European Journal of Operational Research 221 (2012) 275–284

Contents lists available at SciVerse ScienceDirect

European Journal of Operational Research

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

Invited Review

Review of inventory systems with deterioration since 2001

Monique Bakker, Jan Riezebos ⇑, Ruud H. TeunterDepartment of Operations, Faculty of Economics and Business, University of Groningen, Groningen, The Netherlands

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

Article history:Received 20 June 2011Accepted 4 March 2012Available online 20 March 2012

Keywords:Inventory controlDeteriorating itemsPerishabilityShelf lifeDecayReview

0377-2217/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2012.03.004

⇑ Corresponding author.E-mail addresses: j.riezebos@rug.nl (J. Riezebos), r

ter).

This paper presents an up-to-date review of the advances made in the field of inventory control of per-ishable items (deteriorating inventory). The last extensive review on this topic dates back to 2001 (GoyalS.K. and Giri B.C., Recent trends in modeling of deteriorating inventory, European Journal of OperationalResearch, 134, 1–16). Since then, over two hundred articles on this subject have been published in themajor journals on inventory control, indicating the need for a new review. We use the classification ofGoyal and Giri based on shelf life characteristics and demand characteristics. Contributions are high-lighted by discussing main system characteristics, including price discounts, backordering or lost sales,single or multiple items, one or two warehouses, single or multi-echelon, average cost or discounted cashflow, and payment delay.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

In the mathematical modeling of inventory control that startedwith the classical Economic Order Quantity (EOQ) model of Harris[91], the implicit assumption was that stocked items have infiniteshelf lives. Deterioration was first accounted for by Whitin [218],who considered fashion items deteriorating after a prescribed stor-age period. Ghare and Schrader [82] first modeled negative exponen-tial decaying inventory. Almost 50 years later, many variations existthat differ in assumptions on not only the lifetime of an item, but alsoon the type of demand, the presence of price discounts, allowingshortages and backordering, single or multiple items, one or twowarehouses, single or multi-echelon modeling, average cost or dis-counted cash flows, and whether a delay in payment is permissible.

About a decade ago, Goyal and Giri [85] made a distinction be-tween three broad categories of inventory in their excellent reviewon deteriorating inventory models, which included 130 references.They classified models based on the presence of obsolescence,deterioration, or neither. Items are subject to obsolescence if theylose their value over time because of rapid changes of technologyor the introduction of a new product by a competitor, or becausethey go out of fashion. Deterioration is defined as the damage,spoilage, vaporization, dryness etc. of items. Like our review, theone by Goyal and Giri is restricted to deterioration since, as theyargued, ‘‘considerable attention has not yet been given on model-ing of such an inventory system only because once the itemsbecome obsolete they are not reordered’’. More than twenty years

ll rights reserved.

.h.teunter@rug.nl (R.H. Teun-

ago, the first reviews appeared. Prastacos [171] provided a surveyof blood inventory management. Raafat [172] did a survey restrict-ing his study to continuously deteriorating inventory models only,as he had been preceded by Nahmias [157] who had already pro-vided an extensive review on fixed-lifetime models. Recently, sev-eral review contributions have appeared. Some of them focus onspecific areas, such as Pierskalla [170], who discusses blood inven-tory and supply chain management. Nahmias [158] published abook that provides an overview of the modeling approaches usedin the field of perishable inventory systems. Pahl et al. [165], Akk-erman et al. [4], and Karaesman et al. [112] focus on the issue ofdeterioration in production and distribution planning, especiallyof food supply chains. Finally, Li et al. [125] provide an overviewof some 100 recent papers on deterioration inventory manage-ment, including 25 publications that have appeared in Chinesemanagement journals. All these reviews do not aim to provide anoverview similar to Goyal and Giri, who covered the main contri-butions in modeling deteriorating inventory problems in the scien-tific journals in the field of inventory theory.

The objective of our article is therefore to give a comprehensiveliterature review of models for inventory control with deteriorat-ing items that have been published since the review of Goyal andGiri. The classification is consistent with that of Goyal and Giri tofacilitate comparison.

The scope of this paper is limited to deteriorating inventory anddoes not include literature on obsolescence (sudden death) to keepthe length of this article to a containable size. The focus is not onthe details of mathematical derivations, but on the assumptionsand specifications of the models. Section 2 describes the methodthat has been applied for the collection of literature for this review.

Table 1Journals for which keyword search has resulted in relevant articles publishedbetween 2001 and 2011 (subsequent columns present number of additional hits;⁄identifies wildcard).

Journal Keyword

deteriorat⁄ perish⁄ shelflife

decay⁄

Computers & Industrial Engineering 20 4 0 0Computers & Operations Research 23 0 0 1European Journal of Operational

Research28 4 1 0

International Journal of Informationand Management Sciences

13 4 0 0

International Journal of ProductionEconomics

34 4 3 0

International Journal of Retail andDistribution Management

3 0 0 0

International Journal of SystemsScience

28 0 0 0

International Transactions inOperations Research

5 0 0 0

Management Science 1 1 0 0Naval Research Logistics 2 3 0 0Operations Research Letters 2 2 0 0Production and Operations

Management1 5 0 0

Production Planning & Control 6 1 0 0

276 M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284

Section 3 describes the classification of deteriorating inventorymodels, including the allocation of reviewed literature to theseclasses and pertinent additional model characteristics. Section 4gives an overview of the results. The conclusion is given in Section5.

2. Types of inventory models based on deterioration anddemand

2.1. Deterioration

Models for deteriorating inventory can be broadly categorizedaccording to the lifetime of products and characteristics of de-mand. Three categories are distinguished based on shelf lifecharacteristics:

(1) Fixed lifetime, i.e. predetermined deterministic lifetime ofe.g. two days or one season.

(2) Age dependent deterioration rate (which implies a probabi-listic distributed lifetime, e.g. Weibull).

(3) Time or inventory (but not age) dependent deteriorationrate.

Note that models with a constant deterioration rate per stockeditem (and so inventory but not age dependent) belong to class 3.

2.2. Demand

Demand can be modeled as either stochastic or deterministic.From a real life point of view, a stochastic demand distributionis more plausible, although less than 20% of the models reviewedin this paper can be classified as such. Following Goyal and Giri,for the stochastic demand models a further distinction is made be-tween a specific type of probability distribution and an arbitraryprobability distribution. Only ten papers in this review providemodels with an arbitrary probability distribution for demandand two papers include models that treat demand as a fuzzynumber.

In the case of deterministic demand, a sub-categorization ismade into:

(a) Uniform demand, i.e. demand is a constant, fixed number ofitems.

(b) Stock-dependent demand.(c) Time-varying demand.(d) Price-dependent demand.

It must be noted that a combination of the above is alsopossible.

3. Method

3.1. Initial phase

Our study aims to find papers on deteriorating inventory con-trol that have been published between January 2001 and Decem-ber 2011. In that way, it can serve as a follow up of the review ofGoyal and Giri [85], which included papers till 2000. We firststarted with a keyword search in a selection of major journals(listed in Table 1) that publish on this subject. Four different pairsof keywords were entered subsequently, using the wildcard sym-bol ⁄ plus ‘AND inventory’. The keywords are 1.deteriorat⁄ (mean-ing that we search for words in title, abstract, or key word listingsstarting with ‘deteriorat’ that can end differently, e.g. ‘deteriorate’,‘deteriorating’, ‘deterioration’ etc.), 2. perish⁄, 3. shelf life, 4.decay⁄.Articles were judged on relevance by scanning the title first and if

necessary the abstract to see whether it is indeed an article onmathematical modeling of perishable inventory control, publishedafter the year 2000. Articles on obsolescence (sudden death ofitems) were excluded in a later stage. The results of this searchare given in Table 1 below. Note that only the additional articlesfor collection are listed (e.g. for Computers and Operations Re-search: after searching with deteriorat⁄, searching with perish⁄ orshelf life gave no additional papers, but searching with the key-word decay provided an additional paper).

3.2. Second phase

In the second phase, the 199 collected papers (models) wereexamined. We excluded 17 papers because closer examinationproved them not to be relevant for this review. For example, [1]examines the root causes of stock-outs for deteriorating items, withthe purpose of identifying practical implications for management,rather than to develop an inventory model. The other 182 paperswere allocated to the type of deterioration as described in Section2 (fixed lifetime; age dependent deterioration rate; time or inven-tory dependent deterioration rate), and to the type of demand(deterministic: uniform, time-varying, stock-dependent, price-dependent; stochastic: specific or arbitrary probability distribution).

3.3. Final phase

In the final phase, the references of the papers were checked inorder to obtain additional papers from these or other journals. Thisresulted in 63 articles from 25 additional journals. Of these, 18 arti-cles were excluded later on because of irrelevance. The remaining45 papers were added to the set of relevant papers and allocatedaccording to type of deterioration and demand. A further classifica-tion of models was done based on the characteristics described inSection 4.

The European Journal of Operational Research and the Interna-tional Journal of Production Economics both have publishedaround 40 of the 227 papers on deteriorating inventory duringthe last decade. All other journals published less than 30 paperson this subject. See Table 2 for an overview.

Table 2Number of relevant articles on deteriorating inventory between 2001–2011 perjournal.

