European Journal of Operational Research 232 (2014) 64–71

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Production, Manufacturing and Logistics

Reserve stock models: Deterioration and preventive replenishment q

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.06.043

q The authors would like to acknowledge the financial support of the LebaneseNational Council for Scientific Research (LNCSR).⇑ Corresponding author. Tel.: +961 1 350 000x3551; fax: +961 1 744 462.

E-mail address: bacel.maddah@aub.edu.lb (B. Maddah).

Bacel Maddah ⇑, Ali A. Yassine, Moueen K. Salameh, Lama ChatilaEngineering Management Program, Faculty of Engineering and Architecture, American University of Beirut, Lebanon

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 May 2012Accepted 24 June 2013Available online 11 July 2013

Keywords:Reserve stockSupply interruptionDeteriorationPreventive replenishmentInventory control

Reserve stocks are needed in a wide spectrum of industries from strategic oil reserves to tactical (machinebuffer) reserves in manufacturing. One important aspect under-looked in research is the effect of deteri-oration, where a reserve stock, held for a long time, may be depleted gradually due to factors such asspoilage, evaporation, and leakage. We consider the common framework of a reserve stock that is utilizedonly when a supply interruption occurs. Supply outage occurs randomly and infrequently, and its dura-tion is random. During the down time the reserve is depleted by demand, diverted from its main supply.We develop optimal stocking policies, for a reserve stock which deteriorates exponentially. These policiesbalance typical economic costs of ordering, holding, and shortage, as well as additional costs of deterio-ration and preventive measures. Our main results are showing that (i) deterioration significantlyincreases cost (up to 5%) and (ii) a preventive replenishment policy, with periodic restocking, can offsetsome of these additional costs. One side contribution is refining a classical reserve stock model (Hans-mann, 1962).

� 2013 Elsevier B.V. All rights reserved.

1. Introduction and literature review

An implicit assumption in traditional inventory control modelsis the continuity of supply, where the potential for supply interrup-tion is not considered. Supply interruptions can occur due to inter-nal random incidents such as machine breakdown, or due toexternal random events such as worker strikes or political crises.For example, the internal operations of a factory can be disruptedif an upstream operation fails in a tandem factory layout; thus,starving the downstream machine. External random occurrencesof natural disasters, wars, or strikes could also disrupt theincoming flow of material and render simple inventory controltechniques inapt. A natural remedy to such situations is to resortto multi-sourcing solutions or simply carry a strategic/operationalreserve stock that hedges against such random occurrences ofinternal and external disruptions. However, another complicatingfactor that is usually ignored in research is the deterioration of re-serve stock. Deterioration, which could take place to any reservestock (strategic or operational) is more pronounced in situationswhere the storage amounts are large and storage periods are long.Frequent monitoring of such reserves is necessary to ensure thatthe quality and the quantity of these reserves are maintainedduring long storage periods and are available when needed.

Strategic oil reserves are important examples of reserve stockwhere deterioration can be an important factor. For example, theUS government carries a strategic petroleum reserve of about725 million barrels of oil (about 75 days of import protection) inunderground salt caverns at five sites along the Gulf of Mexico.These strategic reserves are kept for long periods and used onlyin case of emergency disruptions. Europe, Asia, and India are alsofollowing US footsteps in developing their own strategic storageprograms (Thomson, 2009). Giles, Joachim Koenig, Neihof, Shay,and Woodward (1991) discussed the deterioration of such reservesand find that refined products tend to have less storage stabilitythan crude oils. This finding is corroborated by data from the Cana-dian Ministry of Agriculture showing that evaporation losses fromgasoline storage tanks is up to 3.2% per summer month (FSHR,2005).

In manufacturing, large reserves are kept, and deteriorationmay occur. For example, according to the US Census Bureau, thetotal value of manufacturing inventories was around $610 billionin December 2011. Among these, there are $240 billion dollars innondurable goods inventory (such as food, beverages, petroleumand chemical products), where deterioration may occur at a signif-icant level.

In this paper, we combine both factors of random disruptionand reserve stock deterioration, to build new models, which aremore sensible in such uncertain environments. Next we present aconcise review of the literature.

Our proposed research relates to three streams of research oninventory models with (i) supply disruption, (ii) deterioration,

B. Maddah et al. / European Journal of Operational Research 232 (2014) 64–71 65

and (iii) corrective and preventive actions. In the following, webriefly review the research in these three streams.

First, on supply disruption, a seminal work is that by Hansmann(1962). He determined the optimal stock level by minimizing acost function composed of holding and shortage costs. Our modelsare based on Hansmann’s work. However, we account for addi-tional costs related to ordering, deterioration and preventivereplenishment. More recent works on supply disruption includeParlar and Berkin (1991) who analyzed an inventory control prob-lem with exponentially distributed on (supplier available) and off(supplier unavailable) periods and deterministic demand, as we as-sume in this paper. In addition, Parlar (1997) introduced a modelwith random demand and random lead-time assuming that supplymay be disrupted.

Few papers model supply disruptions as they relate to strategicreserves. Oren and Hang Wan (1986) determined the optimal size,fill up, and drawdown rates for a strategic petroleum reserve undera variety of supply and demand conditions using numerical meth-ods. They assumed random on/off supply and deterministic de-mand, as we do in this paper. However, our model is different.We consider the additional factor of reserve deterioration, whichis not studied by Oren and Hang Wan (1986). Wu and Weil(2010) developed a reserve stock policy that minimizes the ex-pected insecurity cost to the Chinese economy arising from uncer-tainty in the supply of imported oil using a dynamic programmingmodel. Zhang and Li (2009) developed a multi-objective model formanaging strategic reserves with the objectives of minimizingstorage equipment investment, storage expenses material valueloss, and shortage risk. Both papers by Zhang and Li (2009) andWu and Weil (2010) also ignored deterioration, unlike our modelsin this paper.

