total costs computed by using Iexact and Ia_{differ by no more than $0.01 (only the exact costs are shown}

in the tables), except for three problems (where the expected costs differ by about $0.02 or $0.03). This result agrees withLau and Lau (2002), who found that Ia_{is more accurate than the inventory}

lit-erature implies.

InTables 4 and 5, we investigate the effect of the ratios A0=A and c0_{=c on the performance of the proposed policy. It can be seen that}

Table 1

Effect of theaservice level on performance ofChiang’s model, 2002; D ¼ 500 units=year, 1 year ¼ 250 days, A ¼ $16, A0

¼ $4, r ¼ 25%=year, c ¼ $4:0=unit, c0_{¼ $0:5=unit,}

L ¼ :1 years, M ¼ :08 years, N ¼ :02 years, G ¼ :004 years.

að%Þ Expediting is not allowed Expediting is allowed % Savings

s Q TCðs; Q Þ s Ea

Q TCðs; E; Q Þ

95.0 62 126 138.99 Expediting is not economical

99.9 73 126 149.99 71 12 127 148.79 0.80

99.99 78 126 154.99 74 16 127 152.20 1.80

a

E is called the expediting level in Chiang’s model.

Table 2

Effect of theaservice level on performance of the proposed policy; D ¼ 500 units=year, 1 year ¼ 250 days, A ¼ $16, A0

að%Þ Expediting is not allowed Expediting is allowed % Savings

s Q TCðs; Q Þ D R Q QE A _{TCðD; R; Q Þ}

95.0 62 126 138.99 49 11 128 .252 16.41 138.61 0.27

99.0 67 126 143.99 48 15 129 .345 16.55 142.14 1.28

99.9 73 126 149.99 49 18 128 .252 16.41 145.61 2.92

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Table 3

Effect of the b service level on performance of the proposed policy; D ¼ 500 units=year, 1 year ¼ 250 days, A ¼ $16, A0

bð%Þ Expediting is not allowed Expediting is allowed % Savings

s Q TCðs; Q Þ D R Q QE _A _{TCðD; R; Q Þ}

99.9 63 126 139.99 49 11 130 .252 16.41 138.62 0.98

99.99 69 132 146.11 47 16 131 .464 16.72 142.81 2.26

Table 4

Effect of the ratio A0_{=A on performance of the proposed policy; D ¼ 500 units=year, 1 year ¼ 250 days, A ¼ $16, r ¼ 25%=year, c ¼ $4:0=unit, c}0_{¼ $0:5=unit, L ¼ :1 years,}

M ¼ :08 years, N ¼ :02 years, G ¼ :004 years, andaor b ¼ 99:9%.

A0 _{No-shortage probability constraint} _{% Savings} _{Fill-rate constraint} _{% Savings}

D R Q QE _A _TC _D _R _Q QE _A _TC $16 52 17 128 .091 16.50 147.94 1.37 55 7 134 .029 16.17 139.84 0.11 8 49 18 129 .252 16.69 146.70 2.19 51 10 128 .129 16.38 139.48 0.37 4 49 18 128 .252 16.41 145.61 2.92 49 11 130 .252 16.41 138.62 0.98 2 47 19 128 .464 16.48 144.87 3.41 47 12 128 .464 16.48 137.88 1.51 0 45 20 128 .807 16.42 143.63 4.24 42 14 137 1.663 16.86 136.54 2.46 Table 5

Effect of the ratio c0_{=c on performance of the proposed policy; D ¼ 500 units=year, 1 year ¼ 250 days, A ¼ $16, A}0

¼ $4, r ¼ 25%=year, c ¼ $4:0=unit, L ¼ :1 years, M ¼ :08 years, N ¼ :02 years, G ¼ :004 years, andaor b ¼ 99:9%.

c0 _{No-shortage probability constraint} _{% Savings} _{Fill-rate constraint} _{% Savings}

