Research ArticleIntegrating Dynamic Pricing and Inventory Control forFresh Agriproduct under Multinomial Logit Choice

Xiong-zhi Wang andWenliang Zhou

College of Economics amp Management South China Agriculture University Guangzhou 510642 China

Correspondence should be addressed to Xiong-zhi Wang flyhawkrenscaueducn and Wenliang Zhou zwlxpd126com

Received 16 May 2018 Revised 1 September 2018 Accepted 29 November 2018 Published 25 December 2018

Academic Editor Mustapha Nourelfath

Copyright copy 2018 Xiong-zhi Wang and Wenliang Zhou This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this article we investigate a joint pricing and inventory problem for a retailer selling fresh agriproducts (FAPs) with two-periodshelf lifetime in a dynamic stochastic setting where new and old FAPs are on sale simultaneously At the beginning of each periodthe retailer makes ordering decision for new FAP and sets regular and discount price for new and old inventories respectivelyAfter demand realization the expired leftover is disposed and unexpired inventory is carried to the next period continuing sellingUnmet demand of all FAPs is backorderedThe objective is to maximize the total expected discount profit over the whole planninghorizonWe present a price-dependent stochastic dynamic programmingmodel taking into account zero lead time linear orderingcosts inventory holding and backlogging costs as well as disposal cost Considering the influence of the perishability we integratea Multinomial Logit (MNL) choice model to describe the consumer behavior on purchasing fresh or nonfresh product By way ofthe inverse of the price vector the original formulation can be transferred to be jointly concave and tractable Finally we characterizethe optimal policy and develop effective methods to solve the problem and conduct a simple numerical illustration

1 Introduction

Motivated by local fresh agriproducts (FAPs) store retailingissues in its operations management this paper investigatesa revenue problem in a stochastic setting where a retailerconsiders a jointly pricing and inventory control issue underconsumer choice In retailing practice FAPs such as freshfruits or vegetables are usually delivered to the store ormarket every day in themorning based on the retailerrsquos ordersubmitted in the previous evening and to be selling at aregular price in the following one or two more days As theperishability nature FAPs are always deteriorating along timegoing gradually being perceived nonfresh and less attractiveto customers which weakens the demand and results in theloss of revenue To reduce the amount of mismatch betweensupply and demand the retailer always sets a discount pricefor the nonfresh FAPs at a certain time point which otherwiseare deposed if they remain unsold at the end of selling In themeantime the store shelf is replenished with a new batch ofFAPs ordered by the seller and charged a regular price onselling simultaneously with those unsold nonfresh products

While the regular price and discount price are determinedcentrally on a long-term basis some fundamental decisionsare at the store managerrsquos discretion in such an environmentincluding how much new FAPs to order and what pricingpolicies to follow for selling It is not a trivial problem forthe seller to sell deteriorated items on discount paralleledwith fresh products with regular price As every coin hastwo sides on one hand the price cutting sales can generatesome revenue for the otherwise disposed items since offeringa price discount on those items can induce purchasing whileon the other hand in codisplay selling simultaneously modediscount policy results in competition among consumers topurchase either fresh or nonfresh products on the basis ofits utility from combination of quality and price Some ofthe consumers who can afford the full price may be attractedby the price discount to buy nonfresh products which has anegative impact on the revenue from selling at the full price

In some form or other jointly pricing and inventorycontrol problem arises in a lot of instances For exampleStater Bros a well-known grocery store in America alwayssells nonfresh vegetables or meat at a discount along with

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 5936971 12 pageshttpsdoiorg10115520185936971

2 Mathematical Problems in Engineering

fresh ones and disposes those unsold items at the end of theirshelf lifetime That is the Stater Brothers storersquos managershouldmake optimal order policy for fresh vegetables ormeatand charge for both fresh and nonfresh products differentlyevery day so that the inventory and demand can be extremelymatched In essence these examples exhibit one homogene-ous competition issue which is an important concern andcommon in retailing practice Meanwhile pricing is also adouble-edged sword in that pricing can not only influencethe demand but also have great impact on the sellersrsquo profitespecially a poor pricing policy which can easily damage thesellerrsquos benefit For example Tesco the worldrsquos third-biggestsupermarket chain failed to revive its sales performancedespite spending m500 million on price cuts in Eurozone(httpwwwftcomintlcmss0dee4d20c-2172-11e1-a1d8-00144feabdc0htmlaxzz3Iua2R9jr) On the contrary StaterBros Markets reported its modest profit gains in appro-priate price strategy in fiscal 2011 despite higher commoditycosts and competitive pressures (httpwwwpecomarticlesmillion-634066-store-yearhtml) Therefore instead ofsimple pricing of changing prices dramatically sellers shouldbe more prudent to adjust the price in a dynamic fashionwhich enable price change in line with the availability ofinventory and products residual shelf-life time Neverthelessas consumers may pick any product available and selects hispreferred choice based on the utility of an appealing quality-price combination due to the codisplaying mode on salemanagers need to carefully take account of consumerbehavior when coordinating discount sales with inventoryordering decisions

In this study we formulate a novel model in whichthe retailer makes simultaneous decisions dynamically oncharging appropriate price and optimal inventory policy forperishable products in the presence of homogeneously sell-ing cannibalization and demand uncertainty The retail timeof any batch of FAPs is divided into two periods a regularsale period and a discount pricing period which implies thatFAPs have two periods of shelf lifetime At every period theretailer starts with ordering new FAPs and charging pricepolicy for two different ages of FAPs which compete amongcustomers in their attributes of quality and price and endswith disposing all unsold leftovers with zero shelf lifetimeand carrying the leftover inventory of unexpired products tothe next period for continuing selling Generally new FAPsare perceived to have a higher quality than the old ones Theutility for customers is mainly determined by quality-pricecombination Demand in each period is random and dependson the current price which takes a form of price functionplus an additive random perturbation The sellerrsquos objectiveis to maximize the total expected discount profit over thewhole planning horizon We formulate a stochastic price-dependent demand dynamic programming model takinginto account zero lead time linear ordering costs inventoryholding and backlogging costs Our model assumes a pop-ulation of homogeneous (statistically identical and indepen-dent) customers makingmutually exclusive choices from twodifferent ages of FAPsMeanwhile we integrate aMultinomialLogit Choice model to describe dynamic consumer behavioron choosing new or old products available in inventory

and prices adjustment The retailer who should considerboth the influence of perishable nature of fresh agriculturalproducts and the homogeneous retailing cannibalizationintegrates inventory control and dynamic pricing policy tomaximize his profit on the whole planning horizon To thebest of our knowledge this is the first stochastic and dynamicinventory model on perishable fresh agricultural productstaking account of consumer behavior with joint ordering anddiscount decisions The problem is extremely complicatedeven if the products are not perishable Unlike standardinventory models of general products with same attributesthe perishable property plays an important role in driving thedynamic pricing adjustments which affects the demand shareof new and old FAPs Traditional dynamic inventory modelscannot be suitable to our problem in that even the singleperiod profit function is neither jointly convex nor concave inthe decision variables therefore intractable In addition thehomogeneous competition results in interdependence acrossthe planning horizon which leads to complicated dynamicsin how inventory is controlled over from one period toanother To deal with the complexity we work with thetransformation technique and exploiting specific propertiesby way of the inverse of the price vector We also conductnumerical studies to further characterize the optimal policy

