inventory management under random supply disruptions and partial backorders

Inventory Management Under Random Supply Disruptionsand Partial Backorders

Antonio Arreola-Risa,1 Gregory A. DeCroix2

1 Department of Information & Operations Management, Lowry Mays College &Graduate School of Business, Texas A&M University, College Station, Texas 77843-4217

2 Fuqua School of BusinessDuke University, Durham, North Carolina 27708-0120

Received June 1997; revised March 1998; accepted 30 March 1998

Abstract: We explore the management of inventory for stochastic-demand systems, wherethe product’s supply is randomly disrupted for periods of random duration, and demandsthat arrive when the inventory system is temporarily out of stock become a mix of backordersand lost sales. The stock is managed according to the following modified (s , S) policy: Ifthe inventory level is at or below s and the supply is available, place an order to bring theinventory level up to S . Our analysis yields the optimal values of the policy parameters,and provides insight into the optimal inventory strategy when there are changes in theseverity of supply disruptions or in the behavior of unfilled demands. q 1998 John Wiley &Sons, Inc. Naval Research Logistics 45: 687–703, 1998

1. INTRODUCTION

We consider a stochastic-demand inventory system where the product’s supply is ran-domly disrupted for periods of random duration. The source of supply disruptions could beprocess-related or market-related. In the first case, the supply may become unavailable dueto breakdowns, transportation disruptions, or a strike. In the second case, even though thegoods may be physically available, market conditions may be such that a particular companydoes not have access to the product. One example of such conditions may arise as a resultof price fluctuations, where, during certain periods of time, the price at which supply equalsdemand is so high that purchasing the product becomes prohibitive for some companies.Another example is a scarcity of goods where the supplier gives preference to ‘‘major’’players at the expense of ‘‘minor’’ players. The authors are familiar with a company thatfaces this situation. For excellent motivations of inventory systems with random supplydisruptions, see Parlar and Berkin [18] and Gupta [8] .

Customers that encounter a stockout may decide to backorder or may become a lost sale.Most work in inventory management has considered one of the two extremes of customers’behavior during stockouts: All customers backorder or all customers are lost. In this paperwe consider a more general stockout behavior, commonly called partial backorders, where

Correspondence to: T. Arreola-Risa

q 1998 by John Wiley & Sons, Inc. CCC 0894-069X/98/070687-17

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688 Naval Research Logistics, Vol. 45 (1998)

a fraction of demands that arrive during a stockout is backordered and the remaining fractionis lost.

The costs of managing the system are stationary, and include ordering, holding, lost sales,and backordering costs. The system is controlled according to the following modified(s , S) policy: When the inventory level is at or below s and the supply is available, procurethe necessary amount to bring the inventory level up to S . This modified (s , S) policy wasadopted for several reasons. The optimal policy for the inventory system under study is anopen research question. In the absence of supply disruptions, the proposed policy becomesthe standard (s , S) policy which is known to be optimal for this situation. Placing orderswhen the inventory level is at or below the reorder point and the supply is available naturallymodifies the standard (s , S) policy to accommodate supply disruptions.

For the inventory system under consideration, we first derive the cost function and useit to determine the optimal values of the policy parameters (s , S) . Then we explore theimpact on the optimal values of the policy parameters of variations in the average frequencyand duration of supply disruptions, and of variations in the fraction of stockouts that arebackordered. We use the results of this analysis to obtain insights into the optimal strategyfor using the policy parameters s and S to protect the system against supply disruptionswith different levels of severity, and also to examine the effect of different stockout behaviorson this optimal strategy.

Due to the difficulty of handling partial backorders, the inventory literature in this areais limited. Moinzadeh [12] considers a base-stock level inventory system with Poissondemand, constant resupply times, and partial backorders. Other inventory models with partialbackorders can be found in Montgomery et al. [15], Kim and Park [10], and Posner andYansouni [22].

Treatment of supply uncertainty in a production-inventory system can be traced to Meyeret al. [11], which studied a single-stage production system facing a constant demand wherethe supply source is subject to random failure. The authors assumed zero setup cost andpure lost sales, and developed expressions for the operating characteristics of the system.Other work on production-inventory systems with deterministic demand and supply disrup-tions includes [1] , [4] , [5] , [6] , [7] , and [14].