In a recent work, Shen, Dessouky, and Ordonez (2011) devel-oped a model for a manufacturer of deteriorating medical suppliesused as a strategic reserve by the US government in the event of apublic health emergency. This work is closely related to this paper.However, Shen et al. (2011) research is different in two aspects (i)they took the point of view of the manufacturer, while we take thepoint of view of the reserve keeper (e.g. the government) and (ii)they assumed that deterioration occurs after a fixed lifetime, whilewe assume a random lifetime.

Second, on deteriorating or perishable inventories, it has beenearly recognized that special control policies are needed for suchsystems (see Nahmias (1982) for background and literature re-view). Random lifetime perishables, such as the one we considerin this paper, include oil produce, meats, and many other foodproducts. One main ingredient of the research in this direction isthe probability distribution of lifetime. The exponential lifetimedistribution, which we adopt, is the most popular in the literature.The first known work that considered deterioration is by Ghare andSchrader (1963) who adapted the classic economic order quantitymodel to items with exponentially distributed lifetime. Several pa-pers extended the work of Ghare and Schrader (1963) to accountfor backordering and finite production capacity, e.g., Misra(1975), Shah (1977) and Mak (1982). Reviews on perishable inven-tory models are presented by Raafat (1991) and Goyal and Giri(2001). This literature dealt with perishable products within aninventory control setting under continuous demand. However, inthis paper, we investigate the effect of deterioration on a reservestock where demand occurs sporadically when supply is disrupted,which constitutes a novel contribution.

Finally, inventory models on corrective actions typically relateto repair. For example, Chakrababorty, Giri, and Chaudhuri(2008) considered production lot sizing with machines which areprone to failure. They differentiated between corrective repair,which is done if a breakdown occurs during a production run,and preventive repair, which is done on a regular basis to minimize

the chance of failure. Similar corrective/preventive repair modelsare considered by many other authors, e.g., Abboud (1997, 2001),Giri and Dohi (2005), and Lin and Gong (2006). Lin and Gong(2006) also considered exponentially deteriorated items, similarto our case. However, their focus was on a production lot sizingrather than on a reserve stock, like our case. A recent work whichis more related to this paper is by Lee and Wu (2006) who consid-ered statistical process control (SPC) based replenishment. This isshown to be effective in reducing the bullwhip effect in a two-echelon supply chain. The SPC based replenishment is somewhatsimilar to our proposed preventive replenishment scheme as bothpolicies are triggered by signals related to the supply process. (Inour work, the signal is the reserve stock dropping below a certainlevel.)

The rest of this paper is organized as follows. In Section 2, wepresent a base model adapted from Hansmann (1962) but withan additional variable ordering cost, which we found to have animportant effect on the optimal stock level and the total cost.Our core contribution is in Section 3 where we account for the ef-fect of exponential deterioration on the reserve stock. In Section 4,and in an effort to mitigate the effect of deterioration on reservestock, we assume that a periodic preventive replenishment policyis adopted and we analyze the cost implications. In Section 5, wepresent numerical results and managerial insights. We find thatneglecting deterioration can increase costs up to 5%, and that pre-ventive replenishment can offset a significant part of these addi-tional costs. Finally, Section 6 concludes the paper and discussesfuture research.

2. Base model

Consider a reserve stock which is held to hedge against supplyinterruption. Supply availability time, i.e. ‘‘up time’’, X, is a randomvariable with mean 1/k. Likewise, supply interruption time, i.e.‘‘down time’’, Y, is a random variable with mean 1/l. We makethe reasonable assumption that down time is short relative to uptime, Y� X. A base stock policy is adopted; a reserve stock of levelS is kept at all times when supply is available. In the event of a sup-ply interruption, this reserve stock is consumed at a known rate Dper unit time. In the event that demand during the down time ex-ceeds supply S, excess demand is lost. At the end of the supplyinterruption the reserve stock is replenished instantaneously upto S. The assumption of instantaneous replenishment is not restric-tive. We adopt it to simplify the presentation. Fig. 1 shows a typicalprofile of stock level over time. Table 1 presents the notation usedin this section and the remainder of the paper.

The company’s cost structure entails (i) a holding cost propor-tional to average inventory with a unit cost of h $/unit/unit time,(ii) a shortage cost proportional to shortage time with a unit costp $/unit time, and (iii) a variable ordering cost proportional tothe amount ordered at the end of supply interruption with a unitcost c $/unit time. The variable ordering cost must be accountedfor because excess demand during down time is lost. Previous lit-erature (e.g. Hansmann, 1962) ignores this fact and neglectsaccounting for the ordering cost.

Under the assumption that Y� X, the drop in inventory duringdown time can be ignored. It follows that the expected holding costper unit time is

ChuðSÞ ¼ hS: ð1Þ

We define a cycle as the time between two consecutive supply res-torations. Then, with the assumption that Y� X, the expected cycleduration is approximately 1/k. The shortage cost depends onwhether demand exceeds supply during down time. Therefore,the expected shortage cost per cycle is

Fig. 1. Typical stock level for the base model.