Journal Papers

Advanced Modeling & Optimization 1Annals of Operations Research 1Applied Mathematical Modeling 7Applied Mathematics and Computation 2Computers & Industrial Engineering 21Computers & Mathematics with Applications 2Computers & Operations Research 17European Journal of Industrial Engineering 1European Journal of Operational Research 40Expert Systems with Applications 1Fuzzy Optimization and Decision Making 1IIE Transactions 3International Journal of Automation and Computing 1International Journal of Information and Management Sciences 15International Journal of Mathematical Education in Science and

Technology1

International Journal of Physical Distribution & LogisticsManagement

1

International Journal of Production Economics 39International Journal of Retail and Distribution Management 2International Journal of Systems Science 26International Transactions in Operations Research 5Journal of Applied Mathematics and Computing 1Journal of Computational and Applied Mathematics 1Journal of the Operational Research Society 3Management Science 2Manufacturing & Service Operations Management 1Mathematical and Computer Modeling 1Mathematical Problems in Engineering 1Mathematics of Operations Research 1Naval Research Logistics 6Omega 1Operations Research 2Operations Research Letters 3Optimal Control Applications and Methods 1Orion 1Production and Operations Management 6Production Planning & Control 6Sadhana 1The Engineering Economist 2

Total 227

M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284 277

4. Analysis

The selection process described in the previous section led to227 relevant papers that have been published between January2001 and December 2011. Note that some of them were still inpress at the moment of writing this manuscript, but these paperswere already available online. In Table 2, we show the number ofpapers published by each journal. In Table 3 the papers are classi-fied according to the modeling of the deterioration (lifetime) and ofthe demand, key criteria that were discussed in Section 2.

In what remains, rather than discussing contributions in rela-tive isolation, we will discuss a number of key modeling elementsin separate subsections. These are the inclusion or not of a price in-crease/discount (Section 4.1), treatment of stock-outs (Section 4.2),single- or multi-item (Section 4.3), number of warehouses (Section4.4), single vs. multi-echelon (Section 4.5), cost accounting aspects(Section 4.6) and a permissible delay in payment (Section 4.7).

4.1. Known price increase or price discount on perishables

Very few authors consider a (potential) price increase in theirmodeling of deteriorating inventory control [2,3,17,205,209]. Abad[2], with a profit per period objective, allows for flexible pricing,assuming that the perishables are price sensitive. His procedureis relatively straightforward and can be implemented on a spread

sheet. Similarly, in another inventory model of Abad [3], the sellingprice is the decision variable. Teng et al. [205] extend his model byadding a backlogging cost and a cost of lost goodwill. Tsao andSheen [209], with the same objective to maximize net profit, showthat dynamic decision-making in terms of retail price and promo-tional effort is superior to fixed decision-making.

In practice, it is common to offer deteriorated items at a discountprice. In many supermarkets, for instance, items that are near theirexpiration date are marked down by a fixed percentage to influencethe consumer’s buying behavior. Such price discounts are addressedin many papers [2,3,5,7,17,21,24,29,35,37,44,47,59,60,74,75,78,80,99,101,111,118,126,127,142,161,201,205,208,209,213]. Monitor-ing and control of time-sensitive products can be facilitated by theapplication of radio frequency identification (RFID) [23,26] and apolicy of dynamic pricing [17,21], such as price discounts whenthe item reaches a predetermined age. Ferguson and Koenigsberg[78] discuss the possible cannibalization effect of having bothhigh-priced fresh products and low-priced older products. Ramana-than [173] and Sezen [184] assume two discount rates for the firstand second discount period in their models. Ramanathan extendsthe expected profit approach from Sezen by including decisionson the quantities of the perishable products to be stocked by theretailer.

In recent years, some authors have explicitly modeled the ben-efits and costs of using preservation technology in the supply chainand storage of fresh products. See e.g. [22,24,70,98,117,156]. Inthese models, deterioration rate is not an exogenous variable, butan endogenous variable. This way of modeling seems to becomemore important for gaining insight in the choices with respect todeterioration.

There are four papers, [43,104,153,217], that address the com-bined effect of deterioration and amelioration, where ameliorationis the growth of inventory over time. An example of such inventoryis livestock. In these models, both demand and deterioration causethe inventory to decrease, while the amount of growth depends onthe amelioration rate and the on hand stock.

Arcelus et al. [7] provide a model with payment reductionschemes other than temporary price discounts. They assume thatthe vendor’s trade promotion is a mix of credit and/or pricediscount. For the retailer, the model determines (i) the size of thespecial order to be placed at the vendor, (ii) the price and/or cred-it-terms incentives to be passed onto his own customers and (iii)the quantity to be sold under these one-time-only conditions. Inthe model by Tsao and Sheen [208] the price discount is a functionof both order quantity and time. Lin et al. [137] assume a sellingprice that decreases linearly with time and customer demand lin-early increases with the declining selling price.

Another new development in the field of deterioration modelsthat include price changes can be found in the research towardsclosed loop supply chains. We found three papers [6,49,215] thataddress closed loop supply chain issues, where products can be re-used after consumption, although they deteriorate. It appearedthat in the field of closed loop supply chains, many papers assumeobsolescence or assume that the products are remanufactured(which includes costs related to the deterioration rate) before theyare used again.

4.2. Shortages and backordering

Of all articles analyzed, almost 50% allows for shortages. Thesemodels differ with regard to the fraction backordered and whethera cost parameter for lost sales and/or backorders is accounted for.Models that assume complete backlogging [9,31,34,38,55,56,63,64,69,77,83,84,89,93,94,105,122,129,135,147–149,152,160,176,179,182,183,185,214,216,224,227,232] are often far less applicablein real life situations than models that assume partial backlogging

Table 3Number of papers per category (references in the footnote between brackets).

Deterioration Demand

Deterministic Stochastic

Uniform Stock-dependent

Time-varying

Price-dependent

Known probabilitydistribution

Arbitrary probabilitydistribution

Fixed lifetime 8 (1,1) 4 (1,2) 10 (1,3) 6 (1,4) 20 (1,5) 4 (1,6)Age dependent deterioration rate 5 (2,1) 4 (2,2) 13 (2,3) 3 (2,4) 3 (2,5) 2 (2,6)Time or inventory dependent

deterioration rate41 (3,1) 22 (3,2) 68 (3,3) 23 (3,4) 13 (3,5) 6 (3,6)

(1,1) [26,55,80,87,142,175,227,240](1,2) [12,46,239,241](1,3) [8,12,41,56,65,92,100,107,142,211](1,4) [44,60,81,100,154,184](1,5) [16,21,23,47,58,78,79,90,110,113,114,117,123,127,128,139,160,199,207,226](1,6) [17,88,126,173](2,1) [50,51,129,217,228](2,2) [48,116,140,181](2,3) [14,32,34,38,63,67,84,108,122,179,195,219,221](2,4) [155,169,214](2,5) [89,164,229](2,6) [122,231](3,1) [22,27–30,35,39,49,52,53,62,66,70,93,98,101,102,104,120,124,131–133,135,138,141a]

[159,161,163,174,177,178,197,210,213,215,217,230,232,234,236](3,2) [19,31,66,71,94,95,111,140,145,151,162,166,180,196,201,204,206,213,216,223,224,237](3,3) [6,7,9–11,13,33,36,37,40,42,43,45,54,57,59,64,68,69,72–74,76,83,86,96,97,103,105,106,115,121,134,136,137,143,144,

146–148,152,153,156,167,182,183,185,189–194,198,200,203,206,209,211,212,220,222–224,233,235,237,238](3,4) [2,3,20,36,37,68,72,75,77,94,99,111,137,140,166,168,176,201,202,205,208,209,225](3,5) [15,18,24,25,109,118,130,149,150,168,186–188](3,6) [5,25,39,61,119a, [141]a]

a Reference models a fuzzy demand rate.

278 M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284

[10,25,46,48,62,151,162]. Chakravarty and Daniel [25] allow a pre-determined maximum of N backlogged items within each replen-ishment cycle. Even more realistic are the models that assumethe backlog rate to be some function of waiting time, because inmost real life situations, the customer decides how long he/she iswilling to wait for the order [2,3,20,32,36,40,43,67,68,71,73,76,95,100,103,106,108,120,130,153,159,169,189,192,193,200,203,205,211,219–221,223,233–237].

Manuel et al. [150] propose an interesting model with a waitingroom and a single server. Any arriving customer (Markovian arri-val process) who finds the waiting room full enters into the orbitof infinite space. The orbiting customers compete for service bysending out signals and the duration between successive attemptsis exponentially distributed. They follow the constant retrial pol-icy, as opposed to the classical retrial policy. The former meansthat the probability of a repeated attempt is independent of thenumber of orbiting customers. The latter means that the probabil-ity of a repeated attempt is dependent on the number of orbitingcustomers.

4.3. Single item and multi-item inventory control

Two- and multi-item deteriorating inventory models are veryscarce in the literature of inventory control, as opposed to singleitem models. This imbalance can, for a large part, be directly attrib-uted to the increased complexity of multi-item models comparedto single-item models. From a real-life perspective, multi-iteminventory models are far more realistic in the sense that mostfirms, especially retailers, sell more than one item and controllingthe inventory for more than one item simultaneously, can be verycomplex. In particular, items that are substitutes of one another, orcomplementary items, severely complicate inventory control. Inthis review, only eight articles were found with a multi-item (morethan two items) deteriorating inventory model [5,37,81,124,143,145,216,240]. Two articles discussed two-item models [111,188].