Parlar and Berkin [18] study the classic EOQ problem with supply disruptions. Parlarand Perry [19] extend this analysis to a system where orders may be placed before theinventory level reaches zero, and where there is a fixed cost for determining the state ofthe supplier. Weiss and Rosenthal [25] determine the optimal inventory policy when thetiming (but not the duration) of supply disruptions is known in advance. Parlar and Perry[20] consider a system with two suppliers subject to independent disruptions.

To our knowledge, the only papers dealing with supply disruptions and random demandare [2] , [3] , [8] , [17], [21], and [23]. Posner and Berg [21] extend Meyer et al. [11]to the case of compound Poisson demand. Chao [2] presents a model in which the rate ofinventory accumulation or reduction can be continuously adjusted. Under a linear coststructure the author characterizes the optimal control policy using a single inventory target.Chao et al. [3] present a very general model for managing fuel inventories at electricutilities, and describe solution procedures involving dynamic programming and simulation.Related work can be found in [13] and [16].

The work most similar to that presented here is contained in Gupta [8] , Parlar [17], andSong and Zipkin [23]. One of the models presented in Song and Zipkin is essentially aperiodic-review version of our model, with the additional assumption of complete backorder-ing. In that setting, the authors show that a policy equivalent to our modified (s , S) policy

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689Arreola-Risa and DeCroix: Inventory Management with Supply Disruptions

is optimal, thus providing further motivation for our policy choice. Parlar presents a modelsimilar to ours, with positive delivery lead times and more general assumptions regardingthe frequency and duration of disruptions, but with the more restrictive assumption of fullbackordering. The author derives an approximate cost function and demonstrates a solutionprocedure for the special case when order quantities are large. Gupta studies a model withdemand and disruption assumptions similar to those considered here. His model is morerestrictive in that it assumes that all shortages result in lost sales, but more general sinceit allows for positive delivery lead times. In addition, Gupta proposes using a standard(r , Q) policy, which may result in placing multiple orders during a supply disruption. Theunnecessary ordering costs resulting from such a policy make it generally inferior to themodified (s , S) policy proposed here.

In light of the previous work on inventory systems with supply disruptions, this papermakes a number of primary contributions. First, it is the only work to address the problemof managing a stochastic-demand system with both supply disruptions and partial backor-ders. Since the qualitative behavior of the model presented here remains intact for theextreme cases of full backordering and pure lost sales, our work subsumes those versionsof previous models that agree with our assumptions. It thus provides a unified treatment ofsome of the models considered by Song and Zipkin [23], Parlar [17], and Gupta [8] . Inaddition, we obtain exact closed-form expressions for the system costs. Based on theseexpressions, we are able to develop reliable solution procedures and extract valuable insightsinto the behavior of the system. In particular, our analysis provides guidance to managersregarding the best strategy to use for protection against changes in the severity of supplydisruptions or changes in the behavior of unfilled demands.

The contents of the paper are organized as follows. Section 2 deals with model develop-ment and analysis. The section includes derivation of the cost function and determinationof the optimal values of the policy parameters. Sections 3 and 4 respectively address theoptimal inventory strategy when there are changes in the severity of supply disruptions andin the stockout behavior. The last section summarizes our research findings and offersdirections for future research.

2. MODEL DEVELOPMENT AND ANALYSIS

Let D denote demand per unit time, I denote the interarrival time of supply disruptions,and L denote the length of supply disruptions. We assume that D is a Poisson process withparameter a, and that I and L are exponentially distributed with parameters l and m,respectively. Thus a represents average demand per unit time, 1/l represents average timebetween supply disruptions, and 1/m represents average duration of supply disruptions.

In our experience with systems facing supply disruptions, the delivery lead time after anorder is placed tends to be small compared to the average length of a supply disruption.To reflect this, and to simplify the analysis, our model assumes that delivery lead times arezero. Since zero lead times result in never having orders outstanding, the modified (s , S)inventory policy can be implemented based on the inventory level rather than the moretraditional inventory position.

We will use the following notation and definitions.

N Å long-run average number of orders placed per unit time,OH Å long-run average on-hand inventory level,

q Å probability that a stockout becomes a backorder,

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1 0 q Å probability that a stockout becomes a lost sale,LS Å long-run average number of lost sales per unit time,BR Å long-run average number of units backordered per unit time,BL Å long-run average backorder level,

k Å ordering cost,h Å holding cost rate,p Å unit lost sales cost,p0 Å unit backordering cost,p1 Å backordering cost rate,K Å long-run average total cost per unit time.