Table 1Notations.

X Supply availability time, i.e. ‘‘up time’’ which is a random variablewith mean 1/k

Y Supply interruption time, i.e. ‘‘down time’’ which is a randomvariable with mean 1/l

S Reserve stock levelD Reserve stock consumption per unit timeh Holding cost proportional to average inventory in $/unit/unit timep Shortage cost proportional to shortage time with a unit cost $/unit

timec Variable ordering cost proportional to the amount ordered at the end

of supply interruption in $/unit timefY(�) Probability density function (pdf) of YChu(S) Expected holding cost per unit timeCs(S) Shortage cost per cycleCsu(S) Expected shortage cost per unit timeCou(S) Expected ordering cost per unit timeCbu(S) Expected total cost per unit time in the base modelS⁄ Optimal reserve stock in the base model

F�1Y (�) Inverse of FY(�) which is the cumulative distribution function (CDF) of

Y

SHb

Expression derived by Hansmann for optimal reserve stock level

h Exponential deterioration rate per unit timefX(�) Probability density function (pdf) of XCdu(S) Expected total cost per unit time in the model with deteriorationS�d Optimal reserve stock in the model with deteriorationtr Preventive replenishment period in units of times Level at which stock is replenished up to Skr Fixed replenishment cost due to the preventive replenishmentS Average stock levelCdru(S) Expected total cost per unit time in the model with deterioration and

replenishment

66 B. Maddah et al. / European Journal of Operational Research 232 (2014) 64–71

CsðSÞ ¼ pZ 1

S=Dðy� S=DÞfY ðyÞdy;

where fY(y) is the probability density function (pdf) of Y. Applyingthe renewal reward theorem (Ross, 1983, p. 78), it follows thatthe shortage cost per unit time is

CsuðSÞ ¼ pkZ 1

S=Dðy� S=DÞfYðyÞdy: ð2Þ

Similarly, the variable ordering cost depends on whether de-mand exceeds supply during down time, with an expected valueper unit time given by

CouðSÞ ¼ ckZ S=D

0DyfYðyÞdyþ

Z 1

S=DSfY ðyÞdy

!: ð3Þ

Combining (1)–(3), the total expected cost per unit time is

CbuðSÞ ¼ hSþ pkZ 1

SD

ðy� S=DÞfYðyÞdy

þ ckZ S

D

0DyfYðyÞdyþ

Z 1

SD

SfYðyÞdy

!: ð4Þ

The company’s objective is to minimize expected cost. This canbe written as

minS>0

CbuðSÞ: ð5Þ

The following theorem establishes that the expected cost, Cbu(S), isconvex in the reserve level S, and gives a closed-form expression forthe optimal reserve stock S⁄.

Theorem 1. If (p � cD)k > hD, then Cbu(S) is convex in S and theoptimal reserve stock is

S�b ¼ DF�1Y 1� hD

ðp� cDÞk

� �; ð6Þ

where FY(�) is the cumulative distribution function (CDF) of Y.

Proof. See Appendix A. h

The condition (p � cD)k > hD in Theorem 1 indicates that theshortage cost per unit time exceeds the expected ordering andholding costs. This condition is unrestrictive and allows avoidingthe trivial case where it is optimal not to carry reserve stock. Notefinally, that when the unit ordering cost is negligible, c = 0, ourmodel reduces to that of Hansmann (1962), and the reserve stockin (6) reduces to

SHb ¼ DF�1

Y 1� hDpk

� �; ð7Þ

which is the expression derived by Hansmann (1962). When c – 0,utilizing the reserve stock in (6) instead of the optimal one in (7)(i.e. neglecting the unit ordering cost) can lead to a significant in-crease in the expected cost, as we show in Section 5.

3. Model with deterioration

Considering the base model in Section 2, we now assume fur-ther that reserve stock is subject to deterioration at an exponentialrate h per unit time. That is, starting with the base stock S at thebeginning of a cycle, the stock level decreases to Se�hX at the endof up-time, as a result of deterioration, and then decreases further,during down time, as a result of both demand and deterioration asshown in Fig. 2. To simplify the analysis, we make the reasonableassumption that stock is consumed only by demand during downtime. This assumption is highly justified in practice where thedeterioration rate, h, is commonly small with respect to the de-mand rate, D.

Following a similar analysis to that in Section 2, the expectedholding cost per unit time is

ChuðSÞ ¼ hkZ 1

0

Z x

0Se�ht dt

� �fXðxÞdx

¼ hkZ 1

0ðS=hÞð1� e�hxÞfXðxÞdx; ð8Þ

where fX(�) is the pdf of X. In addition, under the assumption thatthe stock is consumed only by demand during down time, the ex-pected shortage cost per unit time is

CsuðSÞ ¼ pkZ 1

0

Z 1

Se�hxD

y� Se�hx

D

!fYðyÞdy

" #fXðxÞdx; ð9Þ

and the expected ordering cost is

B. Maddah et al. / European Journal of Operational Research 232 (2014) 64–71 67

CouðSÞ¼ ckZ 1

0

Z Se�hxD

0ðS�Se�hxþDyÞfY ðyÞdyþ

Z 1

Se�hxD

SfYðyÞdy

" #fXðxÞdx:

ð10Þ

The expected cost per unit time is Cdu(S) = Chu(S) + Csu(S) + Cou(S).Analyzing Cdu(S), we establish in the following theorem that it isconvex in S under nonrestrictive assumptions similar to those inTheorem 1. The theorem also gives a single-variable equation whichcan be solved with any search method to obtain the optimal reservestock.