The remainder of this paper is organized as follows Inthe next section we provide a literature review and introducethe model and formulate the problems a dynamic programin Section 3 In Sections 4 and 5 we perform the structuralanalysis and present an optimal pricing policy coordinatedwith inventory control strategies respectively A numericalstudy is conducted to show the policy efficiency in Section 6We conclude our work in Section 7

2 Literature Review

Our research ismostly related to jointly pricing and inventorycontrol on perishable products including three streams ofliteratures (1) dynamic inventory control for perishable prod-ucts (2) combined dynamic pricing and inventory controland (3) customer behavior in the context of dynamic pricing

Firstly our work is closely related to dynamic inventorycontrol for perishables As far as we know earliest work datesback to Ghare amp Schrader [1] who first considered inventorycontrol of exponential decay product and generalized anEOQ model by assuming the lifetime of each product isexponentially distributed Other forms of random lifetimemodels can be referred to Weibull distributed deterioration[2] Gamma distributed decay [3] generalized exponen-tial decay [4] etc These works mainly focused on staticinventory control policies instead of dynamic strategies Inretailing the perishable product lifetime was always knowndeterministically Fixed lifetime models are also concernedand can be traced to Nahmias and Pierskalls [5] who firstexplored the problem in a two-period lifetime setting withzero lead time and demand uncertainty On the basis of theirresearch framework Nahmias [6] and Fries [7] extendedthe work on the case with multiperiod lifetime in periodic-review system where only the excess inventory expired atthe end of the current period is disposed of They probed

Mathematical Problems in Engineering 3

the characteristic of optimal policy and pointed out that theoptimal order quantity was a decreasing function of on-handinventory of different ages And a series of papers thereafterfocused on the similar topic see Nahmias [8] Cohen [9]Nandakumar and Morton [10] Liu and Lian [11] Lian andLiu [12] Gurler and Ozkaya [13] Berk and Gurler [14] etcNote that a suitable model might vary even for the sameproblem analysis of which was lengthy and difficult to begeneralized Due to the fact that complexity the literaturethenceforward concentrates more on developing heuristicsexcept for model analysis Nahmias [15 16] and Karaesmenet al [17] have reviewed the early and lately developmentsRecently Xue et al [18] Li et al 2013 and Li and Yu [19]studied a perishable inventory model in a secondary marketsetting where the excess inventory can be cleared with certainsalvage value They provided some structure properties andthen brought about an effective heuristic Recently Chao etal [20] Zhang et al [21] and Chao et al [22] developedapproximation algorithms for perishable inventory systemsOur study generalizes this literature to allow for consumerchoice on two different ages of FAPs in periodic-reviewsystem And we do focus on exploring optimal policy insteadof effective algorithms

Secondly in terms of jointly pricing and inventorymanagement to the best of our knowledge a mountain ofwork contributes to model on nonperishable products inrecent decades under the environment of from deterministicperiodic-review inventory and pricing models to stochasticmodels with different cost structures distinguishing betweensingle period and multiperiod See for example Wagnerand Whitin [23] Kunreuther and Schrage [24] Zabel [25]Thomas [26] Gilbert [27] Petruzzi and Dada [28] Feder-gruen and Heching [29] Feng and Chen [30] Geunes et al[31] Chen and Simchi-Levi [32 33] Raz and Porteus [34]Huh and Janakiraman [35] Chen et al ([36] 2010) Songet al [37] and Simchi-Levi [38] Among them there aresome recent surveys on different operations research andmanagement science perspective which refer to Eliashbergand Steinberg [39] Elmaghraby and Keskinocak [40] Yanoand Gilbert [41] and Chan et al [42] and Chen and Simchi-Levi [43] This substantial body of literature illustrates thatthe retailer can benefit great from the structures of optimalpolicies and effective algorithms of integrated pricing andinventory strategies Dynamic pricing supposed to be aneffective lever to manage the profitability of perishablesretailing was firstly investigated together with inventorycontrol by Rajan et al [4] who studied dynamic pricing andordering decisions for a certain product that experiences ageneral exponential decaying under deterministic demandwhereafter Abad [44] generalized this work by allowing forpartially backlogged demand with the same objective ofmaximizing the long-run average profit As Nahmias [15]pointed out developing pricing policies for perishable itemsis very important in a demand uncertainty context There area few studies pertaining to the case on perishables sellingAmong them Ferguson and Koenigsberg [45] considered ajoint pricing and inventory control problem in a two-periodsetting addressing the competition impact between new andold products inventory They presented a stylized model

and concentrated on deriving managerial insights Withthe assumption of inventory depleted in a first-in-first-out(FIFO) sequence Li et al [46] explored a dynamic joint pric-ing and inventory control for a two-period lifetime perishableproduct over an infinite horizon taking into account linearprice-response demand backlogging and zero lead timeAnd they also extended the problem to a stationary systemwith multiperiod lifetime and developed a base-stocklist-price (BSLP) heuristic policy Then Li et al [47] continuedanalyzing the infinite-horizon lost-sales case where new andold inventory can not be simultaneously sold and sellers candecide whether to dispose of or carry all ending inventoryuntil it expires at the end of each period They proposed astationary structural policy consisting of an inventory order-up-to level state-dependent price and inventory clearingdecisions and developed a fractional programming algorithmto obtain the optimal policy among the class of proposedstructural policies In another relevant work Sainathan [48]studied the two-period lifetime case where new and oldinventory can also be selling simultaneously and the sellermakes decisions on price polices and how much to order atthe beginning of each period Recognizing the influence ofperishability on demand Sainathan modeled the consumerchoice and characterized the structure of the optimal pol-icy under different demand context including determinis-tic demand two-point demand and substitution demandExtending the analysis and results to the case with arbitraryperiod lifetime Chen et al [49] presented a significantgeneralization of these papers by allowing for positive leadtime both backlogging and lost-sales cases and unrestrictedordering decisions They developed the structural resultsof L-concavity and generalize the regularity conditions ofdemand functions for lost-sales inventory-pricing models ofsetting a single price for inventories of different ages Wu etal [50] study a problem that the seller dynamically makesthe joint pricing and inventory replenishment decisions overmultiple periods where each periods consist of two stagesfor ordering and markdown pricing respectively Chen ampChu [51] and Chen amp Cong (2018) integrated a strategiccustomerrsquos behavior into joint pricing and inventory modeland derived an optimal joint pricing delivery and inventorypolicy Recently Peng Hu et al (2016) formulated a modelon a firmrsquos dynamic inventory and markdown decisions forperishable goods where every period consists of regularsales phase and clearance phase under strategic consumerbehavior They showed that the seller should either put theleftover inventory on discount or dispose all of them and thechoice depends on the amount of leftover inventory from theprevious period