In analyzing our modified (s , S) inventory policy, we will find it convenient to workwith s and D å S 0 s instead of s and S . The policy parameter D could be interpreted asthe amount ordered if the supply is available when the inventory levels hits s or, alternatively,as the minimum quantity ordered.

2.1. Cost Function

Since demand for the product is a Poisson process, we restrict s and D to be integers.Naturally D ¢ 1, and following previous research in inventory theory, we assume thats¢ 0. Let b denote the probability of finding the supply unavailable when the inventorylevel hits s . The following proposition establishes the fundamental result of this section.

PROPOSITION 1: In the inventory system under study

K(s , D) Å NHk / hFDa SD / 12

/ sD / b

m Ss 0 (1 0 r s)am DG / r sbag

m J ,

N Å am

mD / ab,

and

b Å l

l / m F1 0 S a

a / l / mDDG

where r Å a / (a / m) and g Å (1 0 q)p / q(p0 / p1 /m) .

All proofs are included in the Appendix. When no confusion arises, K will be used inlieu of K(s , D) . Notice that, in the limiting case q r 0, the inventory system becomes onewith pure lost sales and g r p . In the limiting case q r 1, the inventory system becomesone with full backorders and g r p0 / p1 /m which represents the expected cost perbackorder. Consequently, g can be interpreted as the expected cost per demand arrivingduring a stockout.

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2.2. Optimization Results

Let s* and D* denote the optimal values of the policy parameters. In this section wefocus on finding s* and D*.

Define

s*(D) Å arg mins¢0 K(s , D)

and

D*(s) Å arg minD¢1 K(s , D) .

For ease of exposition, the following development assumes that s*(D) and D*(s) areunique. If this is not the case, the arguments involving unique s*(D) and D*(s) can beadapted by focusing on particular values of s*(D) and D*(s) , say the smallest or thelargest.

PROPOSITION 2: For a fixed value of D, K is convex in s . Moreover, if

D

bú a

m F0ln(r)S r

1 0 rDS h / gm

h D 0 1G ,

then s*(D) Å 0; otherwise

s*(D) Å ln(arb)ln(r)

or s*(D) Å ln(arb)ln(r)

where

a Å m2D / bam

0ba 2 ln(r)

and

b Å h

h / g.

Proposition 2 establishes that K is convex in s for fixed D. However, although visualanalysis of numerical trials suggests that K is in fact jointly convex in s and D, we havenot been able to verify this analytically, or even to show that K is convex in D for fixeds . As a result, a closed-form expression for D*(s) does not seem possible, and thus wecannot rule out potential difficulties in computing s* and D* if K is not well behaved.Despite this possibility, we were able to obtain some qualitative information regarding thebehavior of s*(D) and D*(s) , as indicated in the following proposition.

PROPOSITION 3: s*(D) is nonincreasing in D and D*(s) is nonincreasing in s .

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In addition to providing insight into the behavior of the optimal policy parameters,Proposition 3 allows us to establish the finite convergence of Algorithm A below to theoptimal pair (s*, D*).

ALGORITHM A:

1. Perform a line search on D to compute D*(0).2. For each D √ {1, . . . , D*(0)}, compute s*(D) using Proposition 2.3. Among the D*(0) pairs (s*(D) , D) computed in Step 2, the one that yields

the smallest value of K is the optimal pair (s*, D*).

3. OPTIMAL INVENTORY STRATEGY FOR CHANGES IN THESEVERITY OF SUPPLY DISRUPTIONS

Given the complexity of s*, D*, and K(s*, D*), our characterization of the optimalinventory strategy as m or l changes will be based on analytical results for limiting valuesof m and l, and on results of extensive numerical experiments for their nonlimiting values.

In preparation for the forthcoming analysis, we will take a small digression to discusshow the protection of the inventory system against supply disruptions can be adjusted byvarying the values of s , D, and S . While it is obvious that increasing s would increase theprotection level of the system, and decreasing s would decrease the protection level of thesystem, it is less clear what would be the effect on the protection level when D increasesor decreases. Specifically, increasing D reduces the number of cycles per year and hencedecreases the number of times that a potential supply disruption may be faced, which inturn increases the protection level. On the other hand, it is easy to see from Proposition 1that increasing D increases b and thus increases the probability of encountering a supplydisruption when the inventory level hits s , which in turn decreases the protection level. Thefollowing proposition settles this dilemma.