Theorem 2. If p � cD > (1/u � 1)(hD/h + cD), then the expected costunder exponential deterioration is convex in the reserve level, S, andthe optimal reserve stock, S�d, is the unique solution of

cþhh

� �ð1�uÞ�ðp� cDÞ

D

Z 1

01�FY

Se�hx

D

!" #e�hxfXðxÞdx¼ 0; ð11Þ

where u ¼ E½e�hX � ¼R1

0 e�hxfXðxÞdx.

Proof. See Appendix A. h

Note that the condition, p� cD > 1u� 1� �

hDh þ cD� �

, in Theo-

rem 2 is equivalent to pD >

hh

1u� 1� �

þ cu, which will hold if, with re-

spect to the shortage cost per unit, pD, the deterioration rate, h, is not

too high (as it can be shown that limh?1 hu =1), the unit holdingcost, h, is not too high, and the unit ordering cost, c, is not too high.This condition allows avoiding trivial cases where no stock is car-ried. (It can be shown formally that without the condition the opti-mal reserve stock level is zero.)

Our numerical experimentation (see Section 5) indicates thatsolving the first-order optimality condition in (11) and obtainingthe optimal reserve level S�d is straightforward. Note that whenthere is no deterioration, h = 0, the first order conditions in (11) re-duces to (6), the optimality condition of the base model in Sec-tion 2. Setting h = 0 in the left-hand-side of (11) and utilizingL’Hospital’s rule on the first term, we get

hk� ðp� cDÞ

D1� FY

SD

� �� ¼ 0;

which is equivalent to (6).

4. Model with deterioration and preventive replenishment

Our numerical results (in Section 5) indicate that deteriorationcan significantly increase the cost of maintaining a reserve stock. Inan effort to mitigate the effect of deterioration on cost, we add pre-ventive replenishment, where the stock level is replenished peri-odically up to a base level. This avoids having low reserve whena sudden supply outage occurs. Specifically, during the up-time(when supply is available) the reserve stock level is replenishedto S every tr units of time (e.g., tr = 3 months, tr = 1 quarter, tr = 6 -months, tr = 1 year or tr = 3 years). This is equivalent to the well-known (s, S) policy, with s ¼ Se�htr ; whenever the stock level dropsdown to s, the stock is replenished up to S. The profile of the stocklevel in this situation is shown in Fig. 3. The cost structure is thesame as in Section 3 here, with an additional fixed replenishmentcost due to the preventive replenishment. Specifically, we assumethat each preventive replenishment costs kr.

Finding an exact expression for the expected cost is complex, andleads to intractable results, as one needs to keep track of the residual(remaining) time in the last preventive replenishment cycle beforesupply interruption. In the following, we develop a simplifiedapproximate model which works well under reasonable assump-tions, and, as such it is highly suited for practical applications.

Consider first the case where the up-time is large enough, i.e.,X > tr. We define the average stock level during a replenishment cy-cle as

S ¼R tr

0 Se�hx dxtr

¼ SdðtrÞ; ð12Þ

where dðtrÞ ¼ ð1�e�htr Þhtr

. Our main approximating assumption here isto use the average stock S as a base stock, similar to the base modelin Section 2, for determining the holding, ordering and shortagecosts when X > tr. This is a reasonable assumption as, in practice,the deterioration rate, h, is small. In the following, we derive anapproximate formulation of the expected cost as a function of Sand tr. Once suitable values of these alternate decision variablesare found, an equivalent ‘‘optimal’’ (s, S) policy is easily determinedvia the identities S ¼ S=dðtrÞ and s ¼ Se�htr . Then, the holding andshortage costs per unit time are respectively ChuðS; trÞ ¼ hS and

CsuðS; trÞ ¼ pkR1

SD

y� SD

� �fY ðyÞdy. The ordering cost (at the end of

down time) requires a slight modification from Section 2, as thestock is assumed to drop from S at the beginning of the down timeand is replenished back to S at the end of this time. Therefore, theexpected ordering cost per unit time is

CouðS; sÞ ¼ ckS

dðtrÞ�Z S

D

0ðS� DyÞfYðyÞdy

!:

Finally, we add the expected ordering cost during the up-time.This has two components, a fixed cost component, equal to kr

trper

unit time, and a variable cost component accounting for the vari-able cost of stock purchasing during the up time at the end of eachreplenishment cycle, equal to cSð1� e�htr Þ ¼ cShtr per replenish-ment cycle. Therefore, the expected cost per unit time whenX > tr is approximately

CdruðS;tr jX> trÞ¼ hSþpkZ 1

SD

y� SD

!fY ðyÞdy

þ ckShkþ S

dðtrÞ�Z S

D

0ðS�DyÞfYðyÞdy

!þkr

tr: ð13Þ

Next, consider the case where the up-time is not too large, i.e., X < tr.(As the up-time, X, fluctuates randomly, the same system can exhi-bit both regimes, X > tr and X < tr.) Then, the system behaves similarto the deterioration model is Section 3, but with an upper bound onX, equal to tr. Then, replacing S by S

dðtr Þ and setting the upper boundon X to, the expected cost when X < tr is given from 7, 9 and 10 as