Finally our work involves modeling of consumer choicebehavior which is an important phenomenon within revenuemanagement system The earliest study in the literaturerelated to this research is pioneered by Kincaid and Darling[52] who described the probability of the charging price bywhich the expected demand can be calculated This generalmodel has been extended by other authors Bitran andMondschein [53] who concentrated on applications of themodel to the sale of fashion goods van Ryzin and Mahajan[54] firstly emploied the MNL model to describe the normal

4 Mathematical Problems in Engineering

demand distribution for substitute products Aydin and Ryan[55] used theMNLmodel to study pricing problem assumingthe firm satisfies demand by make-to-order Recent papersrelated stream of literature include Aydin and Porteus (2005)Zhang and Cooper [56] Xu andHopp [57] Perakis and Sood[58] Lin and Sibdari [59] Song and Xue [60] and Martınez-de-Albeniz andTalluri [61] In all those papers customers candecide which product to buy based on prices and inventorylevels at time of purchase with the MNL model for productcategories assuming homogeneous product groups Ourwork is consistent with those characteristics above demanduncertainty two-period shelf lifetime codisplaying sellingand homogeneous cannibalization We use MNL model todescribe consumer discrete choice and present a jointlydynamic pricing and inventory control policy to maximizesellerrsquos profit on the whole planning horizon

In summary our contribution to the literature is asfollows Firstly we generalize the perishable joint pricing andinventory models for perishable products of two-period life-time After characterizing the structure of model we presentan optimal solution to the dynamic pricing of perishableproducts with consumers incurring dynamic substitutionsthroughout the selling season Second we provide sortsof insights on the behavior of the optimal dynamic pricesand highlight the complex interplay of inventory scarcityand product quality difference Thirdly we generalize theregularity conditions of demand functions for integrationof inventory-pricing models using the concept of concavityand characterize the optimal policy and develop effectivemethods

3 The Problem Formulation and Model

31 Problem Formulation Consider a periodic-review singletype of FAP inventory system over a finite planning horizonof T periodsThe product is perishable with two-period shelflifetime fresh in the first period and nonfresh in the nextperiod At the beginning of each period selling the retailerreviews the current inventories and decides simultaneouslyon an order quantity for new FAP as well as charging pricesfor both new delivery and old inventory During the salecustomers arrive at the store or market and make purchasechoices based on FAPs attributes of quality and price whichcauses a drop in the inventory level The demand for thecurrent inventory depends on the newly charged pricesand it is always met to the maximum extent with on-handinventories At the end of the period the leftover inventorywith one period lifetime is carried forward to next periodbecoming old products to sell while the unsold products withzero shelf-life time have to be discardedThen the sale movesto the next period and the above processes repeat

For convenience and tractability we assume that thereplenishment order placed by the retailer arrives immedi-ately before the demand unfolds that is to say replenishmentis instantaneous with zero lead time Meanwhile there couldhave chance to replenish fresh products during the sale fromsecond market since in retailing practice backorder policy isprominent which means unsatisfied demands of all productsduring selling are to be backlogged And there is neither

fixed ordering cost nor constraints on the supply capacityDemand at each period is uncertain and independent acrossall periods generally decreasing on current prices In linewith Petruzzi amp Dada [28] and Chen amp Simchi-Levi [32] wetake a form of a function of prices charged at that periodmultiplies a stochastic variable plus a random disturbanceNevertheless customers are assumed to be homogeneousand decide purchasing based on current prices and productattributes alone instead of acting strategically by adjustingtheir buying behavior in response to the firmrsquos pricing policyIn the whole horizon planning although the retailer getsrevenue from selling new and old products he bears somecosts incurred in the operations such as order decisionfor wholesale each order incurs a variable cost c for pernew FAP And three more other costs are incurred leftoverinventory carrying costs over from one period to the nextbacklogged costs for unmet demand fromon-hand inventoryand disposal costs of leftovers with zero shelf-life time Theobjective is to dynamically determine ordering and pricingdecisions in all periods so as to maximize the total expecteddiscounted profit over the whole planning horizon

32 Demand Model During sales the new and old productscompete among customers by combination of their attributesand price And each customer selects his preferred choicebased on his utility from purchasing a unit of product newor old which is given by 119880119894 = 120572120579119894 minus 119901119894 + 120585119894 (119894 = 1 2)where i=1 for the fresh product and i=2 for the nonfreshproduct respectively In the formula 120579119894 and p119894 are defined asthe average perceptive value and the charged price on producti respectively while 120585119894 is a certain customerrsquos deviation from120579119894 which is influenced by unobservable characteristics Thecoefficient 120572 is denoted as a customerrsquos quality sensitivityNote that all consumers are statistically homogeneous andhave the same sensitivity to price Generally any customerwho visits the retailer has three choices select one unit of newor old FAP or do nothing Here we set a virtual product andlet i=3 with price zero for denoting no purchase choice Let1205853 be a certain customerrsquos deviation from 1205793 Obviously 1205853= 0and 1199013= 0 Therefore under the assumption of rationality acustomer will always purchase one product with the highestutility among these three options that is a customer choosesto buy product i where 119894 = argmax119896=123119880119896 = 120572120579119896minus119901119896+120585119896Suppose random variable 120585119894 be iid distributed with a doubleexponential distribution that is Pr (120585119894 le 120575) = 119890minus119890minus120575 120575 isin(minusinfin +infin) with mean zero and variance 120576120575 See Guadagniand Little [62] and Anderson et al [63] Define 120573119894(1199011 1199012) tobe the probability that an arriving customer in a period selectsfresh or nonfresh agriproduct in response to price (1199011 1199012)It can be shown that a consumerrsquos choice probability has asimple form

120573119894 (1199011 1199012) = exp (120572120579119894 minus 119901119894)1 + sum119896=12 exp (120572120579119896 minus 119901119896) 119894 = 1 2 (1)

which is anMNLmodel often used to study a set of productsdifferentiated in either quality or style a term coined in theliterature of MNL models (see [63]) Let (1199011119905 1199012119905) be thecharged prices at any period 119905 isin 1 2 119879 On the basis

Mathematical Problems in Engineering 5

of analysis above we denote 120573119894119905(1199011119905 1199012119905) the market share ofproduct i at a certain period t which is similar to equation (1)and expressed as follows

120573119894119905 (1199011119905 1199012119905) = exp (120572120579119894 minus 119901119894119905)1 + sum119896=12 exp (120572120579119896 minus 119901119896119905) 119894 = 1 2 (2)

Note that the total market share sum119894=12 120573119894119905(1199011119905 1199012119905) le 1Denote 120596119905 to be the amount of arriving customers to buy

the products at period t which is generally uncertain andindependently distributed with finitemean of 120583119905 and variance120591t across all time periods Note that the expected demandof new or old product is proportional to the market share120573119894119905(1199011119905 1199012119905) So provided with price (1199011119905 1199012119905) as well as ademand disturbance on product i at period t 120576119894119905 we can writeout the demand function of product i denoted as 119863119894119905 belowan additive form which is commonly used in the literature(see eg [28 32])