D R Q _QE _A _TC _D _R _Q QE A _TC $2.0 49 18 130 .252 16.79 147.08 1.94 54 8 127 .043 16.14 139.55 0.32 1.0 49 18 129 .252 16.54 146.10 2.59 49 11 130 .252 16.54 139.11 0.63 0.5 49 18 128 .252 16.41 145.61 2.92 49 11 130 .252 16.41 138.62 0.98 0 47 19 128 .464 16.49 144.90 3.39 47 12 128 .464 16.48 137.90 1.50 Table 6

Effect of G on performance of the proposed policy; D ¼ 500 units=year, 1 year ¼ 250 days, A ¼ $16, A0

¼ $4, r ¼ 25%=year, c ¼ $4:0=unit, c0_{¼ $0:5=unit, L ¼ :1 years, M ¼ :08 years,}

N ¼ :02 years, andaor b ¼ 99:9%.

G No-shortage probability constraint % Savings Fill-rate constraint % Savings

D R Q QE _A _TC _D _R _Q

QE _A _TC

.016 52 19 127 .091 16.16 148.61 0.92 Expediting is not economical

.008 48 19 129 .345 16.55 146.13 2.57 51 10 134 .129 16.22 139.03 0.69

.004 49 18 128 .252 16.41 145.61 2.92 49 11 130 .252 16.41 138.62 0.98

.001 46 19 130 .616 16.92 145.58 2.94 49 11 128 .252 16.41 138.61 0.99

Table 7

Effect of M on performance of the proposed policy (withaservice level); D ¼ 500 units=year, 1 year ¼ 250 days, A ¼ $16, A0

N ¼ :02 years, G ¼ 0:004 years, anda¼ 99:9%.

M Expediting is not allowed Expediting is allowed % Savings

s Q TCðs; Q Þ D R Q QE A _{TCðD; R; Q Þ} 0.02 35 126 141.99 17 17 127 .028 16.07 141.27 0.51 0.04 48 126 144.99 27 18 128 .141 16.28 143.10 1.30 0.08 73 126 149.99 49 18 128 .252 16.41 145.61 2.92 0.16 121 126 157.99 95 18 128 .199 16.28 151.10 4.36 Table 8

Effect of M on performance of the proposed policy (with b service level); D ¼ 500 units=year, 1 year ¼ 250 days, A ¼ $16, A0

N ¼ :02 years, G ¼ 0:004 years, and b ¼ 99:9%.

M Expediting is not allowed Expediting is allowed % Savings

s Q TCðs; Q Þ D R Q QE A _{TCðD; R; Q Þ}

0.02 27 141 134.74 18 9 131 .013 16.03 134.21 0.39

0.04 39 142 136.84 27 11 128 .141 16.28 136.10 0.54

0.08 63 126 139.99 49 11 130 .252 16.41 138.62 0.98

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the smaller the ratio A0_{=A or c}0_{=c, the more cost-effective the}

pro-posed policy and the savings could be larger than 4%. This result is also expected without explanation.

Next, we examine the effect of G and M. It is seen fromTable 6 that as G is shorter, the proposed policy becomes more attractive. Moreover, it appears fromTables 7 and 8that the longer the man-ufacturing lead-time M, the more cost-effective the proposed pol-icy. This result is intuitively reasonable, since with a longer M the lead-time demand becomes more volatile and expediting part of an order at the end of M is more beneﬁcial.

Finally, we consider a high-volume item for which demand is normally distributed. We ﬁnd that as demand variability is larger, the proposed policy yields a larger percentage savings (detailed computational results are available from the author upon request), which also intuitively makes sense.

5. Conclusion

This paper proposes an order expediting inventory control pol-icy. It consists of three operational parameters: s; Q , and R, or equiv-alently,D;Q , and R whereDequals s R, such that if inventory falls below R at the end of the manufacturing lead-time, part of an out-standing order is delivered via a fast supply mode. We derive the average total cost per unit time and minimize it subject to a service level constraint. Either the ﬁll rate or the probability-of-no-short-age-during-a-cycle service measure is utilized. We show that the average total cost is a function ofD. Thus, one can perform a simple search onDto obtain optimal operational parameters. Computa-tional results show that the proposed policy is worthwhile to use, especially if service level is high, demand variability is large, the ex-tra expediting cost is small, or the manufacturing lead-time is long. References

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