CdruðS; tr jX < trÞ ¼ hkZ tr

0

ShdðtrÞ

!ð1� e�hxÞfXðxÞdx

þ pkZ tr

0

Z 1

Se�hxdðtr ÞD

y� Se�hx

dðtrÞD

!fYðyÞdy

" #fXðxÞdx

þ ckZ tr

0

Z Se�hxdðtr ÞD

0

SdðtrÞ

� Se�hx

dðtrÞþ Dy

!fY ðyÞdy

"

þZ 1

Se�hxdðtr ÞD

SdðtrÞ

fYðyÞdy

#fXðxÞdx: ð14Þ

Finally, the expected cost under deterioration and preventivereplenishment is given from (13) and (14) as

CdruðS; trÞ ¼ CdruðS; trjX < trÞPfX < trg þ CdruðS; tr jX > trÞPfX > trg:ð15Þ

While the expected cost in (15) appears convoluted at first sight, itsanalysis, however, is relatively easy, by building on our previous re-sults, when the replenishment time tr is exogenously determined. In

Fig. 2. Reserve stock with deterioration.

Fig. 3. Model with deterioration and preventive replenishment.

Table 2Results for base model and comparison with Hansmann. Base case: 1/k = 1 year,1/l = 7/365 year, c = $1/unit, h = 0.15 $/unit/year, p = $45,000/year, h = 5%.

Case Variation from basecase

S�b Cbu S�b� �

SHb Cbu SH

b

� �DH (%)

1 None 795 $516 971 $522 1.02 h = $0.05 1174 $421 1350 $423 0.53 h = $1 140 $830 316 $869 5.04 c = $0 971 $198 971 $198 0.05 c = 0.8 838 $454 971 $457 0.86 c = $1.8 532 $753 971 $782 4.07 c = $2.3 99 $861 971 $944 10.0

68 B. Maddah et al. / European Journal of Operational Research 232 (2014) 64–71

practice, one would experiment with a handful of values of tr andpick an appropriate one. This is especially plausible since ournumerical analysis shows that the expected cost is not too sensitiveto tr. When tr is given, the expected cost in (15) is convex in theaverage stock level S as indicated in the following theorem. Thisgreatly simplifies the analysis, as S could be found with any numer-ical search technique.

Theorem 3. If p � cD > 0, then the expected cost under exponentialdeterioration and preventive replenishment is convex in the averagestock level S.

8 p = $25,000 329 $446 768 $474 6.09 p = $20,700 0 $397 703 $457 15.20

10 p = $65,000 986 $545 1098 $547 0.40

Proof. See Appendix A. h

5. Numerical results and insights

In this section, we present the results of our numerical study.Much of the numerical study is a sensitivity analysis from a basecase. The base case has the following reasonable parameter values.The demand rate is D = 18,000 units/year. The cost structure entailsa unit ordering cost c = $1/unit, a holding cost h = $0.15/unit/year,and a shortage cost p = $45,000/year. In the case of preventivereplenishment (see Section 4), each preventive replenishmentcosts kr = $1. The supply process involves an up-time which isexponentially distributed with mean 1/k = 1 year, and a down timewhich is also exponentially distributed with mean 1/l = 7/365 year(1 week). Finally, the deterioration rate is h = 5%.

For the base model in Section 2, where deterioration is assumedaway, the optimal reserve stock is given from (6), withF�1

Y ðzÞ ¼ �ð1=kÞ lnð1� zÞ, as S�b ¼ 795, with an expected costCbu S�b� �

¼ $516/year, given from (5). If instead of using our optimalbase model, one utilizes Hansmann’s (1962), which ignores thevariable ordering cost as discussed in Section 2, the resulting re-serve stock level from (7) is SH

b ¼ 971 with a corresponding actual

cost of Cbu SHb

� �¼ $522/year. That is, the regret (penalty) from using

Hansmann’s model instead of our optimal model is DH � 1%, where

DH ¼ Cbu SHb

� �� Cbu S�b

� �h i=Cbu S�b

� �. Table 2 shows further compari-

son between our base model and Hansmann’s model. Each entryin Table 2 involves the variation of one or more parameters fromthe base case while keeping other parameters at their base values.The major insight from Table 2 is that the penalty from utilizingHansmann’s model instead of our base model can be significant reach-ing 15%in some cases where the shortage cost is low, e.g. Case 9 inTable 2. Table 2 also indicates that Hansmann’s model always rec-ommend a higher reserve stock, SH

b , than the optimal one, S�b . Ta-ble 3 shows similar results as Table 2 but with a longer up-timewith a mean of 5 years, and shows similar insights as Table 2.The penalty from using Hansmann’s model is significant (reaching24% here) and this model recommends high reserve levels. Finally,by comparing Tables 2 and 3, we observe that the optimal reservelevels and their corresponding costs decrease with the duration ofup-time, which is intuitive. (Note that when the up-time is infinite,

Table 3Results for base model and comparison with Hansmann. Base case: 1/k = 5 years,1/l = 7/365 year, c = $1/unit, h = 0.15 $/unit/year, p = $45,000/year, h = 5%.