Akçay et al. [5] and Maity and Maiti [145] account for substituteand complimentary items in their multi-item inventory model. Inexplaining the model, Maity and Maiti [145] give a real life exam-ple of a merchant who sells two types of bananas and apples. Onetype of bananas is said to be a substitute for the other and applesare argued to be complementary to the bananas. A customer whoarrives at the shop to buy bananas is assumed to be motivated tobuy apples as well, and vice versa. Indeed, one can think of manyreal-world business settings where this is the case. The modelwould become very complicated if more than three items areincluded that can either be substitutes or complements or perhapsboth. In practice, some objective decision rule would be needed tosystematically determine what items are substitutes for oneanother and what items complement each other.

4.4. Two warehouses for deteriorating inventory

A number of authors model the case in which two warehousesare deployed for the storage of deteriorating inventory [14,52,65,76,97,106,120,121,134,159,176,232,234]. In these models, onewarehouse is owned (OW) and the other is rented (RW). In almostall cases, inventory in the RW is depleted first, before the OWinventory is used. Costs, deterioration rates, demands and otherparameters can differ between warehouses and frequently theRW bears higher costs, which is plausible from a real-life point ofview. Kar et al. [111] consider the case of a primary and a second-ary shop. Items are purchased in lots and received at the primaryshop, where fresh and deteriorated units are separated. Only thefresh items are sold from the primary shop; the deteriorated itemsare transferred to the secondary shop and offered at a reducedprice. As the items deteriorate continuously with time, the unitsconsidered fresh at a particular point in time again deteriorateand are transferred. This procedure is common for fruit and vege-tables in developing countries where there is a big income gapbetween the rich and the poor.

M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284 279

4.5. Multi-echelon inventory control

Most multi-echelon deteriorating inventory models arerestricted to two echelons, and with three exceptions [24,48,238]these consider the single supplier – single buyer setting [24,49,52,59,76,97,107,110,114,120,135,138,159,174,176,226,232,234,239,240]. Cai et al. [24] study a single producer-single distributor sys-tem, Chung and Wee [48] analyse a single producer - single retailermodel, while Yang and Wee [238] consider the situation of singlesupplier – multiple buyers.

Some models include a third echelon [100,137,174,175,210] ora return facility (closed loop supply chains) [6,49,215]. We didnot found any papers analyzing situations with more than threeechelons. Rau et al. [174] consider the three echelon setting for asingle item, but limit their paper to a deterministic model in whichshortages are not allowed and no price discount, price increase orcost accounting assumptions (e.g. inflation, permissible delay inpayment) are present. In a follow-up paper, Rau et al. [175] intro-duce the shortage effect on downstream partners and introduceJust-In-Time (JIT) production in their model. According to their cal-culations, JIT production results in lower holding costs and lowerjoint total costs in the supply chain, compared to their previouswork. Hsu, Wee and Teng [100] limit complexity in their single-item three-echelon model, by assuming deterministic demandand a fixed lifetime and not allowing for a price discount, price in-crease, time value of money or permissible delay in payment. Theydo allow for shortages which are partially backordered.

4.6. Deteriorating inventory with inflation and time value of money

Until 1970, inflation was not included in the modeling of inven-tory, perhaps because inflation was not believed to influence theinventory policy to a significant degree [85]. That thought seemsto have changed with the double-digit inflation in the 1970s andearly 80s. In many real-life situations, the practical experiences re-veal that the inflation is non-deterministic and a variable [152].Nowadays, accounting for inflation and time value of money iscommon in the modeling of deteriorating inventory, to make themodels more realistic. Many authors emphasize the need not toignore this phenomenon any longer [10,11,13,27,29,34,38,39,43,48,54–56,62,65,66,74,77,93,94,97,99,105,152,153,162,180,185,191,198,213,214,217,222,232,234,236,237]. Dey et al. [65] attributethis development to high inflation in countries like Argentina, Bra-zil, India, Bangladesh etc. With Brazil and India growing in globaleconomic importance (being two of the BRIC countries with Russiaand China), the financial situation has been changing, making itimpossible to ignore inflation any longer. Similarly, time value formoney cannot be ignored, especially when interest rates are high[74,77] and because inventory represents a capital investment.

Chen and Ouyang [39], De and Goswami [62], and Roy et al.[180] go one step further and treat inflation as a fuzzy number intheir inventory model. Other papers also address fuzzy parameters,but focus on cost. For example, Katagiri and Ishi [113] use fuzzyoutdating and shortage costs, while Roy et al. [178] use fuzzy costcoefficients, storage space and budgetary costs.

Mirzazadeh et al. [152] make a distinction between stochasticinternal and external inflation rates to incorporate a more realisticsituation than previous deteriorating inventory models. Demand ismodeled as a linear function of these rates because in real life, theconsumption rate may change with variations in prices and pricechanges may very well be induced due to inflation.

4.7. Deteriorating inventory with permissible delay in payment

When the traditional EOQ inventory model was derived, it wasimplicitly assumed that payments are made to the supplier

instantly. In reality, however, we often find that the supplier offersthe buyer a delay in payment for settling the account, often with-out charging interest. The main purpose is to stimulate the buyerto buy more, to increase market share or to deplete inventoriesof certain items. Especially when inventories are deteriorating, apermissible payment delay may boost demand and thereby pre-vent the loss of value. In many cases, the length of the delay perioddepends on the purchase quantity [27,29,53,54,102,132,163], butthere are alternatives [39,72,156,177,196,197,202,204].

A few inventory models not only take into account a permissi-ble delay in payment to the retailer, but also from the retailer tohis customer. These models include two- or even three-echelonsin the form of the wholesaler, the retailer and the consumer[30,52,96,133]. Hsieh et al. [96] adopt a trade credit policy wherethe supplier offers the retailer the permissible delay period Mand the retailer offers the trade credit period N to his/her customerwhere M P N. An extensive, rigorous mathematical analysis is usedto prove that a unique, global-optimal minimal cost solution exists.Liao [133] takes a similar approach where the assumption is thatM > N. Chang et al. [30] extend her EPQ model and, in addition, re-lax this assumption.

5. Conclusion

In this paper we have provided an up-to-date review of deteri-orating inventory literature succeeding the work of Goyal and Giri[85]. It is eminent that deteriorating inventory models with time-dependent or time-varying demand are well represented in thecurrent literature. A great majority of models assume a determin-istic setting. From a practical viewpoint, this makes these modelsless readily applicable in a business environment. On the otherhand, deterministic models can contribute in the development ofintuition with regard to deteriorating inventory modeling.

A discussion of key model characteristics reveals that some ofthese characteristics are well represented in the literature, whileothers are hardly studied. For instance, incorporating a knownprice discount is very common and corresponds well with realworld situations for deteriorating items. A few authors considertwo deterioration stages linked to two subsequent price discounts.With regard to the key characteristics in dealing with stock-outs,the majority of the models allow for shortages and the assumptionof time-dependent backlogging rate – a realistic way to treat un-met demand – is well covered in the modeling of deterioratinginventory.

Multi-echelon inventory control is gaining importance due tothe need for supply chain integration in today’s competitive envi-ronment. More readily available information, such as radio fre-quency identification (RFID) facilitates integration down thesupply chain, with great opportunities for inventory control of per-ishables. However, only two authors take detailed informationabout the age of the inventory into account, enabled by RFID tech-nology [23,26]. Key cost accounting aspects such as accounting forinflation and permissible delays in payments, can no longer be dis-regarded in an environment where inflation and interest rates arehigh. This area is therefore well covered in the deteriorating inven-tory literature. Taking these financial aspects into account can, ashas been proven by a number of authors here, significantly influ-ence costs, thus profits.

Although modeling in a deterministic setting is more straight-forward, this review suggests a stronger focus on stochastic mod-eling of deteriorating inventory, in order to better representinventory control in practice. In addition, multi-item approaches– complex as they are – will make a substantial contribution totheory and practice since this is still a scarcely represented sub-category of deteriorating inventory.

280 M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284

Finally, a remark has to be made concerning the modeling of sub-stitutability in inventory control of perishable items. Only very fewarticles addressed this issue while it is definitely of interest toinventory managers. The modeling of substitutability in inventorycontrol would surely make a contribution to existing literaturebuilding on, and extending the marketing decision models on prod-uct assortment and shelf-space allocation towards inventory con-trol. The fact that a replenishment decision is dependent on theavailable stock of substitute items should not be disregarded anylonger. Future research should direct efforts towards this very com-plex, very poorly covered aspect of deteriorating inventory control.

References

[1] J. Aastrup, H. Kotzab, Analyzing out-of-stock in independent grocery stores:an empirical study, International Journal of Retail & Distribution Management37 (9) (2009) 765–789.

[2] P.L. Abad, Optimal price and order size for a reseller under partialbackordering, Computers & Operations Research 28 (2001) 53–65.

[3] P.L. Abad, Optimal pricing and lot-sizing under conditions of perishability,finite production and partial backordering and lost sale, European Journal ofOperational Research 144 (2003) 677–685.

[4] R. Akkerman, P. Farahani, M. Grunow, Quality, safety and sustainability infood distribution: a review of quantitative operations managementapproaches and challenges, OR Spectrum 32 (4) (2010) 863–904.

[5] Y. Akçay, H.P. Natarajan, S.H. Xu, Joint dynamic pricing of multiple perishableproducts under consumer choice, Management Science 56 (8) (2010) 1345–1361.

[6] A.A. Alamri, Theory and methodology on the global optimal solution to ageneral reverse logistics inventory model for deteriorating items and time-varying rates, Computers & Industrial Engineering 60 (2) (2011) 236–247.

[7] F.J. Arcelus, N.H. Shah, G. Srinivasan, Retailer’s pricing, credit and inventorypolicies for deteriorating items in response to temporary price/creditincentives, International Journal of Production Economics (2003) 153–162.