119863119894119905 = 120573119894119905 (1199011119905 1199012119905) 120596119905 + 120576119894119905 119894 = 1 2 (3)

where 120576119894119905 is assumed to be identically distributed over timewith zero mean and finite variance as well as cdf 119865(sdot) and

pdf 119891(sdot) Denote 119889119894119905(1199011119905 1199012119905) as the expected demand levelthen 119889119894119905(1199011119905 1199012119905) = 120573119894119905(1199011119905 1199012119905)119864(120596119905)+119864(120576119894119905) = 120573119894119905(1199011119905 1199012119905)120583119905The selling price is generally restricted to an interval [119901

119905 119901119905]

and 1199011119905 ge 1199012119905Note that the monotonicity of the expected demand

function implies a one-to-one correspondence between theselling price and the expected demand And we observe thatif we use the inverse of the market share function to expressthe price variables as a function of the market share thenthe revenue function becomes jointly concave under somecertain condition allowing us to analyze the structure ofmodel Accordingly we have the following

Proposition 1 Under the MNL market share model the pricefunction at period t is

119901119894119905 = 119875119894119905 (1205731119905 1205732119905) = 120572120579119894 + ln((1 minus sum119894 120573119894119905)120573119894119905 ) 119894 = 1 2

(4)

e corresponding market share space Ω119905 = (1205731119905 1205732119905) 119901119905 le1198752119905(1205731119905 1205732119905) le 1198751119905(1205731119905 1205732119905) le 119901119905 119894 = 1 2 can be expressed as

Ω119905 =

(1205731119905 1205732119905)

exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) le 1205731119905 leexp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905)sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905) le 1205731119905 + 1205732119905 lesum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)

exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905) le 1205732119905 leexp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)

(5)

which is convex and compact

Proof Equation (4) comes from (2) by transforming andtaking ln operation on both sides The range is derived fromthe fact that the prices should be nonnegative Consider anyproduct new or old FAP If we charge the lowest price 119901

119905

for one product and the highest price 119901119905 for the other thenthis product should occupy the highest possiblemarket share(sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))33 Dynamic Programming Formulation Based on the de-scription aforementioned we define some other parameterswith respect to period t below

119868119905 = initial inventory level of old FAP before orderingat the beginning of period t119876119905 = order quantity of new FAP at the beginning ofperiod t119888119905 = unit procurement cost of new FAPℎ1119905 = unit holding cost for the excess inventory

ℎ2119905 = unit disposal cost of remaining inventory withzero shelf-life time119897119894119905 = backordered cost for unit shortage of new andold FAP 119894 = 1 2120574 = profit discount factor 120574 isin (0 1]

Here we do not factor dynamic demand substitution thatthe customer may buy the other product type when hispreferred one is out of stock At the beginning of period 119905 119905 isin1 2 119879 the retailer reviews the current inventory 119868119905 of oldFAPwhich comes from the unsold inventory of new productsat the previous period and places replenishment order 119876119905 fornew FAP as well as setting product prices 1199011119905amp1199012119905 for newand old FAPs respectively Obviously under zero lead timeassumption the inventory of new FAPs is instantaneouslyreplenished to be 119876119905 After demand realization and theretailer satisfies the demand to the maximum extent theremaining inventory levels of the new and old products are(119876119905 minus 1198631119905)+ and (119868119905 minus 1198632119905)+ respectively The (119876119905 minus 1198631119905)+of leftover that remains one period of shelf-life time will becarried to next period for sale as old FAP incurring holding

6 Mathematical Problems in Engineering

cost and becoming the initial inventory of nonfresh product119868119905+1 at period t+1 while the (119868119905 minus 1198632119905)+ of old products willbe disposed of on account of zero shelf-life time incurringdisposal cost Otherwise the relevant backordered amountof new FAPs and old FAPs for period t is (119876119905 minus 1198631119905)minus and(119868119905minus1198632119905)minus respectively whendemand is greater than on-handinventories where 119909+ = max(0 119909) and 119909minus = max(0 minus119909)

At period t the retailer revenue 119905((1199011119905 1199012119905)) can becalculated as follows

119905 ((1199011119905 1199012119905)) = (1199011119905 1199012119905) (1198631119905 1198632119905)119879 (6)

Substitute the price function (5) into (6) we can obtain theexpected revenue 119877119905((1205731t 1205732119905))119877119905 ((1205731t 1205732119905)) = 119864 (119905 (1199011119905 1199012119905))= sum119894=12

(120572120579119894 + ln((1 minus sum119894 120573119894119905)120573119894119905 ))120573it (1199011119905 1199012119905) 120583119905 (7)

After demand realization at period t somemore costs incursbackordered cost of 1198971119905(119876119905 minus1198631119905)minus or holding cost of ℎ1119905(119876119905 minus1198631119905)+ for new FAPs and backlogged cost of 1198972119905(119868119905 minus 1198632119905)minus ordisposal cost of ℎ2119905(119868119905 minus1198632119905)+ for old FAPs totally denoted as119905(119876119905 (1205731119905 1205732119905))119905 (119876119905 (1205731119905 1205732119905)) = (ℎ1119905 ℎ2119905) sdot ((119876119905 minus 1205731t120596119905 minus 1205761119905)

+

(119868119905 minus 1205732t120596119905 minus 1205762119905)+)+ (1198971119905 1198971119905)sdot ((119876119905 minus 1205731t120596119905 minus 1205761119905)minus(119868119905 minus 1205732t120596119905 minus 1205762119905)minus)

(8)

And the corresponding expected function is 119867119905(119876119905 (12057311199051205732119905)) = 119864(119905(119876119905 (1205731119905 1205732119905)))Let 119866119905(119876119905 (1205731119905 1205732119905)) be the profit of period t we have

119866119905 (119876119905 (1205731119905 1205732119905)) = 119905 (119876119905 (1205731119905 1205732119905))minus 119905 (119876119905 (1205731119905 1205732119905)) minus 119888119905119876119905 (9)

Accordingly the expected profit function is as below

119866119905 (119876119905 (1205731119905 1205732119905)) = 119864119866119905 (119876119905 (1205731119905 1205732119905))= 119877119905 (1205731119905 1205732119905) minus 119867119905 (119876119905 (1205731119905 1205732119905))minus 119888119905119876119905

(10)

Denote 120587119905(119868119905) to be the maximum total expected profit-to-go function from period t to T in state 119868119905 witha discount factor 120574 We have the recursive equation120587119905(119868119905) = max119876119905ge0(1205731t 1205732119905)isinΩ119905119866119905(119876119905 (1205731119905 1205732119905)) + 120574119864120587119905+1(119868119905+1)With transfer function 119868119905+1 = (119876119905 minus 1205731119905120596119905)+ by definingΦ119905(119876119905 (1205731119905 1205732119905))

Φ119905 (119876119905 (1205731119905 1205732119905)) = 119866119905 (119876119905 (1205731119905 1205732119905))+ 120574119864120587119905+1 (119868119905+1) (11)

we have the following dynamic programming model

120587119905 (119868119905) = max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731119905 1205732119905)) (12)

119904119905 120587119879+1 = minusℎ2119879+1119868119879+1 (13)

where (13) is the boundary condition since all leftovers will bedisposed of at the end of horizon planningThat is to say thereare no second markets where some unexpired inventory canbe cleared with some salvage value On the contrary the firmneeds to pay on deposing expired inventories even unexpiredleftovers Our analysis can be readily extended to address thissituation