Case Variation from base case S�b Cbu S�b� �

SHb Cbu SH

b

� �DH (%)

1 None 239 $157 416 $162 4.02 h = $0.05 619 $117 795 $119 2.03 h = $0.3 0 $173 156 $184 6.04 c = $0 416 $114 416 $114 0.05 c = $0.8 283 $150 416 $114 24.06 c = $1.74 5 $173 416 114 13.07 p = $31,500 0 $121 293 135 11.08 p = $65,000 431 $185 543 188 1.3

Table 4Results for Model 2 and comparison with base model. Base case: 1/k = 1 year, 1/l =7/365 year, c = $1/unit, h = 0.15 $/unit/year, p = $45,000/year, h = 5%

Case Variation from base case S�d Cud S�d� �

S�b Cud S�b� �

Db (%)

1 None 730 $553 795 $555 0.22 h = $0.05 928 $465 1174 $471 1.33 h = $1 148 $826 140 $826 0.04 c = $0 986 $190 971 $190 0.05 c = $1.8 386 $776 532 $783 0.96 p = $25,000 261 $452 329 $453 0.27 p = $65,000 891 $578 986 $580 0.38 h = 1% 780 $524 795 $524 0.09 h = 2.5% 743 $530 795 $530 0.1

10 h = 10% 633 $557 795 $565 1.511 h = 15% 636 $618 795 $625 1.2

Table 5Results for Model 2 and Comparison with Base Model. Base case: 1/k = 5 years,1/l = 7/365 year, c = $1/unit, h = 0.15 $/unit/year, p = $45,000/year, h = 5%

Case Variation from base case S�d Cud S�d� �

S�b Cud S�b� �

Db (%)

1 None 169 $166 239 $167 0.52 h = $0.05 463 $142 619 $145 1.43 h = $0.3 0 $173 0 $173 0.04 c = $0.8 243 $159 282 $160 0.25 c = $1.74 0 $173 0 $173 06 p = $65,000 404 $206 431 $205 0.17 h = 1% 227 $159 239 $159 0.08 h = 2.5% 207 $162 239 $162 0.09 h = 10% 84 $171 239 $175 2.0

10 h = 15% 0.4 $173 239 $181 5.0

B. Maddah et al. / European Journal of Operational Research 232 (2014) 64–71 69

no reserve stock is needed.) Case 4 with zero ordering cost showsthat the model converges to Hansmann’s model when the unitordering cost is ignored.

Next, we analyze the effect of exponential deterioration on thereserve stock policy. For the base case given at the beginning of this

Table 6Results for Model 3 and comparison with Model 2. Base case: 1/k = 5 years, k=$1, 1/l = 7/

Case Change No preventive replenishment Preventive replen

Half year

S�d C�ud S�drC�udr

1 None 169 $169 202 $1732 p = $100,000 644 $247 527 $2353 p = $150,000 854 $283 692 $2664 p = $200,000 999 $307 804 $2885 p = $250,000 1110 $327 888 $3046 p = $300,000 1200 $343 956 $3177 p = $350,000 1276 $356 1013 $3288 p = $400,000 1343 $368 1062 $3379 p = $450,000 1401 $378 1105 $345

10 p = $500,000 1453 $387 1143 $353

section, the optimal reserve stock under deterioration is obtainedby solving (11) as S�d ¼ 730 with a corresponding annual costCud S�d� �

¼ $554. If one ignores deterioration, and utilize the optimalreserve level of the base model, S�b ¼ 795, the actual annual cost isCud S�b� �

¼ $555. That is, the regret (penalty) from using the basemodel instead of the optimal deterioration model (Model 1) isDb � 0.2%, where Db ¼ Cdu S�b

� �� Cdu S�d

� � �=Cdu S�d

� �. In this case,

the penalty from ignoring deterioration Db, is not too high, becausethe deterioration rate is somewhat small (at 5%) and the reserve le-vel is not high. This is not always the case.

Table 4 shows several other cases where the penalty fromignoring deterioration, Db, is high especially when the holding costis low (e.g. Case 2 with Db = 1.3%) or the deterioration rate h is highthe penalty from ignoring deterioration (e.g. Case 11 withDb = 1.2%). Table 5 shows similar results to Table 4 but with a long-er up-time having a mean of 5 years. This table reveals similar in-sights as Table 4, while highlighting the effect of deterioration at amore pronounced level. That is, deterioration is more damaging atlonger up-times, which is intuitive, and this is even most severe forlow holding cost (Case 2, with Db = 1.4%) and high deterioration(Case 10, with Db = 5%). The main insight from Tables 4 and 5 isthat the penalty of ignoring deterioration can be significant, reaching5%.

In terms of the effect of deterioration on the reserve level, bothTables 4 and 5 indicate that the reserve level decreases with dete-rioration, as S�d < S�b in all cases of these tables. This may seemcounter-intuitive at first sight, as one expects the reserve stockto increase to avoid shortages resulting from the reserve stockdepletion as a result of deterioration. However, keeping a high re-serve stock will significantly increase the deterioration cost, as un-der exponential deterioration, the deterioration amount is apercent of the on-hand reserve. Therefore, the reserve stock is de-creased to avoid excessive deterioration costs. A similar behavior isobserved in the classic inventory control literature (e.g. Nahmias,1982).