[8] Q.-G. Bai, Y.-Z. Zhang, G.-L. Dong, A note on an economic lot-sizing problemwith perishable inventory and economies of scale costs: approximationsolutions and worst case analysis, International Journal of Automation andComputing 7 (1) (2010) 132–136.

[9] Z.T. Balkhi, On a finite horizon production lot size inventory model fordeteriorating items: an optimal solution, European Journal of OperationalResearch 132 (4) (2001) 210–223.

[10] Z.T. Balkhi, An optimal solution of a general lot size inventory model withdeteriorated and imperfect products, taking into account inflation and timevalue of money, International Journal of Systems Science 35 (2) (2004) 87–96.

[11] Z.T. Balkhi, Optimal economic ordering policy with deteriorating items underdifferent supplier trade credits for finite horizon case, International Journal ofProduction Economics 133 (1) (2011) 216–223.

[12] Z.T. Balkhi, L. Benkherouf, On an inventory model for deteriorating items withstock dependent and time-varying demand rates, Computers & OperationsResearch 31 (2004) 223–240.

[13] Z.T. Balkhi, L. Tadj, A generalized economic order quantity model withdeteriorating items and time varying demand, deterioration, and costs,International Transactions in Operational Research 15 (4) (2008) 509–517.

[14] S. Banerjee, S. Agrawal, A two-warehouse inventory model for items withthree-parameter Weibull distribution deterioration, shortages and lineartrend in demand, International Transactions in Operational Research 15 (6)(2008) 755–775.

[15] L. Benkherouf, A. Boumenir, L. Aggoun, A diffusion inventory model fordeteriorating items, Applied Mathematics and Computation 138 (1) (2003)21–39.

[16] E. Berk, Ü. Gürler, Analysis of the (Q, r) inventory model for perishables withpositive lead times and lost sales, Operations Research 56 (5) (2008) 1238–1246.

[17] E. Berk, Ü. Gürler, G. Yıldırım, On pricing of perishable assets with menu costs,International Journal of Production Economics 121 (2) (2009) 678–699.

[18] O. Berman, K.P. Sapna, Optimal service rates of a service facility withperishable inventory items, Naval Research Logistics 49 (5) (2002) 464–482.

[19] D.K. Bhattacharya, On multi-item inventory, European Journal of OperationalResearch 162 (3) (2005) 786–791.

[20] A.K. Bhunia, S. Kundu, T. Sannigrahi, S.K. Goyal, An application of tournamentgenetic algorithm in a marketing oriented economic production lot-sizemodel for deteriorating items, International Journal of Production Economics119 (1) (2009) 112–121.

[21] A. Bisi, M. Dada, Dynamic learning, pricing, and ordering by a censorednewsvendor, Naval Research Logistics 54 (4) (2007) 448–461.

[22] J. Blackburn, G. Scudder, Supply chain strategies for perishable products: thecase of fresh produce, Production and Operations Management 18 (2) (2009)129–137.

[23] R.A.C.M. Broekmeulen, K.H. Van Donselaar, A heuristic to manageperishable inventory with batch ordering, positive lead-times, andtime-varying demand, Computers & Operations Research 36 (11) (2009)3013–3018.

[24] X. Cai, J. Chen, Y. Xiao, X. Xu, Optimization and coordination of fresh productsupply chains with freshness-keeping effort, Production and OperationsManagement 19 (3) (2009) 261–278.

[25] S.R. Chakravarthy, J.K. Daniel, A Markovian inventory system with randomshelf time and back orders, Computers & Industrial Engineering 47 (2004)315–337.

[26] A. Chande, S. Dhekane, N. Hemachandra, N. Rangaraj, Perishable inventorymanagement and dynamic pricing using RFID technology, Sadhana 30 (2–3)(2005) 445–462.

[27] C.-T. Chang, An EOQ model with deteriorating items under inflation whensupplier credits linked to order quantity, International Journal of ProductionEconomics 88 (3) (2004) 307–316.

[28] C.-T. Chang, L.-Y. Ouyang, J.-T. Teng, An EOQ model for deteriorating itemsunder supplier credits linked to ordering quantity, Applied MathematicalModelling 27 (12) (2003) 983–996.

[29] C.-T. Chang, L.-Y. Ouyang, J.-T. Teng, M.-C. Cheng, Optimal ordering policiesfor deteriorating items using a discounted cash-flow analysis when a tradecredit is linked to order quantity, Computers & Industrial Engineering 59 (4)(2010) 770–777.

[30] C.-T. Chang, J.-T. Teng, M.-S. Chern, Optimal manufacturer’s replenishmentpolicies for deteriorating items in a supply chain with up-stream and down-stream trade credits, International Journal of Production Economics 127 (1)(2010) 197–202.

[31] C.-T. Chang, J.-T. Teng, S.K. Goyal, Optimal replenishment policies for non-instantaneous deteriorating items with stock-dependent demand,International Journal of Production Economics 123 (1) (2010) 62–68.

[32] H.-J. Chang, C.-Y. Dye, An inventory model for deteriorating items with partialbacklogging and permissible delay in payments, International Journal ofSystems Science 32 (3) (2001) 345–352.

[33] H.-J. Chang, C.-H. Hung, C.-Y. Dye, An inventory model for deteriorating itemswith linear trend demand under the condition of permissible delay inpayments, Production Planning & Control 12 (3) (2001) 274–282.

[34] H.-J. Chang, C.-H. Hung, C.-Y. Dye, A finite time horizon inventory model withdeterioration and time-value of money under the conditions of permissibledelay in payments, International Journal of Systems Science 33 (2) (2002)141–151.

[35] H.-J. Chang, W.-F. Lin, J.-F. Ho, Closed-form solutions for Wee’s and Martin’sEOQ models with a temporary price discount, International Journal ofProduction Economics 131 (2) (2011) 528–534.

[36] H.-J. Chang, J.-T. Teng, L.-Y. Ouyang, C.-Y. Dye, Retailer’s optimal pricing andlot-sizing policies for deteriorating items with partial backlogging, EuropeanJournal of Operational Research 168 (1) (2006) 51–64.

[37] J.-M. Chen, L.-T. Chen, Pricing and production lot-size/scheduling with finitecapacity for a deteriorating item over a finite horizon, Computers &Operations Research 32 (11) (2005) 2801–2819.

[38] J.-M. Chen, C.-S. Lin, An optimal replenishment model for inventory itemswith normally distributed deterioration, Production Planning & Control 13 (5)(2002) 470–480.

[39] L. Chen, L.-Y. Ouyang, Fuzzy inventory model for deteriorating items withpermissible delay in payment, Applied Mathematics and Computation 182(1) (2006) 711–726.

[40] L.-H. Chen, L.-Y. Ouyang, J.-T. Teng, On an EOQ model with ramp type demandrate and time dependent deterioration rate, International Journal ofInformation and Management Sciences 17 (4) (2006) 51–66.

[41] M. Cheng, G. Wang, A note on the inventory model for deteriorating itemswith trapezoidal type demand rate, Computers & Industrial Engineering 56(4) (2009) 1296–1300.

[42] M. Cheng, B. Zhang, G. Wang, Optimal policy for deteriorating items withtrapezoidal type demand and partial backlogging, Applied MathematicalModelling 35 (7) (2011) 3552–3560.

[43] M.-S. Chern, H.-L. Yang, J.-T. Teng, S. Papachristos, Partial backlogginginventory lot-size models for deteriorating items with fluctuating demandunder inflation, European Journal of Operational Research 191 (1) (2008)127–141.

[44] E.P. Chew, C. Lee, R. Liu, Joint inventory allocation and pricing decisions forperishable products, International Journal of Production Economics 120 (1)(2009) 139–150.

[45] L.Y. Chu, V.N. Hsu, Z.-J.M. Shen, An economic lot-sizing problem withperishable inventory and economies of scale costs: approximationsolutions and worst case analysis, Naval Research Logistics 52 (6) (2005)536–548.

[46] P. Chu, P.S. Chen, A note on inventory replenishment policies for deterioratingitems in an exponentially declining market, Computers & OperationsResearch 29 (13) (2002) 1827–1842.

[47] Y.H. Chun, Optimal pricing and ordering policies for perishable commodities,European Journal of Operational Research 144 (1) (2003) 68–82.

[48] C.J. Chung, H.-M. Wee, An integrated production-inventory deterioratingmodel for pricing policy considering imperfect production, inspectionplanning and warranty-period- and stock-level-dependant demand,International Journal of Systems Science 39 (8) (2008) 823–837.

[49] C.-J. Chung, H.-M. Wee, Short life-cycle deteriorating productremanufacturing in a green supply chain inventory control system,International Journal of Production Economics 129 (1) (2011) 195–203.

[50] K.-J. Chung, A complete proof on the solution procedure for non-instantaneous deteriorating items with permissible delay in payment,Computers & Industrial Engineering 56 (1) (2009) 267–273.

M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284 281

[51] K.-J. Chung, Using the convexities of total annual relevant costs to determinethe optimal cycle times of inventory models for deteriorating items withpermissible delay in payments, Computers & Industrial Engineering 58 (4)(2010) 801–808.

[52] K.-J. Chung, T.S. Huang, The optimal retailer’s ordering policies fordeteriorating items with limited storage capacity under trade creditfinancing, International Journal of Production Economics 106 (1) (2007)127–145.

[53] K.-J. Chung, J.-J. Liao, Lot-sizing decisions under trade credit depending on theordering quantity, Computers & Operations Research 31 (6) (2004) 909–928.