PROPOSITION 4: The fraction of time that the inventory system is incurring stockouts( lost sales or backorders) is decreasing in D.

While the above discussion makes it clear that increasing either s or D increases thesystem’s protection against supply disruptions, it is not obvious to what extent each of theseparameters should be used to achieve such protection. For example, if an increase in theprotection level is needed, should s be increased, should D be increased, should both beincreased, or should one be increased and one be decreased? In addition, if an increase inprotection level is needed, will the adjustments in s and D always result in larger valuesof S? Given Proposition 3, which states that all else being equal, increases in s tend to yielddecreases in D and vice versa, one should not be surprised to learn that the answers to theabove questions are complex. The following two subsections go a long way toward answer-ing these questions, and provide significant insight into the impact of supply disruptionson the inventory system being studied.

3.1. Changes in m

Consider the limiting case m r 0. Using Eq. (13) in the Appendix, it is not hard todemonstrate that limmr0 ( 0

iÅ0` P( i , 1) Å 1. Thus for any values of (s , D) all demands will

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Figure 1. Optimal behavior of the inventory system as m varies: (l) s*; (j) D*; (n) S*.

be lost or backordered. This indicates that when the average duration of supply disruptionsgrows without bound, the product should not be inventoried.

Consider now the limiting case m r ` . Using Proposition 1, it is not difficult to showthat

limmr`

K Å ka

D/ h(D / 1)

2/ hs . (1)

Hence, if m Å ` , then

s* Å 0 (2)

and

D* Å√

2ka

hor D* Å

√2ka

h. (3)

Therefore, as the average duration of supply disruptions goes to zero, the optimal behaviorof the inventory system approaches that of the basic EOQ model.

For nonlimiting values of m, to study the behavior of s*, D*, and S* as m changes, weformulated 128 test problems. These problems were generated by fixing h Å 1, and thentaking all combinations of the following parameter values: a Å 100, 500; q Å 0.3, 0.7;k Å 50, 200; p Å 10, 50; p0 Å 2, 5; p1 Å 50, 200; l Å 1, 12.

Based on the optimization results of Section 2.2, we solved each of the 128 test problemsfor a large number of values of m, and produced graphs of s* vs. m, D* vs. m, and S* vs.m. The behavior of s*, D*, and S* as m changes was consistent across all test problems,and this behavior is captured by the example in Figure 1, where a Å 100, q Å 0.3, k Å 50,h Å 1, p Å 10, p0 Å 2, p1 Å 50, l Å 1, and EOQ Å 100. Note the following:

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Á S*, s*, and D* are decreasing in m.Á As m r ` , s* r 0 and D* r EOQ.

The managerial implications of these research findings are several. When the severity ofthe supply disruptions diminishes because m is increasing, less protection is needed, and,as a result, the optimal values of the policy parameters should decrease. Conversely, whenthe severity of the supply disruptions increases because m is decreasing, more protection isneeded, and the optimal values of the policy parameters should increase. However, what isnot necessarily intuitive is that, even though, by Proposition 3, s*(D) and D*(s) tend tomove in opposite directions, an increase in the system’s protection level is best achievedby increasing both s* and D*. With regard to the second finding, we know from theanalytical results in Eqs. (1) – (3) that, as m r ` , the limiting behavior of the system isthat of a basic EOQ system. Nevertheless, it is interesting to see that in some circumstanceseven for moderate values of m, the inventory system starts to exhibit EOQ behavior.

3.2. Changes in l

Consider the limiting case l r 0. Using Proposition 1, it is not difficult to show that

limlr0

K Å ka

D/ h(D / 1)

2/ hs . (4)

Comparing (1) to (4) , we conclude that the results in (2) and (3) apply here as well, andhence when the average time between supply disruptions goes to infinity, the optimalbehavior of the inventory system also becomes that of the basic EOQ model.