Finally, we analyze Model 3 on preventive replenishment aimedat curbing the effect of deterioration. As discussed in Section 4, gi-ven the complexity of the problem, we analyze this model by opti-mizing over the average stock level S for a set of practical values ofthe replenishment time, tr. Here, we choose values of one quarter,half-year, three quarters, and one year. For each one of thesereplenishment times, we solve for S, and we get the following re-sults for the base case. For tr = 0.25 year (a quarter), the optimalaverage stock is S�dr ¼ 514 and the corresponding annual cost is

$576. (This gives an order-up-to level of S�dr ¼S�

drdðtrÞ ¼ 192 and a reor-

der point s�dr ¼ S�dre�htr ¼ 190.) Since the (s, S) policy follows imme-

diately from the optimal average stock, S, and given replenishmenttime, tr, we report only on S in the sequel. Recall that the expected

365 year, c = $1/unit, h = 0.15 $/unit/year, p = $45,000/year, h = 5%.

ishment every

Year Two years and a half

Dd (%) S�drC�udr Dd (%) S�dr

C�udr Dd (%)

�2 275 $175 �3.6 868 $167 15 593 $228 7.7 1135 $184 266 757 $256 9.5 1297 $194 316 869 $275 10.4 1409 $201 357 954 $289 11.6 1494 $206 378 1023 $301 12.2 1564 $210 398 1080 $310 12.9 1622 $213 408 1129 $319 13.3 1672 $216 419 1173 $326 13.8 1717 $219 429 1211 $332 14.2 1756 $221 43

70 B. Maddah et al. / European Journal of Operational Research 232 (2014) 64–71

cost with no preventive replenishment, from the model inSection 3, is $554, with a base-stock level S�d ¼ 730. This indicatesthat preventive replenishment is not useful in this case, with an

‘‘improvement’’ over Model 2, Dd ¼ Cdu S�dð Þ�Cdru S�drð Þ½ �

Cdu S�dð Þ, of around �4%.

That is, the savings on shortage do not outweigh the additionalordering costs accrued from preventive replenishment. Similaranalysis with different values of the replenishment time, namely,tr = 0.5, 0.75 and 1 year, was carried-out and a similar result is ob-served. Preventive replenishment is not useful in this case. This isperhaps due to the somewhat short up-time (1 year, on average)and low shortage cost ($45,000). In an attempt to identify caseswhere preventive replenishment works, we systematicallyincrease the shortage cost, p, up to $500,000. However, we con-tinue to observe no significant improvement from preventivereplenishment.

We then increased the up-time to 5 years, on average. This re-vealed significant improvements from preventive replenishment,

as shown in Table 6, where C�ud ¼ Cdu S�d� �

and C�udr ¼ Cdru S�dr

� �.

The improvement ranges from 26%, when p = $100,000 andtr = 2.5 years, in Case 2 of Table 5, to 43%, when p = $500,000 andtr = 2.5 years, in Case 10 of Table 6. We point out, that in some sit-uations, e.g., in airlines, when shortages of fuel or other supplieslead to grounding a plane, such high shortage costs are not unu-sual. We have observed such high shortage penalties at a US-basedairline few years ago. The main insight here is that preventivereplenishment is most useful for (i) infrequent supply interruption(ii) high deterioration and (iii) high shortage cost.

6. Conclusion

This paper presents three models to set the optimal reserve stocklevel that hedges against supply interruption. The first model is arefinement of a classical model by Hansmann (1962), which leadsto a significant cost reduction, reaching 15% in some cases. Thetwo other models are our core contribution and address the effectof exponential deterioration on reserve stock, which has not beeninvestigated before. In our second model, we determine the optimalreserve stock under exponential deterioration. We use this model tonumerically show that ignoring deterioration can significantly in-crease costs, up to 5%. We also observe numerically that deteriora-tion increases the reserve stock in order to reduce deteriorationeffects. Our third model proposes a policy to cope with the effectof deterioration, via a preventive replenishment scheme, wherethe stock is replenished periodically, while supply is available, to re-place deteriorated items. Our numerical results indicate that thismodel can offset much of the additional costs incurred due to dete-rioration, especially when supply interruptions are infrequent andthe shortage cost is high. The improvement from adopting preven-tive replenishment reaches 40%. It is worth mentioning that wehave derived rigorous convexity results for all three models.

An immediate extension to our work is addressing the highlypressing issue of managing reserves with products undergoingfixed-life deterioration (e.g. as in government strategic healthcarereserves). This came to our attention while reading the excellentrecent work by Shen et al. (2011), who took the viewpoint of themanufacturer producing the reserve, and not the reserve keeper(e.g. the government) as we do here. While the mathematical anal-ysis of the fixed-life deterioration case is expected to be more in-volved (as illustrated, for example, in Nahmias, 1982), theframework and ideas set-forth in this paper are expected to yieldreasonable control policies under fixed-life deterioration. We arecurrently investigating this extension. Another natural extensionof our work is accounting for the stochasticity of demand duringdowntime (our models assumed deterministic demand).

Appendix A. Proof of convexity results

Proof of Theorem 1. Differentiating the expected cost in (4) gives

@CbuðSÞ@S

¼ h� kðp=DÞZ 1

SD

fY ðyÞdy

þ ck ðS=DÞfYðS=DÞ þZ 1

SD

fY ðyÞdy� ð1=DÞSfY ðS=DÞ !