[54] K.-J. Chung, J.-J. Liao, The optimal ordering policy in a DCF analysis fordeteriorating items when trade credit depends on the order quantity,International Journal of Production Economics 100 (1) (2006) 116–130.

[55] K.-J. Chung, C.-N. Lin, Optimal inventory replenishment models fordeteriorating items taking account of time discounting, Computers &Operations Research 28 (1) (2001) 67–83.

[56] K.-J. Chung, S.-F. Tsai, Inventory systems for deteriorating items withshortages and a linear trend in demand-taking account of time value,Computers & Operations Research 28 (9) (2001) 915–934.

[57] K.-J. Chung, S.-L. Chang, W.-D. Yang, The optimal cycle time for exponentiallydeteriorating products under trade credit financing, The EngineeringEconomist 46 (3) (2001) 232–242.

[58] W.L. Cooper, Pathwise properties and performance bounds for a perishableinventory system, Operations Research 49 (3) (2001) 455–466.

[59] K. Das, T.K. Roy, M. Maiti, Buyer–seller fuzzy inventory model for adeteriorating item with discount, International Journal of Systems Science35 (8) (2004) 457–466.

[60] S. Dasu, C. Tong, Dynamic pricing when consumers are strategic: analysis ofposted and contingent pricing schemes, European Journal of OperationalResearch 204 (3) (2010) 662–671.

[61] L.N. De, A. Goswami, Probabilistic EOQ model for deteriorating items undertrade credit financing, International Journal of Systems Science 40 (4) (2009)335–346.

[62] S.K. De, A. Goswami, An EOQ model with fuzzy inflation rate and fuzzydeterioration rate when a delay in payment is permissible, InternationalJournal of Systems Science 37 (5) (2006) 323–335.

[63] P.S. Deng, Improved inventory models with ramp type demand and weibulldeterioration, International Journal of Information and Management Sciences16 (4) (2005) 79–86.

[64] P.S. Deng, R.H.-J. Lin, P. Chu, A note on the inventory models for deterioratingitems with ramp type demand rate, European Journal of Operational Research178 (1) (2007) 112–120.

[65] J. Dey, S. Mondal, M. Maiti, Two storage inventory problem with dynamicdemand and interval valued lead-time over finite time horizon underinflation and time-value of money, European Journal of OperationalResearch 185 (1) (2008) 170–194.

[66] S.M. Disney, R.D.H. Warburton, On the Lambert W function: economic orderquantity applications and pedagogical considerations, International Journal ofProduction Economics, in press, doi:10.1016/j.ijpe.2011.02.027.

[67] C.-Y. Dye, A note on an EOQ model for items with weibull distributeddeterioration, shortages and power demand pattern, International Journal ofInformation and Management Sciences 15 (2) (2004) 81–84.

[68] C.-Y. Dye, Joint pricing and ordering policy for a deteriorating inventory withpartial backlogging, Omega 35 (2) (2007) 184–189.

[69] C.-Y. Dye, H.-J. Chang, A replenishment policy for deteriorating items withlinear trend demand and shortages when payment periods are offered,International Journal of Information and Management Sciences 14 (2) (2003)31–45.

[70] C.-Y. Dye, T.-P. Hsieh, An optimal replenishment policy for deterioratingitems with effective investment in preservation technology. European Journalof Operational Research, in press, doi:10.1016/j.ejor.2011.10.016.

[71] C.-Y. Dye, L.-Y. Ouyang, An EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging, EuropeanJournal of Operational Research 163 (3) (2005) 776–783.

[72] C.-Y. Dye, L.-Y. Ouyang, A particle swarm optimization for solving jointpricing and lot-sizing problem with fluctuating demand and trade creditfinancing, Computers & Industrial Engineering 60 (1) (2011) 127–137.

[73] C.-Y. Dye, H.-J. Chang, J.-T. Teng, A deteriorating inventory model with time-varying demand and shortage-dependent partial backlogging, EuropeanJournal of Operational Research 172 (2) (2006) 417–429.

[74] C.-Y. Dye, H.-J. Chang, C.-H. Wu, Purchase-inventory decision models fordeteriorating items with a temporary sale price, International Journal ofInformation and Management Sciences 18 (1) (2007) 17–35.

[75] C.-Y. Dye, T.-P. Hsieh, L.-Y. Ouyang, Determining optimal selling price and lotsize with a varying rate of deterioration and exponential partial backlogging,European Journal of Operational Research 181 (2) (2007) 668–678.

[76] C.-Y. Dye, L.-Y. Ouyang, T.-P. Hsieh, Deterministic inventory model fordeteriorating items with capacity constraint and time-proportionalbacklogging rate, European Journal of Operational Research 178 (3) (2007)789–807.

[77] C.-Y. Dye, L.-Y. Ouyang, T.-P. Hsieh, Inventory and pricing strategies fordeteriorating items with shortages: a discounted cash flow approach,Computers & Industrial Engineering 52 (1) (2007) 29–40.

[78] M.E. Ferguson, O. Koenigsberg, How should a firm manage deterioratinginventory?, Production and Operations Management 16 (3) (2009) 306–321

[79] M. Ferguson, M.E. Ketzenberg, Information sharing to improve retail productfreshness of perishables, Production and Operations Management 15 (1)(2006) 57–73.

[80] M. Ferguson, V. Jayaraman, G.C. Souza, Note: an application of the EOQ modelwith nonlinear holding cost to inventory management of perishables,European Journal of Operational Research 180 (1) (2007) 485–490.

[81] K.C. Frank, H.-S. Ahn, R.Q. Zhang, Inventory policies for a make-to-ordersystem with a perishable component and fixed ordering cost, Naval ResearchLogistics 56 (2) (2009) 127–141.

[82] P.N. Ghare, G.F. Schrader, A model for exponentially decaying inventories, TheJournal of Industrial Engineering (1963) 238–243.

[83] S.K. Ghosh, K.S. Chaudhuri, An EOQ model with a quadratic demand, time-proportional deterioration and shortages in all cycles, International Journal ofSystems Science 37 (10) (2006) 663–672.

[84] B.C. Giri, A.K. Jalan, K.S. Chaudhuri, Economic order quantity model withWeibull deterioration distribution, shortage and ramp-type demand,International Journal of Systems Science 34 (4) (2003) 237–243.

[85] S.K. Goyal, B.C. Giri, Recent trends in modeling of deteriorating inventory,European Journal of Operational Research 134 (1) (2001) 1–16.

[86] S.K. Goyal, B.C. Giri, The production–inventory problem of a product withtime varying demand, production and deterioration rates, European Journalof Operational Research 147 (3) (2003) 549–557.

[87] Y. Gupta, P.S. Sundararaghavan, M.U. Ahmed, Ordering policies for items withseasonal demand, International Journal of Physical Distribution & LogisticsManagement 33 (6) (2003) 500–518.

[88] Ü. Gürler, B. Yüksel Özkaya, A note on continuous review perishableinventory systems: models and heuristics, IIE Transactions 35 (3) (2003)321–323.

[89] Ü. Gürler, B.Y. Özkaya, Analysis of the (s, S) policy for perishables with arandom shelf life, IIE Transactions 40 (8) (2008) 759–781.

[90] R. Haijema, A new class of stock-level dependent ordering policies forperishables with a short maximum shelf life. International Journal ofProduction Economics, in press, doi:10.1016/j.ijpe.2011.05.021.

[91] F.W. Harris, How many parts to make at once, Factory, The Magazine ofManagement 10 (2) (1913) 135–136, 15.

[92] Y. He, S.-Y. Wang, K.K. Lai, An optimal production-inventory model fordeteriorating items with multiple-market demand, European Journal ofOperational Research 203 (3) (2010) 593–600.

[93] K.-L. Hou, An inventory model for deteriorating items with stock-dependentconsumption rate and shortages under inflation and time discounting,European Journal of Operational Research 168 (2) (2006) 463–474.

[94] K.-L. Hou, L.-C. Lin, An EOQ model for deteriorating items with price- andstock-dependent selling rates under inflation and time value of money,International Journal of Systems Science 37 (15) (2006) 1131–1139.

[95] T.-P. Hsieh, C.-Y. Dye, Optimal replenishment policy for perishable items withstock-dependent selling rate and capacity constraint, Computers & IndustrialEngineering 59 (2) (2010) 251–258.

[96] T.-P. Hsieh, H.-J. Chang, C.-Y. Dye, M.-W. Weng, Optimal lot size under tradecredit financing when demand and deterioration are fluctuating with time,International Journal of Information and Management Sciences 20 (2) (2009)191–204.

[97] T.-P. Hsieh, C.-Y. Dye, L.-Y. Ouyang, Determining optimal lot size for a two-warehouse system with deterioration and shortages using net present value,European Journal of Operational Research 191 (1) (2008) 182–192.

[98] P.H. Hsu, H.-M. Wee, H.M. Teng, Preservation technology investment fordeteriorating inventory, International Journal of Production Economics 124(2) (2010) 388–394.

[99] P.-H. Hsu, H.-M. Wee, Discounting decision for enterprises with high fixedcost and low variable cost, International Transactions in Operational Research13 (2) (2006) 111–124.

[100] P.-H. Hsu, H.-M. Wee, H. Teng, Optimal ordering decision for deterioratingitems with expiration date and uncertain lead time, Computers & IndustrialEngineering 52 (4) (2007) 448–458.

[101] K. Huang, J. Liao, A simple method to locate the optimal solution forexponentially deteriorating items under trade credit financing, Computers &Mathematics with Applications 56 (4) (2008) 965–977.