Consider now the limiting case l r ` . It is not hard to demonstrate that

limlr`

K

Å S am

mD / aDHk / hFD(D / 1)2a

/ sD

a/ 1

m Ss 0 (1 0 r s)am DG / r sag

m J . (5)

As l r ` , the inventory system being studied approaches one where the supply is alwaysunavailable at the time the inventory level hits s . As a result, every ordering cycle experi-ences an exponential inventory-replenishment time with mean 1/m. However, since an orderis not placed until the supply becomes available (i.e., at the end of the inventory-replen-ishment time), this inventory system is distinctly different from classical inventory systemswith exponential inventory-replenishment times (see, for example, Hadley and Whitin [9]) .In particular, each order always returns the on-hand inventory to S , successive inventory-replenishment times are clearly independent, and orders cannot cross.

Using Eq. (5) , it is not difficult to show that, as l r ` , if

D ú a

m F0ln(r)S r

1 0 rDS h / gm

h D 0 1G ,

then s*(D) Å 0; otherwise,

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s*(D) Å ln(arb)ln(r)

or s*(D) Å ln(arb)ln(r)

, (6a)

where

a Å m2D / am

0a 2 ln(r)(6b)

and

b Å h

h / g. (6c)

Therefore, as the average interarrival time of supply disruptions goes to zero, s* and D*clearly converge to finite values. These values can be obtained using an obvious variationof Algorithm A.

We conducted a second numerical experiment with two objectives. First, we sought togain additional insight into the limiting behavior of s*, D*, and S* as l r ` . Second, wewished to study, for nonlimiting values of l, the behavior of s*, D*, and S* as l changes.We formulated 128 test problems by fixing h Å 1, and then taking all combinations of thefollowing parameter values: a Å 100, 500; q Å 0.3, 0.7; k Å 50, 200; p Å 10, 50;p0 Å 2, 5; p1 Å 50, 200; m Å 1, 12.

Using the optimization results of Section 2.2, we then solved each of the 128 test problemsfor a large number of values of l, and produced graphs of s* vs. l, D* vs. l, and S* vs.l. Once more, the behavior of s*, D*, and S* as l changes was consistent across all testproblems, and this behavior is captured by the three examples in Figure 2, where a Å 100,q Å 0.3, k Å 50, h Å 1, p Å 10, p0 Å 2, p1 Å 50, m Å 1, 6, 12, and EOQ Å 100. Notethe following:

Á S* is increasing in l.Á There exists lO ú 0 such that, on l õ lO , s* Å 0 and D* is increasing in l, and,

on l ¢ lO , s* is increasing in l and D* is decreasing in l.Á lO is increasing in m—e.g., lO (m Å 1) Å 0.2, lO (m Å 6) Å 1.0, and lO (m Å 12)Å 2.0.

Á As l r 0, s* r 0 and D* r EOQ.Á As l r ` , D* r EOQ and s* r s* (D Å EOQ).

Several managerial implications can be derived from the above research findings. Notsurprisingly, when the severity of the supply disruptions diminishes because l is decreasing,S* should decrease; and conversely, when the severity of the supply disruptions increasesbecause l is increasing, S* should increase.

A second implication is definitely less intuitive: Depending on the value of l, the systemfollows two different strategies to increase the protection level as l increases. Whenl √ [0, lO ) , the increases in S* should be the result of increasing D* alone. Whenl √ [lO , `) , the increases in S* should be the net result of increasing s* and decreasingD*. Note also that lO , the value of l that triggers the change of strategy from ‘‘increase S*

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Figure 2. Optimal behavior of the inventory system as l varies: (l) s*; (j) D*; (n) S*.

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solely by increasing D*’’ to ‘‘increase S* by increasing s* and decreasing D*,’’ is affectedby m. The smaller the value of m the sooner the strategy shift.

The final finding above is surprising and, even with hindsight, difficult to explain: Thefinite value to which D* converges as l r ` is EOQ, and, as a result, the finite value towhich s* converges is s* (D Å EOQ). We confess that, from a theoretical standpoint, wewere even more surprised when attempts to justify the behavior D* r EOQ as l r ` byusing Eqs. (5) and (6) were fruitless.

4. OPTIMAL INVENTORY STRATEGY FOR CHANGESIN STOCKOUT BEHAVIOR

In this section we characterize the optimal inventory strategy as stockout behaviorchanges. The approach used parallels that of Section 3. To explore the response of the systemto varying moderate values of q , a third extensive numerical experiment was performed on128 test problems. The test problems were formulated by fixing h Å 1, and then taking allcombinations of the following parameter values: a Å 100, 500; k Å 50, 200; p Å 10, 50;p0 Å 2, 5; p1 Å 50, 200; l Å 1, 12; and m Å 1, 12.