¼ h� kðp=D� cÞð1� FYðS=DÞÞ;

@2CbuðSÞ@S2 ¼ ðk=DÞðp=D� cÞfY ðS=DÞ:

Therefore, under the inequality condition in the theorem, p > cDand, accordingly, Cbu(S) is convex. The optimal reserve stock levelis therefore the solution to the first-order optimality condition@CbuðSÞ@S ¼ 0, which implies that

S�b ¼ DF�1Y 1� hD

ðp� cDÞk

� �;

where by the inequality condition in the theorem (p � cD)k > hDwhich implies that 0 < hD

ðp�cDÞk < 1, and such a solution exists. h

Proof of Theorem 2. The ordering per unit time in (10) can bewritten as

CouðSÞ¼ ck SþDZ 1

0

Z Se�hxD

0y�Se�hx

D

!fY ðyÞdy

" #fX ðxÞdx

( )

¼ ck SþDZ 1

0

Z 1

0y�Se�hx

D

!fY ðyÞdy

" #fX ðxÞdx�D

Z 1

0

Z 1

Se�hxD

y�Se�hx

D

!fY ðyÞdy

" #fX ðxÞdx

( ):

From this and (7) and (9), the total expected cost per unit time canbe expressed as

CduðSÞ ¼ hkZ 1

0ðS=hÞð1� e�hxÞfXðxÞdxþ pk

�Z 1

0

Z 1

Se�hxD

y� Se�hx

D

!fYðyÞdy

" #fXðxÞdx

þ ck Sþ DZ 1

0

Z 1

0y� Se�hx

D

!fY ðyÞdy

" #fXðxÞdx

(

�DZ 1

0

Z 1

Se�hxD

y� Se�hx

D

!fY ðyÞdy

" #fXðxÞdx

):

Differentiation gives

@CduðSÞ@S

¼ hkhð1�uÞ � k

p� cDD

Z 1

0

Z 1

Se�hxD

e�hxfYðyÞdy

" #fXðxÞdx

þ ckð1�uÞ ¼ k c þ hh

� �ð1�uÞ

� kp� cD

D

Z 1

0

Z 1

Se�hxD

e�hxfYðyÞdy

" #fXðxÞdx:

Upon further differentiation,

@2CduðSÞ@S2 ¼ k

p� cDD

� �Z 1

0e�hx e�hx

DfY

Se�hx

D

!" #fXðxÞdx:

The inequality condition in the theorem implies that p � cD >(1/u � 1)(hD/h + cD) > 0, where the second inequality follows since

u < 1. This implies that p > cD. It follows that @2CduðSÞ@S2 > 0 and Cdu(S) is

convex. The optimal stock level can then be found from the first-or-der optimality condition, @CduðSÞ

@S ¼ 0, provided that this condition has asolution for S 2 (0,1). Therefore, we still need to verify that the con-

B. Maddah et al. / European Journal of Operational Research 232 (2014) 64–71 71

dition @CduðSÞ@S ¼ 0 has a solution defining a global minimum. (This al-

lows avoiding cases where Cdu(S) is convex and monotone.) Theremainder of the proof handles this.

Setting @CduðSÞ@S ¼ 0 gives (11). In the following, we show that the

condition, @CduðSÞ@S ¼ 0, and equivalently (11), always has a solution.

This is achieved by showing that Cdu(S) is decreasing for S = 0 andincreasing for S ?1, which implies that a local minimum of Cdu(S)exists for S 2 (0, 1).

@CduðSÞ@S

����S¼0¼ hk

hð1�uÞ � ku

p� cDD

� �þ ckð1�uÞ

¼ kuD

1u� 1

� �hDhþ cD

� �� ðp� cDÞ

� < 0;

limS!1

@CduðSÞ@S

¼ k c þ hh

� �ð1�uÞ > 0;

where the first inequality follows from the condition in the theo-rem. Specifically, the condition in the theorem, ðp� cDÞ >

1u� 1� �

hDh þ cD� �

, implies that 1u� 1� �

hDh þ cD� �

� ðp� cDÞ < 0. h

Proof of Theorem 3. The proof is based on showing thatCdruðS; tr jX > trÞ and CdruðS; trjX < trÞ in (13) and (14) are both con-vex since a convex combination of two convex functions is alsoconvex. First, consider CdruðS; tr jX > trÞ. This is convex as

@CdruðS; trjX > trÞ@S

¼ h� pkD

Z 1

SD

fY ðyÞdyþ ckdðtrÞ

� ckZ S

D

0fY ðyÞdyþ ch

¼ hþ ckdðtrÞ

þ ch� pkDþ ðp� cDÞ

DkFY

SD

!;

@2CdruðS; tr jX > trÞ@S2

¼ ðp� cDÞD2 kfY

SD

!> 0;

where the inequality follows from the condition in the theorem.Second, consider CdruðS; trjX < trÞ in (14). This is also convex basedon the following derivations. Note first that (14) can be rewritten as

CdruðS;trjX< trÞ¼hkZ tr

0

ShdðtrÞ

!ð1�e�hxÞfXðxÞdx

þpkZ tr

0

Z 1

Se�hxdðtr ÞD

y� Se�hx

dðtrÞD

!fY ðyÞdy

" #fXðxÞdx

þckZ tr

0

SdðtrÞ

þZ Se�hx

dðtr ÞD

0�Se�hx

dðtrÞþDy

!fY ðyÞdy

" #fXðxÞdx:

Then,

@CdruðS; trjX < trÞ@S

¼ hkhdðtrÞ

1�Z tr

0e�hxfXðxÞdx

� �� pk

dðtrÞD

�Z tr

0

Z 1

Se�hxdðtr ÞD

e�hxfYðyÞdy

" #fXðxÞ þ

ckdðtrÞ

�Z tr

01�

Z Se�hxdðtr ÞD

0e�hxfYðyÞdy

" #fXðxÞdx;

@2CdruðS; trjX < trÞ@S2

¼ kdðtrÞD

ðp� cDÞZ tr

0

e�2hx

dðtrÞDfY

Se�hx

dðtrÞD

!" #fXðxÞdx

> 0;

where the inequality follows from the condition in the theorem. h

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