[102] K.-N. Huang, J.-J. Liao, Bounds on the optimum order quantity of the EOQmodel with deteriorating items under supplier credit linked to orderquantity, International Journal of Information and Management Sciences 17(3) (2006) 105–116.

[103] K.-C. Hung, An inventory model with generalized type demand, deteriorationand backorder rates, European Journal of Operational Research 208 (3) (2011)239–242.

[104] H. Hwang, A stochastic set-covering location model for both ameliorating anddeteriorating items, Computers & Industrial Engineering 46 (2) (2004) 313–319.

[105] C.K. Jaggi, K.K. Aggarwal, S.K. Goel, Optimal order policy for deterioratingitems with inflation induced demand, International Journal of ProductionEconomics 103 (2) (2006) 707–714.

[106] C.K. Jaggi, A. Khanna, P. Verma, Two-warehouse partial backlogging inventorymodel for deteriorating items with linear trend in demand under inflationaryconditions, International Journal of Systems Science 42 (7) (2011) 1185–1196.

[107] J. Jia, Q. Hu, Dynamic ordering and pricing for a perishable goods supplychain, Computers & Industrial Engineering 60 (2) (2011) 302–309.

282 M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284

[108] S.-T. Jung, J.S.-J. Lin, J.P.C. Chuang, A note on an EOQ model for items withweibull distributed deterioration, shortages and power demand pattern,International Journal of Information and Management Sciences 19 (4) (2008)667–672.

[109] S. Kalpakam, S. Shanthi, A perishable inventory system with modified (S-1,S)policy and arbitrary processing times, Computers & Operations Research 28(5) (2001) 453–471.

[110] K. Kanchanasuntorn, A. Techanitisawad, An approximate periodic model forfixed-life perishable products in a two-echelon inventory–distributionsystem, International Journal of Production Economics 100 (1) (2006) 101–115.

[111] S. Kar, A.K. Bhunia, M. Maiti, Inventory of multi-deteriorating items sold fromtwo shops under single management with constraints on space andinvestment, Computers & Operations Research 28 (12) (2001) 1203–1221.

[112] I.Z. Karaesmen, A. Scheller–Wolf, B. Deniz, K.G. Kempf, P. Keskinocak, R.Uzsoy, Planning production and inventories in the extended enterprise, in:K.G. Kempf, P. Keskinocak, R. Uzsoy (Eds.), Planning Production andInventories in the Extended Enterprise, A State of the Art Handbook,Volume 1, vol. 151, 2011, pp. 393–436.

[113] H. Katagiri, H. Ishii, Fuzzy inventory problems for perishable commodities,European Journal of Operational Research 138 (3) (2002) 545–553.

[114] M. Ketzenberg, M.E. Ferguson, Managing slow-moving perishables in thegrocery industry, Production and Operations Management 17 (5) (2008) 513–521.

[115] S. Khanra, K.S. Chaudhuri, A note on an order-level inventory model for adeteriorating item with time-dependent quadratic demand, Computers &Operations Research 30 (12) (2003) 1901–1916.

[116] I. Konstantaras, K. Skouri, A note on a production-inventory model understock-dependent demand, Weibull distribution deterioration, and shortage,International Transactions in Operational Research 18 (4) (2011) 527–531.

[117] C. Kouki, E. Sahin, Z. Jemaı̈, Y. Dallery, Assessing the impact of perishabilityand the use of time temperature technologies on inventory management.International Journal of Production Economics, in press, doi:10.1016/j.ijpe.2010.09.032.

[118] H. Krishnan, R.A. Winter, Inventory dynamics and supply chain coordination,Management Science 56 (1) (2010) 141–147.

[119] S. Kumar, P.K. Kundu, A. Goswami, An economic production quantityinventory model involving fuzzy demand rate and fuzzy deterioration rate,Journal of Applied Mathematics and Computing 12 (1–2) (2003) 251–260.

[120] C.C. Lee, Two-warehouse inventory model with deterioration under FIFOdispatching policy, European Journal of Operational Research 174 (2) (2006)861–873.

[121] C. Lee, S. Hsu, A two-warehouse production model for deteriorating inventoryitems with time-dependent demands, European Journal of OperationalResearch 194 (3) (2009) 700–710.

[122] W.-C. Lee, J.-W. Wu, An EOQ model for items with Weibull distributeddeterioration, shortages and power demand pattern, International Journal ofInformation and Management Sciences 13 (2) (2002) 19–34.

[123] T. Levina, Y. Levin, J. McGill, M. Nediak, V. Vovk, Weak aggregating algorithmfor the distribution-free perishable inventory problem, Operations ResearchLetters 38 (6) (2010) 516–521.

[124] J. Li, T.C. Edwin Cheng, S.-Y. Wang, Analysis of postponement strategy forperishable items by EOQ-based models, International Journal of ProductionEconomics 107 (1) (2007) 31–38.

[125] R. Li, H. Lan, J.R. Mawhinney, A review on deteriorating inventory study,Journal of Service Science and Management 03 (01) (2010) 117–129.

[126] Y. Li, B. Cheang, A. Lim, Grocery Perishables Management, Production andOperations Management, 2011.

[127] Y. Li, A. Lim, B. Rodrigues, Pricing and inventory control for a perishableproduct, Manufacturing & Service Operations Management 11 (3) (2008)538–542.

[128] Z. Lian, L. Liu, Continuous review perishable inventory systems: models andheuristics, IIE Transactions 33 (9) (2001) 809–822.

[129] Z. Lian, L. Liu, M.F. Neuts, A discrete-time model for common lifetimeinventory systems, Mathematics of Operations Research 30 (3) (2005) 718–732.

[130] Z. Lian, X. Liu, N. Zhao, A perishable inventory model with Markovian renewaldemands, International Journal of Production Economics 121 (1) (2009) 176–182.

[131] J. Liao, On an EPQ model for deteriorating items under permissible delay inpayments, Applied Mathematical Modelling 31 (3) (2007) 393–403.

[132] J. Liao, A note on an EOQ model for deteriorating items under supplier creditlinked to ordering quantity, Applied Mathematical Modelling 31 (8) (2007)1690–1699.

[133] J.-J. Liao, An EOQ model with noninstantaneous receipt and exponentiallydeteriorating items under two-level trade credit, International Journal ofProduction Economics 113 (2) (2008) 852–861.

[134] J.-J. Liao, K.-N. Huang, Deterministic inventory model for deteriorating itemswith trade credit financing and capacity constraints, Computers & IndustrialEngineering 59 (4) (2010) 611–618.

[135] C. Lin, Y. Lin, A cooperative inventory policy with deteriorating items for atwo-echelon model, European Journal of Operational Research 178 (1) (2007)92–111.

[136] S.-W. Lin, Inventory models with managerial policy independent of demand,European Journal of Operational Research 211 (3) (2011) 520–524.

[137] Y.S. Lin, J.C.P. Yu, K.-J. Wang, An efficient replenishment model ofdeteriorating items for a supplier–buyer partnership in hi-tech industry,Production Planning & Control 20 (5) (2009) 431–444.

[138] Y.-H. Lin, C. Lin, B. Lin, On conflict and cooperation in a two-echeloninventory model for deteriorating items, Computers & Industrial Engineering59 (4) (2010) 703–711.

[139] E. Lodree Jr., B.M. Uzochukwu, Production planning for a deteriorating itemwith stochastic demand and consumer choice, International Journal ofProduction Economics 116 (2) (2008) 219–232.

[140] N.K. Mahapatra, M. Maiti, Decision process for multiobjective, multi-itemproduction-inventory system via interactive fuzzy satisficing technique,Computers & Mathematics with Applications 49 (5–6) (2005) 805–821.

[141] G.C. Mahata, A. Goswami, An EOQ model for deteriorating items under tradecredit financing in the fuzzy sense, Production Planning & Control 18 (8)(2007) 681–692.

[142] R. Maihami, I. Nakhai Kamalabadi, Joint pricing and inventory control fornon-instantaneous deteriorating items with partial backlogging and time andprice dependent demand. International Journal of Production Economics, inpress, doi:10.1016/j.ijpe.2011.09.020.

[143] A. Maity, K. Maity, S. Mondal, M. Maiti, A Chebyshev approximation forsolving the optimal production inventory problem of deteriorating multi-item, Mathematical and Computer Modelling 45 (1–2) (2007) 149–161.

[144] K. Maity, M. Maiti, Numerical approach of multi-objective optimal controlproblem in imprecise environment, Fuzzy Optimization and Decision Making4 (4) (2005) 313–330.

[145] K. Maity, M. Maiti, Optimal inventory policies for deterioratingcomplementary and substitute items, International Journal of SystemsScience 40 (3) (2009) 267–276.

[146] N.K. Mandal, T.K. Roy, M. Maiti, Inventory model of deteriorated items with aconstraint: a geometric programming approach, European Journal ofOperational Research 173 (1) (2006) 199–210.

[147] S.K. Manna, K.S. Chaudhuri, An economic order quantity model fordeteriorating items with time-dependent deterioration rate, demand rate,unit production cost and shortages, International Journal of Systems Science32 (8) (2001) 1003–1009.

[148] S.K. Manna, K.S. Chaudhuri, An EOQ model with ramp type demand rate, timedependent deterioration rate, unit production cost and shortages, EuropeanJournal of Operational Research 171 (2) (2006) 557–566.

[149] P. Manuel, A. Shophia Lawrence, G. Arivaraginan, A stochastic perishableinventory system with random supply quantity, International Journal ofInformation and Management Sciences 18 (4) (2007) 317–334.