4 Analysis of Model Structure

To analyze the structure of the optimal policy we needto know the concavity property of Φ119905(119876119905 (1205731t 1205732119905)) whichis strictly relative to revenue function 119877119905((1205731t 1205732119905)) andinventory cost function119867119905(119876119905 (1205731t 1205732119905))Lemma 2 119877119905((1205731t 1205732119905)) is continuously twice differentiableand jointly concave in (1205731t 1205732119905)Proof The continuity and twice differentiability of 119877119905((1205731t1205732119905)) is quite intuitive since it is expressed by some elementaryoperations of 1205731t and 1205732119905 To show the concavity we needto prove the negative semidefinite property of its HessianMatrix Taking the second-order derivative with respect to1205731t yields 120597211987711990512059712057321t = minus(1(1 minus 1205731119905 minus 1205732119905) + 11205731119905 + 1(1 minus1205731119905 minus 1205732119905)2)120583119905 le 0 Thus 119877119905 is concave on 1205731t Taking thesecond-order derivative with respect to 1205732t yields 120597211987711990512059712057322t =minus(1(1 minus 1205731119905 minus 1205732119905) + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 le 0which implies the concavity of 119877119905 on 1205732t And taking thecross-partial derivative with respect to 1205731t and 1205732t yields1205972119877119905(1205971205731t1205971205732t) = minus(1(1 minus 1205731119905 minus 1205732119905) + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905Pairing up the above terms leads to Hessian Matrix

119867essian (119877119905) = [[[[[

120597211987711990512059712057321t 1205972119877119905(1205971205731t1205971205732t)1205972119877119905(1205971205732t1205971205731t) 120597211987711990512059712057322t]]]]]

Mathematical Problems in Engineering 7

= [[[[[[minus( 11 minus 1205731119905 minus 1205732119905 + 11205731119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905

]]]]]]

(14)

And the corresponding determinant is as follows

1003816100381610038161003816119867 (119877119905)1003816100381610038161003816 = 120597211987711990512059712057321t sdot 120597211987711990512059712057322t minus ( 1205972119877119905(1205971205731t1205971205732t))

2

= ( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)(11205731119905 + 11205732119905)

+ 112057311199051205732119905 gt 0(15)

which implies that 119867119890119904119904119894119886119899 matrix is seminegative defi-nite On the basis of that 119877119905 is jointly concave in (1205731t1205732119905)Lemma 3 119867119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly convex in (1205731t 1205732119905) and 119876119905Proof We observe that 119905(119876119905 (1205731t 1205732119905)) is continuously andtwice differentiable for any realization of 120596119905 and 120576119894119905 And theexpectation 119867119905(119876119905 (1205731t 1205732119905)) is easily justified to be con-tinuously twice differentiable using the canonical argument(Durrett 2010)

To show the convexity we define some column vectorsexpressed by boldface lowercase letters such as h119905 =(ℎ1119905 ℎ2119905)119879 l119905 = (1198971119905 1198972119905)119879 and 120576119905 = (1205761119905 1205762119905)119879 Then thedemand and cost function can be expressed of vector formas follows

D119905 = (1198631119905 1198632119905) = 120573119905120596119905 + 120576119905 (16)119905 (119876119905 (1205731t 1205732119905)) = 119905 (120573119905 119876119905)= h119905 ((119876119905 119868119905)119879 minusD119905)++ l119905 ((119876119905 119868119905)119879 minusD119905)minus

= h119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)++ l119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)minus

(17)

Define a new vector variable 119906119905 = (119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905 andwe can get 119905(120573119905 119876119905) = 119905(u119905) = h119905(u119905)+ + l119905(u119905)minus which isconvex onu119905 (SundaranmR K 1996) Fix the randompart120596119905to its realization and consider any pair (120573119905 119876119905) and (1205731015840119905 1198761015840119905)with the convexity of 119905(u119905) we have119905 (120572 (120573119905 119876119905) + (1 minus 120572) (1205731015840119905 1198761015840119905)) = 119905 (120572120573119905+ (1 minus 120572)1205731015840119905 120572119876119905 + (1 minus 120572)1198761015840119905)= h119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)

minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)++ l119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)minus= h119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879119905 minus 1205731015840119905120596119905 minus 120576119905))++ l119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879 minus 1205731015840119905120596119905 minus 120576119905))minus = h119905 (120572u119905+ (1 minus 120572)u1015840119905)+ + l119905 (120572u119905 + (1 minus 120572) u1015840119905)minus = 119905 (120572u119905+ (1 minus 120572)u1015840119905) le 120572119905 (u119905) + (1 minus 120572) 119905 (u1015840119905)= 120572119905 (120573119905 (119876119905 119868119905)119879) + (1 minus 120572) 119905 (1205731015840119905 (1198761015840119905 119868119905)119879)

(18)

where 0 le 120572 le 1 Therefore 119905(119876119905 (1205731t 1205732119905)) = 119905(120573119905(119876119905 119868119905)119879) = 119905(u119905) is jointly convex in (1205731t 1205732119905) and 119876119905 sodoes the expectation function119867119905(119876119905 (1205731t 1205732119905))Theorem 4 At any time period t with initial inventory 119868119905the expected revenue function119866119905(119876119905 (1205731t 1205732119905)) is continuouslytwice differentiable and jointly concave in (1205731t 1205732119905) and 119876119905Proof With Lemmas 2 and 3 we can observe that119866119905(119876119905 (1205731t 1205732119905)) is continuously twice differentiable We firstobserve that for any t the three terms of 119866119905(119876119905 (1205731119905 1205732119905)) in(10) are jointly concave The first term is concave in (1205731119905 1205732119905)according to Lemma 2 The second term is jointly concavein (1205731119905 1205732119905) and 119876119905 (see Lemma 3) and the third term islinear in 119876119905 which implies the concavity With the propertythat a nonnegative weighted sum of concave functionsis concave (Stephen Boyd Lieven Vandenberghe 2004)119866119905(119876119905 (1205731119905 1205732119905)) is continuously twice differentiable andjointly concave in (1205731t 1205732119905) and 119876119905

On the basis of the concavity of 119866119905(119876119905 (1205731t 1205732119905)) we canhaveTheorem 5

Theorem 5 Consider any time period t with initial inventory119868119905 the functionΦ119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly concave in (119876119905 (1205731t 1205732119905)) and the function120587119905(119868119905) is concave and nonincreasing in 119868119905Proof By induction first we prove that Φ119879(119876119879 (1205731T 1205732119879))is continuously twice differentiable According to Lemma 2