Using the optimization results of Section 2.2, we solved each of the 128 test problemsfor a large number of values of q with 0 ° q ° 1, and produced graphs of s*, D*, andS* vs. q . The behavior of s*, D*, and S* as q changes was consistent across all testproblems, and this behavior is captured by the two examples in Figure 3, where a Å 500,k Å 50, h Å 1, p Å 50, p0 Å 2, p1 Å 200, l Å 1, and m Å 1 (Example 1) and 12 (Example2). Hence for Example 1 we have that g Å 50 / 152q , and for Example 2 we have thatg Å 50 0 31.33q . Note the following:

Á D* is nearly constant in q .Á s* and S* are increasing/decreasing in q when g is increasing/decreasing in q .

A number of managerial implications are suggested by these numerical results. First,since D* seems to be quite insensitive to the fraction of stockouts that are backordered,the inventory manager can choose a good minimum order quantity D without obtaining aprecise estimate for q . This may be quite useful in practice, since estimating q may bedifficult. On a related vein, this result also implies that the varying levels of protectionagainst supply disruptions required by different stockout behaviors are achieved almostentirely by changing the optimal reorder point s*.

Finally, the second research finding confirms our intuition that, as stockout behaviorshifts toward more backorders, more protection against supply disruptions is desirable whenthe expected cost per backorder is greater than the expected cost per lost sale, and lessprotection against supply disruptions is desirable when the expected cost per backorder isless than the expected cost per lost sale.

5. CONCLUSION

Our research interest in this paper has been to explore the optimal management ofinventory for stochastic-demand systems with random supply disruptions and partial backor-ders. Our results suggest the following managerial insights:

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Figure 3. Optimal behavior of the inventory system as q varies: (l) s*; (j) D*; (n) S*.

Á The optimal inventory strategy to manage a change in the severity of supplydisruptions should be to move the order-up-to level S* in the same directionas the change. However, the specific strategy to accomplish such a move in S*through the variables s* and D* depends on whether the change in the severityof supply disruptions is due to a change in m or is due to a change in l.

Á The optimal inventory strategy to manage a change in stockout behavior shouldbe to move the order-up-to level S* in the same direction as the change inexpected cost per demand arriving during a stockout g. The adjustment in thevalue of S* should be achieved almost exclusively through the optimal reorderpoint s*.

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Due to the lack of previous research on the subject of random supply disruptions instochastic-demand inventory systems with partial backorders, the work presented here mustbe regarded as just one step forward in understanding such systems. Some obvious directionsfor extending this work parallel earlier advances in inventory management without supplydisruptions or partial backorders. Such directions include searching for the optimal inventorypolicy or allowing for nonstationary costs—perhaps depending on the state of the supply.Although we expect that the basic flavor of our results will hold for much more generalsystems than the one considered here, significant additional insights may be gained byexplicitly addressing some of these issues.

APPENDIX

PROOF OF PROPOSITION 1: Let us first derive the expression for b. Modeling the evolution of supplydisruptions as a two-state continuous-time Markov chain, it is not difficult to demonstrate that

Pr{supply is unavailable at time t} Å l

l / m[1 0 e0(l/m) t] .

Define X as the time when the inventory level hits s and let fX (r) denote its probability density function. Then

b Å *`

0S l

l / mD[1 0 e0(l/m)x] fX ( x) dx .

Because D is a Poisson process, X is distributed Erlang with parameters a and D, where

fX ( x) Å a(ax)D01e0ax

G(D), x ú 0.

We thus have

b Å S l

l / mD a

G(D) *`

0

[1 0 e0(l/m)x](ax)D01e0ax dx Å l

l / m F1 0 S a

a / l / mDDG .

Let i denote the inventory level, j Å 0 if the supply is available and j Å 1 otherwise, and P( i , j) be the limitingor steady-state probability of being in state ( i , j) . Using stochastic balance arguments, we constructed the transition-rate diagram depicted in Figure 4. Basic procedures plus tedious algebraic manipulations lead to

P(S , 0) Å m(a / m)(mD / ab)(a / l / m)

, (7)

P(S , 1) Å ml

(mD / ab)(a / l / m), (8)

P( i , 0) Å S m

mD / abDF S l

l / mDS a

a / l / mDS/10 i

/ m

l / mG , i Å s / 1, . . . , S 0 1, (9)

P( i , 1) Å F ml

(mD / ab)(l / m) GF1 0 S a

a / l / mDS/10 iG , i Å s / 1, . . . , S 0 1, (10)

P( i , 0) Å 0, i ° s , (11)

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Figure 4. Steady-state probability and transition-rate diagram for the inventory system under study.