[150] P. Manuel, B. Sivakumar, G. Arivaraginan, A perishable inventory system withservice facilities and retrial customers, Computers & Industrial Engineering54 (3) (2008) 484–501.

[151] J. Min, Y.-W. Zhou, A perishable inventory model under stock-dependentselling rate and shortage-dependent partial backlogging with capacityconstraint, International Journal of Systems Science 40 (1) (2009) 33–44.

[152] A. Mirzazadeh, M.M. Seyyed Esfahani, S.M.T. Fatemi Ghomi, An inventorymodel under uncertain inflationary conditions, finite production rate andinflation-dependent demand rate for deteriorating items with shortages,International Journal of Systems Science 40 (1) (2009) 21–31.

[153] I. Moon, Economic order quantity models for ameliorating/deterioratingitems under inflation and time discounting, European Journal of OperationalResearch 162 (3) (2005) 773–785.

[154] S. Mukhopadhyay, R.N. Mukherjee, K.S. Chaudhuri, Joint pricing and orderingpolicy for a deteriorating inventory, Computers & Industrial Engineering 47(4) (2004) 339–349.

[155] S. Mukhopadhyay, R.N. Mukherjee, K.S. Chaudhuri, An EOQ model with two-parameter Weibull distribution deterioration and price-dependent demand,International Journal of Mathematical Education in Science and Technology36 (1) (2005) 25–33.

[156] A. Musa, B. Sani, Inventory ordering policies of delayed deteriorating itemsunder permissible delay in payments. International Journal of ProductionEconomics, in press, doi:10.1016/j.ijpe.2011.09.013.

[157] S. Nahmias, Perishable inventory theory: a review, Operations Research 30(4) (1982) 680–708.

[158] S. Nahmias, in: F.S. Hillier (Ed.), Perishable Inventory Systems, vol. 160,Springer US, Boston, MA, 2011, p. 80.

[159] B. Niu, J. Xie, A note on two-warehouse inventory model with deteriorationunder FIFO dispatch policy, European Journal of Operational Research 190 (2)(2008) 571–577.

[160] F. Olsson, P. Tydesjö, Inventory problems with perishable items: fixedlifetimes and backlogging, European Journal of Operational Research 202 (1)(2010) 131–137.

[161] L.-Y. Ouyang, C.-T. Chang, J.-T. Teng, An EOQ model for deteriorating itemsunder trade credits, Journal of the Operational Research Society 56 (6) (2005)719–726.

[162] L.-Y. Ouyang, T.-P. Hsieh, C.-Y. Dye, H.-C. Chang, An inventory model fordeteriorating items with stock-dependent demand under the conditions ofinflation and time-value of money, The Engineering Economist 48 (1) (2003)52–68.

[163] L.-Y. Ouyang, J.-T. Teng, S.K. Goyal, C.-T. Yang, An economic order quantitymodel for deteriorating items with partially permissible delay in paymentslinked to order quantity, European Journal of Operational Research 194 (2)(2009) 418–431.

M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284 283

[164] L.-Y. Ouyang, K. Wu, C.-T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments,Computers & Industrial Engineering 51 (4) (2006) 637–651.

[165] J. Pahl, S. Voss, D.L. Woodruff, Production planning with deteriorationconstraints: a survey, in: 19th International Conference on ProductionResearch, 2007, p. 6.

[166] A.K. Pal, A.K. Bhunia, R.N. Mukherjee, Optimal lot size model for deterioratingitems with demand rate dependent on displayed stock level (DSL) and partialbackordering, European Journal of Operational Research 175 (2) (2006) 977–991.

[167] S. Panda, S. Senapati, M. Basu, Optimal replenishment policy for perishableseasonal products in a season with ramp-type time dependent demand,Computers & Industrial Engineering 54 (2) (2008) 301–314.

[168] Z. Pang, Optimal dynamic pricing and inventory control with stockdeterioration and partial backordering, Operations Research Letters 39 (5)(2011) 375–379.

[169] S. Papachristos, K. Skouri, An inventory model with deteriorating items,quantity discount, pricing and time-dependent partial backlogging,International Journal of Production Economics 83 (3) (2003) 247–256.

[170] W.P. Pierskalla, Supply chain management of blood banks, in: M.L. Brandeau,F. Sainfort, W.P. Pierskalla (Eds.), Operations Research and Health Care, vol.70, 2005, pp. 103–145.

[171] G.P. Prastacos, Blood inventory management: an overview of theory andpractice, Management Science 30 (7) (1984) 777–800.

[172] F. Raafat, Survey of literature on continuously deteriorating inventorymodels, Journal of the Operational Research Society 42 (1) (1991) 27–37.

[173] R. Ramanathan, Stocking and discounting decisions for perishablecommodities using expected profit approach, International Journal of Retail& Distribution Management 34 (2) (2006) 172–184.

[174] H. Rau, M.Y. Wu, H.-M. Wee, Integrated inventory model for deterioratingitems under a multi-echelon supply chain environment, International Journalof Production Economics 86 (2) (2003) 155–168.

[175] H. Rau, M.-Y. Wu, H.-M. Wee, Deteriorating item inventory model withshortage due to supplier in an integrated supply chain, International Journalof Systems Science 35 (5) (2004) 293–303.

[176] M. Rong, N.K. Mahapatra, M. Maiti, A two warehouse inventory model for adeteriorating item with partially/fully backlogged shortage and fuzzy leadtime, European Journal of Operational Research 189 (1) (2008) 59–75.

[177] A. Roy, G.P. Samanta, Inventory model with two rates of production fordeteriorating items with permissible delay in payments, International Journalof Systems Science 42 (8) (2011) 1375–1386.

[178] A. Roy, S. Kar, M. Maiti, A deteriorating multi-item inventory model withfuzzy costs and resources based on two different defuzzification techniques,Applied Mathematical Modelling 32 (2) (2008) 208–223.

[179] A. Roy, S. Kar, M. Maiti, A volume flexible production-policy for randomlydeteriorating item with trended demand and shortages, International Journalof Production Economics 128 (1) (2010) 188–199.

[180] A. Roy, M. Maiti, S. Kar, An inventory model for a deteriorating item withdisplayed stock dependent demand under fuzzy inflation and timediscounting over a random planning horizon, Applied MathematicalModelling 33 (2) (2009) 744–759.

[181] T. Roy, K.S. Chaudhuri, A production-inventory model under stock-dependentdemand, Weibull distribution deterioration and shortage, InternationalTransactions in Operational Research 16 (3) (2009) 325–346.

[182] S. Sana, K.S. Chaudhuri, B. Mahavidyalaya, On a volume flexible productionpolicy for a deteriorating item with time-dependent demand and shortages,Advanced Modeling & Optimization 6 (1) (2004) 57–74.

[183] S. Sana, S.K. Goyal, K.S. Chaudhuri, A production-inventory model for adeteriorating item with trended demand and shortages, European Journal ofOperational Research 157 (2) (2004) 357–371.

[184] B. Sezen, Expected profit approach used in discount pricing decisions forperishable products, International Journal of Retail & DistributionManagement 32 (4) (2004) 223–229.

[185] N.H. Shah, Inventory model for deteriorating items and time value of moneyfor a finite time horizon under the permissible delay in payments,International Journal of Systems Science 37 (1) (2006) 9–15.

[186] D. Shen, K.K. Lai, S.C.H. Leung, L. Liang, Modelling and analysis of inventoryreplenishment for perishable agricultural products with buyer–sellercollaboration, International Journal of Systems Science 42 (7) (2011) 1207–1217.

[187] B. Sivakumar, A perishable inventory system with retrial demands and afinite population, Journal of Computational and Applied Mathematics 224 (1)(2009) 29–38.

[188] B. Sivakumar, N. Anbazhaqan, G. Arivaraginan, Two commodity continuousreview perishable inventory system, International Journal of Information andManagement Sciences 17 (3) (2006) 47–64.

[189] K. Skouri, I. Konstantaras, Order level inventory models for deterioratingseasonable/fashionable products with time dependent demand andshortages, Mathematical Problems in Engineering 2009 (2009) 1–25.

[190] K. Skouri, S. Papachristos, A continuous review inventory model, withdeteriorating items, time-varying demand, linear replenishment cost,partially time-varying backlogging, Applied Mathematical Modelling 26 (5)(2002) 603–617.

[191] K. Skouri, S. Papachristos, Note on ‘‘deterministic inventory lot-size modelsunder inflation with shortages and deterioration for fluctuating demand’’ byYang et al, Naval Research Logistics 49 (5) (2002) 527–529.

[192] K. Skouri, S. Papachristos, Four inventory models for deteriorating items withtime varying demand and partial backlogging: a cost comparison, OptimalControl Applications and Methods 24 (6) (2003) 315–330.

[193] K. Skouri, S. Papachristos, Optimal stopping and restarting productiontimes for an EOQ model with deteriorating items and time-dependentpartial backlogging, International Journal of Production Economics (2003)525–531.

[194] K. Skouri, I. Konstantaras, S.K. Manna, K.S. Chaudhuri, Inventory models withramp type demand rate, time dependent deterioration rate, unit productioncost and shortages, Annals of Operations Research 191 (1) (2011) 73–95.

[195] K. Skouri, I. Konstantaras, S. Papachristos, I. Ganas, Inventory models withramp type demand rate, partial backlogging and Weibull deterioration rate,European Journal of Operational Research 192 (1) (2009) 79–92.