8 Mathematical Problems in Engineering

119864(119876119879 minus 1205731119879120596119879 minus 120576119879)+ as a part of 119867119905(119876119905 (1205731t 1205732119905)) isthus continuously twice differentiable which implies thatΦ119879(119876119879 (1205731119879 1205732119879)) = 119866119879(119876119879 (1205731119879 1205732119879)) minus 120574ℎ2119879+1119864(119876119879 minus1205731119879120596119879 minus 1205762119879)+ is continuously and twice differentiableAssuming now that Φ119905(119876119905 (1205731119905 1205732119905)) is continuouslytwice differentiable for 119905 = 1 2 119879 we prove thatΦ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is also continuously twice differen-tiable according to Theorem 4 Let Λ 119879(119876119879 1205731119879) = 119864(119876119879 minus1205731119879120596119879minus1205762119879)+ = ∬1205731119879119909+119910le119876119879(119876119879minus1205731119879119909minus119910)119891120596(119909)119891120576(119910)119889119909119889119910where 119891120596(119909) and 119891120576(119910) is the pdf of random variable120596119879 and 120576119879 respectively which are generally assumed to beindependent For any realization of 1205762119879 such as 1205762119879 taking thesecond-order derivative ofΛ 119879(119876119879 1205731119879)with respective to119876119879yields 1205972Λ 119879(119876119879 1205731119879)1205971198762119879 = (11205731119879)119891((119876119879 minus 1205762119879)1205731119879) gt 0which implies that Λ 119879(119876119879 1205731119879) is convex in 119876119879 Takingthe second-order derivative with respect to 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)12059712057321119879 = (119876211987912057331119879)119891(1198761198791205731119879) gt 0 whichimplies the concavity of Λ 119879(119876119879 1205731119879) in 1205731119879 And takingthe cross-partial derivative with respect to 119876119879 and 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)1205971198761198791205971205731119879 = minus((119876119879 minus 1205762119879)12057321119879)119891((119876119879 minus1205762119879)1205731119879) Pairing up the above three derivative terms leadsto Hessian Matrix

119867(Λ 119879) = [[[[[

1205972Λ 11987912059711987621198791205972Λ 11987912059711987611987912059712057311198791205972Λ 1198791205971205731119879120597119876119879 1205972Λ 11987912059712057321119879

]]]]]

= [[[[[

11205731119879119891(119876119879 minus 12057621198791205731119879 ) minus119876119879 minus 120576211987912057321119879

119891(119876119879 minus 12057621198791205731119879 )minus119876119879 minus 120576211987912057321119879 119891(119876119879 minus 12057621198791205731119879 ) (119876119879 minus 1205762119879)212057331119879 119891(119876119879 minus 12057621198791205731119879 )

]]]]](19)

And the corresponding determinant 119867(Λ 119879) = 0 whichimplies matrix 119867119890119904119904119894119886119899 is semipositive definite On thebasis of analysis above we can conclude that Λ 119879(119876119879 1205731119879)is jointly convex in (119876119879 1205731119879) Meanwhile the last termminus120574ℎ2119879+1119864(119876119879 minus 1205731119879120596119879 minus 1205762119879)+ is obviously justified to bejointly concave in (119876119879 1205731119879) According to the concavityproperty (Stephen Boyd Lieven Vandenberghe 2004)Φ119879(119876119879 (1205731T 1205732119879)) is jointly concave in (1205731t 1205732119905) and 119876119905

120587119879(119868119879) is easily verified to be concave as well and it isobviously nonincreasing

Assume now that Φ119905(119876119905 (1205731t 1205732119905)) is jointly concave forsome 119905 = 1 2 119879 minus 1 and that 120587119905(119868119905) is also concaveand nonincreasing Then Φ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is jointlyconcave joint concavity of the first term of (10) is verified forthe case 119905 = 119879 above For any pair of points (119876119905 1205731119905) and(1198761015840119905 12057310158401119905) with given value of 120596119905 we have120587119905+1 ((120572119876119905 + (1 minus 120572)1198761015840119905)119905 minus (1205721205731119905 + (1 minus 120572) 12057310158401119905) 120596119905minus (1205721205761119905 + (1 minus 120572) 1205761119905)) = 120587119905+1 (120572 (119876119905 minus 1205731119905120596119905 minus 1205761119905)+ (1 minus 120572) (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)) ge 120572120587119905+1 (119876119905 minus 1205731119905120596119905minus 1205761119905) + (1 minus 120572) 120587119905+1 (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)

(20)

by the concavity of 120587119905+1(sdot) which implies that 119864120587119905+1(119876119905 minus1205731119905120596119905 minus 1205761119905) is jointly concave in 119876119905 and 1205731119905 With aboveargument the desired result holds

5 Optimal Policy

Following the structure property of model we can obtain thefollowing result

Theorem 6 Consider any period t with the initial inventory119868119905 there is a unique maximizer (119876lowast119905 (120573lowast1119905 120573lowast2119905)) to maximizethe constrained expected periodical profit function

(119876lowast119905 (120573lowast1119905 120573lowast2119905)) isin arg max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731t 1205732119905)) (21)

Proof It can be easily proved on the basis of Theorem 5

On the basis of the dynamic programming analysisincluding the concavity property we can construct an opti-mization problem for any period t with constrained condi-tions as follows

119872119894119899119894119898119894119911119890 minus Φ119905 (119876119905 (1205731t 1205732119905))

119904119905

(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905))(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) 119905) le 0(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905))(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)) le 0(1205731t + 1205732119905 minus sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905))(1205731t + 1205732119905 minus

sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)) le 0minus119876119905 le 0

(22)

Mathematical Problems in Engineering 9

The Lagrangian and the Karush-Kuhn-Tucker(KTT) condi-tions provide an efficient solution approach to this optimiza-tion problemDenote119860119905 = (exp(1205721205791minus119901119905))(1+exp(1205721205791minus119901119905))119860119905 = (exp(1205721205791 minus 119901119905))(1 + exp(1205721205791 minus 119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119862119905 = (sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))

and 119862119905 = (sum119894=12 exp(120572120579119894 minus119901119905))(1 +sum119894=12 exp(120572120579119894 minus119901119905)) Weconstruct for each period t theLagrangian of the optimizationproblem as

119871 119905 (119876119905 (1205731t 1205732119905) (1205821199051 1205821199052 1205821199053 1205821199054))= minusΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905)+ 1205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905)+ 1205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) minus 1205821199054119876119905

(23)

where the Lagrange multipliers are 120582119905119894 ge 0 119894 = 1 2 3 4 Let(119876lowast119905 (120573lowast1119905 120573lowast2119905)) be the optimizing solution we can constructthe KKT conditions as follows

minus nablaΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (0 21205731t minus 119860119905 minus 119860119905 0)+ 1205821199052 (0 0 21205732t minus 119861119905 minus 119861119905) + 1205821199053 (0 21205731t + 21205732119905minus 119862119905 minus 119862119905 21205731t + 21205732119905 minus 119862119905 minus 119862) minus 1205821199054 (1 0 0) = 0

1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905) = 01205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905) = 01205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) = 01205821199054119876119905 = 0

(24)

where the first equation represents FOC of Lagrangianand the latter four equations represent the conditions ofcomplementary slackness

With solution of theKKT conditions (119876lowast119905 (120573lowast1119905 120573lowast2119905)) and(4) we can get the optimization pricing and ordering policyof (119876lowast119905 (119901lowast1119905 119901lowast2119905)) at period t where (119901lowast1119905 119901lowast2119905) is as follows119901lowast119894119905 = 119875119894119905 (120573lowast1119905 120573lowast2119905) = 120572120579119894 + ln((1 minus sum119894 120573lowast119894119905)120573lowast119894119905 )