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P( i , 1) Å S mb

mD / abDS a

a / mDs/10 i

, i Å 1, . . . , s , (12)

P( i , 1) Å S m

aq / mDS ab

mD / abDS a

a / mDsS aq

aq / mD0 i

, i ° 0. (13)

From basic principles

K Å k N / h OH / p LS / p0 BR / p1 BL .

Define a cycle as the time interval between order receipts and denote its length by T . Let T1 be the cycle lengthif the supply is available when the inventory level hits s , let T2 be the cycle length if the supply is unavailablewhen the inventory level hits s , and let A be an indicator random variable defined by

A Å H1 if the supply is available when the inventory level hits s ,

0 otherwise.

Also define X1 to be an Erlang random variable with parameters a and D, and define X2 to be an exponentialrandom variable with parameter m. Then T Å T1 / T2 , T1 Å AX1 , and T2 Å (1 0 A1)(X1 / X2) . Therefore,

E(T1) Å E(E(T1ÉA)) Å Pr{A Å 1}E(T1ÉA Å 1) / Pr{A Å 0}E(T1ÉA Å 0)

Å (1 0 b)E(X1) / br0

Å (1 0 b)D

a

and

E(T2) Å E(E(T2ÉA)) Å Pr{A Å 1}E(T2ÉA Å 1) / Pr{A Å 0}E(T2ÉA Å 0)

Å (1 0 b)r0 / b(E(X1) / E(X2))

Å bSDa / 1mD .

Therefore,

E(T ) Å (1 0 b)D

a/ bD

a/ b

mÅ D

a/ b

mÅ mD / ab

am.

Hence, since N Å 1/E(T ) , we have

N Å am

mD / ab.

The expressions for OH , LS , BR , and BL follow from Eqs. (7) – (13) and algebra.

PROOF OF PROPOSITION 2: It is easy to show that Ì 2K /Ìs 2 ¢ 0 for any D, thus establishing convexity ins . The remainder of the result follows from algebraic manipulation of the condition

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ÌK

Ìs ZsÅ0

ú 0

(which would imply s* Å 0) and the condition

ÌK

ÌsÅ 0

(which yields the optimal s* if (ÌK /Ìs)ÉsÅ0 ° 0).

PROOF OF PROPOSITION 3: We will first show that the ratio D /b is increasing in D. From Proposition 1

D

bÅ D

( ll / m)[1 0 ( a

a / l / m)D].

Let g(D) Å D / [1 0 (a / (a / l / m))D] . Then

g *(D) Å1 / ( a

a / l / m)D[D ln( aa / l / m) 0 1]

[1 0 ( aa / l / m)D]2

.

If 0 õ a õ 1 and x ¢ 0, clearly ax ¢ 1 / x ln(a) . Hence

S a

a / l / mDD

¢ 1 / D lnS a

a / l / mD ,

and thus

1/ S a

a/ l/ mDDFD lnS a

a/ l/ mD0 1G¢ 1/ F1/D lnS a

a/ l/ mDG FD lnS a

a/ l/ mD0 1GÅ D 2F lnS a

a / l / mDG2

¢ 0.

Therefore, D /b is increasing in D.The result in Proposition 3 follows from the fact that K is superadditive on the sublattice {0, 1, rrr} 1 {1,

2, rrr}. (For a summary of the necessary lattice programming results, see Topkis [24].) Superadditivity of Kis equivalent to the condition

Ì 2K

ÌsÌD¢ 0.

This condition on K holds term by term because D /b is increasing in D.

PROOF OF PROPOSITION 4: Let e denote the fraction of time that the inventory system is incurring stockouts( lost sales or backorders) . Using basic principles, it is not hard to show that

e Å r s

(mD /ab) / 1.

Combining the above equation with the result that D /b is increasing in D (demonstrated in the proof of Proposition3) yields Proposition 4.

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ACKNOWLEDGMENTS

We want to thank the associate editor and the referees for their comments, which led toa much improved paper.

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inventory management under random supply disruptions and partial backorders

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