[196] K. Skouri, I. Konstantaras, S. Papachristos, J.-T. Teng, Supply chain models fordeteriorating products with ramp type demand rate under permissible delayin payments, Expert Systems with Applications 38 (12) (2011) 14861–14869.

[197] X. Song, X. Cai, On optimal payment time for a retailer under permitted delayof payment by the wholesaler, International Journal of Production Economics103 (1) (2006) 246–251.

[198] L. Tadj, M. Bounkhel, Y. Benhadid, Optimal control of a production inventorysystem with deteriorating items, International Journal of Systems Science 37(15) (2006) 1111–1121.

[199] E. Tekin, Ü. Gürler, E. Berk, Age-based vs. stock level control policies for aperishable inventory system, European Journal of Operational Research 134(2) (2001) 309–329.

[200] J. Teng, H.-L. Yang, Deterministic economic order quantity models withpartial backlogging when demand and cost are fluctuating with time, Journalof the Operational Research Society 55 (5) (2004) 9.

[201] J.-T. Teng, C.-T. Chang, Economic production quantity models fordeteriorating items with price- and stock-dependent demand, Computers &Operations Research 32 (2) (2005) 297–308.

[202] J.-T. Teng, C.-T. Chang, S.K. Goyal, Optimal pricing and ordering policy underpermissible delay in payments, International Journal of ProductionEconomics 97 (2) (2005) 121–129.

[203] J.-T. Teng, H.-J. Chang, C.-Y. Dye, C.-H. Hung, An optimal replenishment policyfor deteriorating items with time-varying demand and partial backlogging,Operations Research Letters 30 (6) (2002) 387–393.

[204] J.-T. Teng, I.-P. Krommyda, K. Skouri, K.-R. Lou, A comprehensive extension ofoptimal ordering policy for stock-dependent demand under progressivepayment scheme, European Journal of Operational Research 215 (1) (2011)97–104.

[205] J.-T. Teng, L.-Y. Ouyang, L.-H. Chen, A comparison between two pricing andlot-sizing models with partial backlogging and deteriorated items,International Journal of Production Economics 105 (1) (2007) 190–203.

[206] J.-T. Teng, L.-Y. Ouyang, M.-C. Cheng, An EOQ model for deteriorating itemswith power-form stock-dependent demand, International Journal ofInformation and Management Sciences 16 (1) (2005) 1–16.

[207] P.-C. Tsai, J.-Y. Huang, Two-stage replenishment policies for deterioratingitems at Taiwanese convenience stores, Computers & Operations Research 39(2) (2012) 328–338.

[208] Y.-C. Tsao, G.-J. Sheen, Joint pricing and replenishment decisions fordeteriorating items with lot-size and time-dependent purchasing costunder credit period, International Journal of Systems Science 38 (7) (2007)549–561.

[209] Y.-C. Tsao, G.-J. Sheen, Dynamic pricing, promotion and replenishmentpolicies for a deteriorating item under permissible delay in payments,Computers & Operations Research 35 (11) (2008) 3562–3580.

[210] K.-J. Wang, Y.S. Lin, J.C.P. Yu, Optimizing inventory policy for products withtime-sensitive deteriorating rates in a multi-echelon supply chain,International Journal of Production Economics 130 (1) (2011) 66–76.

[211] S.-P. Wang, An inventory replenishment policy for deteriorating items withshortages and partial backlogging, Computers & Operations Research 29 (14)(2002) 2043–2051.

[212] T.-Y. Wang, L.-H. Chen, A production lot size inventory model fordeteriorating items with time-varying demand, International Journal ofSystems Science 32 (6) (2001) 745–751.

[213] R.D.H. Warburton, EOQ extensions exploiting the Lambert W function,European Journal of Industrial Engineering 3 (1) (2009) 45.

[214] H.-M. Wee, S.-T. Law, Replenishment and pricing policy for deterioratingitems taking into account the time-value of money, International Journal ofProduction Economics 71 (1–3) (2001) 213–220.

[215] H.-M. Wee, M.-C. Lee, J.C.P. Yu, C. Edward Wang, Optimal replenishmentpolicy for a deteriorating green product: life cycle costing analysis,International Journal of Production Economics 133 (2) (2011) 608–611.

[216] H.-M. Wee, C.-C. Lo, P.-H. Hsu, A multi-objective joint replenishmentinventory model of deteriorated items in a fuzzy environment, EuropeanJournal of Operational Research 197 (2) (2009) 620–631.

[217] H.-M. Wee, S.-T.T. Lo, J.C.P. Yu, H.C. Chen, An inventory model forameliorating and deteriorating items taking account of time value ofmoney and finite planning horizon, International Journal of SystemsScience 39 (8) (2008) 801–807.

[218] T.M. Whitin, The Theory of Inventory Management, Princeton UniversityPress, Princeton, NJ, USA, 1953.

[219] K.-S. Wu, An EOQ inventory model for items with Weibull distributiondeterioration, ramp type demand rate and partial backlogging, ProductionPlanning & Control 12 (8) (2001) 787–793.

284 M. Bakker et al. / European Journal of Operational Research 221 (2012) 275–284

[220] K.-S. Wu, Note on an EQQ model for deteriorating items with exponentialtime-varying demand and partial backlogging, International Journal ofInformation and Management Sciences 12 (4) (2001) 55–60.

[221] K.-S. Wu, EOQ inventory model for items with Weibull distributiondeterioration, time-varying demand and partial backlogging, InternationalJournal of Systems Science 33 (5) (2002) 323–329.

[222] K.-S. Wu, Discounted-cash-flow EOQ model for a deteriorating item,International Journal of Information and Management Sciences 13 (4)(2002) 55–67.

[223] K.-S. Wu, L.-Y. Ouyang, C.-T. Yang, An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partialbacklogging, International Journal of Production Economics 101 (2) (2006)369–384.

[224] K.-S. Wu, L.-Y. Ouyang, C.-T. Yang, Retailer’s optimal ordering policy fordeteriorating items with ramp-type demand under stock-dependentconsumption rate, International Journal of Information and ManagementSciences 19 (2) (2008) 245–262.

[225] K.-S. Wu, L.-Y. Ouyang, C.-T. Yang, Coordinating replenishment and pricingpolicies for non-instantaneous deteriorating items with price-sensitivedemand, International Journal of Systems Science 40 (12) (2009) 1273–1281.

[226] T. Xiao, J. Jin, G. Chen, J. Shi, M. Xie, Ordering, wholesale pricing and lead-timedecisions in a three-stage supply chain under demand uncertainty,Computers & Industrial Engineering 59 (4) (2010) 840–852.

[227] Y. Xu, B.R. Sarker, Models for a family of products with shelf life, andproduction and shortage costs in emerging markets, Computers & OperationsResearch 30 (6) (2003) 925–938.

[228] V.S.S. Yadavalli, C. van Schoor, J.J. Strasheim, S. Udayabaskaran, A singleproduct perishing inventory model with demand interaction, Orion 20 (2)(2004) 109–124.

[229] V.S.S. Yadavalli, B. Sivakumar, G. Arivarignan, O. Adetunji, A multi-serverperishable inventory system with negative customer, Computers & IndustrialEngineering 61 (2) (2011) 254–273.

[230] C. Yan, A. Banerjee, L. Yang, An integrated production–distribution model fora deteriorating inventory item, International Journal of Production Economics133 (1) (2011) 228–232.

[231] G.K. Yang, R. Lin, J. Lin, K.-C. Hung, P. Chu, W. Chouhuang, Note on inventorymodels with Weibull distribution deterioration, Production Planning &Control 22 (4) (2011) 437–444.

[232] H.-L. Yang, Two-warehouse inventory models for deteriorating items withshortages under inflation, European Journal of Operational Research 157 (2)(2004) 344–356.

[233] H.-L. Yang, A comparison among various partial backlogging inventory lot-size models for deteriorating items on the basis of maximum profit,International Journal of Production Economics 96 (1) (2005) 119–128.

[234] H.-L. Yang, Two-warehouse partial backlogging inventory models fordeteriorating items under inflation, International Journal of ProductionEconomics 103 (1) (2006) 362–370.

[235] H.-L. Yang, A partial backlogging production-inventory lot-size model fordeteriorating items with time-varying production and demand rate over afinite time horizon, International Journal of Systems Science 42 (8) (2011)1397–1407.

[236] H.-L. Yang, J.-T. Teng, M.-S. Chern, Deterministic inventory lot-size modelsunder inflation with shortages and deterioration for fluctuating demand,Naval Research Logistics 48 (2) (2001) 144–158.

[237] H.-L. Yang, J.-T. Teng, M.-S. Chern, An inventory model under inflation fordeteriorating items with stock-dependent consumption rate and partialbacklogging shortages, International Journal of Production Economics 123 (1)(2010) 8–19.

[238] P.-C. Yang, H.-M. Wee, A single-vendor and multiple-buyers production–inventory policy for a deteriorating item, European Journal of OperationalResearch 143 (3) (2002) 570–581.

[239] P.-C. Yang, H.-M. Wee, An integrated multi-lot-size production inventorymodel for deteriorating item, Computers & Operations Research 30 (5) (2003)671–682.

[240] S. Zanoni, L. Zavanella, Single-vendor single-buyer with integrated transport-inventory system: models and heuristics in the case of perishable goods,Computers & Industrial Engineering 52 (1) (2007) 107–123.

[241] Y.-W. Zhou, S.-L. Yang, An optimal replenishment policy for items withinventory-level-dependent demand and fixed lifetime under the LIFO policy,Journal of the Operational Research Society 54 (6) (2003) 585–593.