119894 = 1 2(25)

Note that (119876lowast119905 (120573lowast1119905 120573lowast2119905)) depends on state variable 119868119905 andparameters of (ℎ1119905 ℎ2119905) and (1198971119905 1198972119905) as well as 120583119905 which

implies that the retailer decides the optimal policy on thebasis of these parameters at every period t With back-ward induction we get the optimal strategy a sequence of[(119876lowast1 (119901lowast11 119901lowast21)) (119876lowast2 (119901lowast12 119901lowast22)) (119876lowast119879 (119901lowast1119879 119901lowast2119879))]6 Numerical Experiment

We now conduct a numerical example to illustrate our resultsand select the parameters as follows The demand followsthe multiplicative form with iid terms 120596119905 sim U(0 100)for all periods We assume (1205791 1205792)119879 = (10 5)119879 sincefresh product has a higher perceptive value than nonfreshones The cost structure is symmetric h119905 = (05 05)119879l119905 = (45 45) and 119888119905 = 10 For simplicity to compute theoptimal prices we consider a finite horizon with 119879 = 5the discount rate is 120574 = 09 and the initial inventory level119868119905 = 0 Using Matlab to program the dynamic model andget the following results with KKT condition method wecan obtain a sequence of order quantity and optimal pricessuch as [(78 (148 97)) (68 (126 86)) (92 (175 114))(43 (109 62)) (52 (115 102))] at which the maximumprofit 120587lowast = 4 2356 occurs

In Table 1 we compare the performance betweendynamic pricing policy and a general policy under oneperiod lifetime setting where all products are sold at aregular price at any period of time Similarly we assume(1205791 1205792)119879 = (10 5)119879 in dynamic pricing policy while(1205791 1205792)119879 = (10 10)119879 in general policy Note that in generalsetting the optimal pricing and ordering policy can beobtained by way of general dynamic programming methodThree examples of different finite horizon time (119879 = 51020)are conducted with the same cost structure and discount ratementioned above Accordingly the optimal solutions suchas pricing and order quantity as well as dynamic inventoryare shown in Table 1 where superscript g and d representregular policy and dynamic policy respectively

A summary of interesting observations is as follows Thedynamic policy is obviously better than general pricing andinventory policy with which the retailerrsquos profit increasesby 15-20 Further the demand is also matched to a morecertain extent by the supply since the order quantity is almostmore than that under general policy while the Based on theresults we can conclude that dynamic pricing and inventorypolicy is beneficial for retailerrsquos revenue management inselling fresh agriculture products Furthermore from theperspective of agriculture industrial development it also canmaximally reduce the operating costs such as holding costsor lost costs since the optimal match between productssupply and demand

7 Concluding Remark

In this paper we address a joint pricing and inventorycontrol problem for stochastic fresh agriproduct inventorysystems under consumer choice over a finite horizon in thebacklogging case In the model the products are perishablealong time and the customers are sensitive to the ages ofinventories which result in consumer choice behavior based

10 Mathematical Problems in Engineering

Table 1 The optimal solution under dynamic pricing and regular pricing

Periods T Order quantity Pricing Inventory 120587d 120587g (120587119889 minus 120587119892)120587119892Qd Qg (1199011198891 1199011198892) 119901119892 (1198681198891 1198681198892 ) 119868119892

5

78 72 (14897) 125 (490) 9

42356 35782 18468 65 (12686) 104 (312) 892 80 (175114) 128 (5611) 1343 45 (10962) 81 (220) 352 45 (11582) 93 (121) 5

10

81 70 (156107) 122 (520) 10

83438 69844 195

72 61 (13684) 103 (386) 783 58 (165104) 132 (569) 876 59 (11978) 85 (322) 470 73 (12592) 81 (346) 456 41 (16598) 146 (301) 968 59 (12288) 95 (312) 865 47 (154104) 138 (339) 758 56 (12473) 92 (230) 349 39 (12864) 97 (191) 2

20

90 86 (15598) 131 (540) 16

175328 149632 172

72 65 (14385) 104 (366) 883 57 (165101) 111 (409) 966 75 (13276) 86 (314) 572 54 (12995) 92 (346) 456 51 (16392) 132 (353) 768 62 (13287) 98 (313) 465 45 (151122) 136 (335) 457 69 (13879) 97 (280) 348 46 (15592) 102 (211) 281 72 (13871) 98 (430) 270 58 (13992) 104 (386) 778 50 (15468) 119 (492) 571 48 (13288) 95 (333) 061 63 (14582) 84 (356) 856 52 (16578) 146 (373) 1076 48 (11388) 99 (323) 666 64 (14594) 128 (395) 852 49 (13778) 98 (230) 350 46 (14569) 91 (220) 1

on product freshness By using MNL model to describeconsumer discrete choice we construct a stochastic anddynamic inventory model facing strategic customer behaviorwith joint ordering and discounting decisions We are ableto characterize the firmrsquos optimal policy with the use ofa transformation technique and employing the concept ofthe concavity and finally develop a Lagrangian and theKarush-Kuhn-Tucker (KTT) conditions to provide an efficientsolution approach to this optimization problem

A limitation in our model is that we assume that thelifetime of the product is only two periods Although this istrue for some fresh agriproducts in themarket there are otherproducts that are perishable but yet have longer lifetimes Toextend ourmodel to the products withmore than two periods

lifetimes the state space has to be increased more to specifythe amounts of inventory that are going to expire at differentpoints of time in the futureThis definitelymakes the problemquite hard Another issue is to incorporate an operatingcost for discounting pricing practice which however willintroduce additional technical complexity to the dynamicprogram There are several future research directions Oneis to extend the model to the environment where all thedemand can be lost or only non-FAP demand can be lostwhile the fresh FAP can be backlogged instead Another is toconsider the case where the product has a fixed finite lifetimeof exactly 119899(119899 ge 3) periods or random lifetime Finally itwould be interesting to account for stock-out based substitu-tions

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

demand distribution for substitute products Aydin and Ryan[55] used theMNLmodel to study pricing problem assumingthe firm satisfies demand by make-to-order Recent papersrelated stream of literature include Aydin and Porteus (2005)Zhang and Cooper [56] Xu andHopp [57] Perakis and Sood[58] Lin and Sibdari [59] Song and Xue [60] and Martınez-de-Albeniz andTalluri [61] In all those papers customers candecide which product to buy based on prices and inventorylevels at time of purchase with the MNL model for productcategories assuming homogeneous product groups Ourwork is consistent with those characteristics above demanduncertainty two-period shelf lifetime codisplaying sellingand homogeneous cannibalization We use MNL model todescribe consumer discrete choice and present a jointlydynamic pricing and inventory control policy to maximizesellerrsquos profit on the whole planning horizon

In summary our contribution to the literature is asfollows Firstly we generalize the perishable joint pricing andinventory models for perishable products of two-period life-time After characterizing the structure of model we presentan optimal solution to the dynamic pricing of perishableproducts with consumers incurring dynamic substitutionsthroughout the selling season Second we provide sortsof insights on the behavior of the optimal dynamic pricesand highlight the complex interplay of inventory scarcityand product quality difference Thirdly we generalize theregularity conditions of demand functions for integrationof inventory-pricing models using the concept of concavityand characterize the optimal policy and develop effectivemethods

3 The Problem Formulation and Model

31 Problem Formulation Consider a periodic-review singletype of FAP inventory system over a finite planning horizonof T periodsThe product is perishable with two-period shelflifetime fresh in the first period and nonfresh in the nextperiod At the beginning of each period selling the retailerreviews the current inventories and decides simultaneouslyon an order quantity for new FAP as well as charging pricesfor both new delivery and old inventory During the salecustomers arrive at the store or market and make purchasechoices based on FAPs attributes of quality and price whichcauses a drop in the inventory level The demand for thecurrent inventory depends on the newly charged pricesand it is always met to the maximum extent with on-handinventories At the end of the period the leftover inventorywith one period lifetime is carried forward to next periodbecoming old products to sell while the unsold products withzero shelf-life time have to be discardedThen the sale movesto the next period and the above processes repeat

For convenience and tractability we assume that thereplenishment order placed by the retailer arrives immedi-ately before the demand unfolds that is to say replenishmentis instantaneous with zero lead time Meanwhile there couldhave chance to replenish fresh products during the sale fromsecond market since in retailing practice backorder policy isprominent which means unsatisfied demands of all productsduring selling are to be backlogged And there is neither

fixed ordering cost nor constraints on the supply capacityDemand at each period is uncertain and independent acrossall periods generally decreasing on current prices In linewith Petruzzi amp Dada [28] and Chen amp Simchi-Levi [32] wetake a form of a function of prices charged at that periodmultiplies a stochastic variable plus a random disturbanceNevertheless customers are assumed to be homogeneousand decide purchasing based on current prices and productattributes alone instead of acting strategically by adjustingtheir buying behavior in response to the firmrsquos pricing policyIn the whole horizon planning although the retailer getsrevenue from selling new and old products he bears somecosts incurred in the operations such as order decisionfor wholesale each order incurs a variable cost c for pernew FAP And three more other costs are incurred leftoverinventory carrying costs over from one period to the nextbacklogged costs for unmet demand fromon-hand inventoryand disposal costs of leftovers with zero shelf-life time Theobjective is to dynamically determine ordering and pricingdecisions in all periods so as to maximize the total expecteddiscounted profit over the whole planning horizon

32 Demand Model During sales the new and old productscompete among customers by combination of their attributesand price And each customer selects his preferred choicebased on his utility from purchasing a unit of product newor old which is given by 119880119894 = 120572120579119894 minus 119901119894 + 120585119894 (119894 = 1 2)where i=1 for the fresh product and i=2 for the nonfreshproduct respectively In the formula 120579119894 and p119894 are defined asthe average perceptive value and the charged price on producti respectively while 120585119894 is a certain customerrsquos deviation from120579119894 which is influenced by unobservable characteristics Thecoefficient 120572 is denoted as a customerrsquos quality sensitivityNote that all consumers are statistically homogeneous andhave the same sensitivity to price Generally any customerwho visits the retailer has three choices select one unit of newor old FAP or do nothing Here we set a virtual product andlet i=3 with price zero for denoting no purchase choice Let1205853 be a certain customerrsquos deviation from 1205793 Obviously 1205853= 0and 1199013= 0 Therefore under the assumption of rationality acustomer will always purchase one product with the highestutility among these three options that is a customer choosesto buy product i where 119894 = argmax119896=123119880119896 = 120572120579119896minus119901119896+120585119896Suppose random variable 120585119894 be iid distributed with a doubleexponential distribution that is Pr (120585119894 le 120575) = 119890minus119890minus120575 120575 isin(minusinfin +infin) with mean zero and variance 120576120575 See Guadagniand Little [62] and Anderson et al [63] Define 120573119894(1199011 1199012) tobe the probability that an arriving customer in a period selectsfresh or nonfresh agriproduct in response to price (1199011 1199012)It can be shown that a consumerrsquos choice probability has asimple form

120573119894 (1199011 1199012) = exp (120572120579119894 minus 119901119894)1 + sum119896=12 exp (120572120579119896 minus 119901119896) 119894 = 1 2 (1)

which is anMNLmodel often used to study a set of productsdifferentiated in either quality or style a term coined in theliterature of MNL models (see [63]) Let (1199011119905 1199012119905) be thecharged prices at any period 119905 isin 1 2 119879 On the basis

Mathematical Problems in Engineering 5

of analysis above we denote 120573119894119905(1199011119905 1199012119905) the market share ofproduct i at a certain period t which is similar to equation (1)and expressed as follows

120573119894119905 (1199011119905 1199012119905) = exp (120572120579119894 minus 119901119894119905)1 + sum119896=12 exp (120572120579119896 minus 119901119896119905) 119894 = 1 2 (2)

Note that the total market share sum119894=12 120573119894119905(1199011119905 1199012119905) le 1Denote 120596119905 to be the amount of arriving customers to buy

the products at period t which is generally uncertain andindependently distributed with finitemean of 120583119905 and variance120591t across all time periods Note that the expected demandof new or old product is proportional to the market share120573119894119905(1199011119905 1199012119905) So provided with price (1199011119905 1199012119905) as well as ademand disturbance on product i at period t 120576119894119905 we can writeout the demand function of product i denoted as 119863119894119905 belowan additive form which is commonly used in the literature(see eg [28 32])

119863119894119905 = 120573119894119905 (1199011119905 1199012119905) 120596119905 + 120576119894119905 119894 = 1 2 (3)

where 120576119894119905 is assumed to be identically distributed over timewith zero mean and finite variance as well as cdf 119865(sdot) and

pdf 119891(sdot) Denote 119889119894119905(1199011119905 1199012119905) as the expected demand levelthen 119889119894119905(1199011119905 1199012119905) = 120573119894119905(1199011119905 1199012119905)119864(120596119905)+119864(120576119894119905) = 120573119894119905(1199011119905 1199012119905)120583119905The selling price is generally restricted to an interval [119901

119905 119901119905]

and 1199011119905 ge 1199012119905Note that the monotonicity of the expected demand

function implies a one-to-one correspondence between theselling price and the expected demand And we observe thatif we use the inverse of the market share function to expressthe price variables as a function of the market share thenthe revenue function becomes jointly concave under somecertain condition allowing us to analyze the structure ofmodel Accordingly we have the following

Proposition 1 Under the MNL market share model the pricefunction at period t is

119901119894119905 = 119875119894119905 (1205731119905 1205732119905) = 120572120579119894 + ln((1 minus sum119894 120573119894119905)120573119894119905 ) 119894 = 1 2

(4)

e corresponding market share space Ω119905 = (1205731119905 1205732119905) 119901119905 le1198752119905(1205731119905 1205732119905) le 1198751119905(1205731119905 1205732119905) le 119901119905 119894 = 1 2 can be expressed as

Ω119905 =

(1205731119905 1205732119905)

exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) le 1205731119905 leexp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905)sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905) le 1205731119905 + 1205732119905 lesum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)

exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905) le 1205732119905 leexp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)

(5)

which is convex and compact

Proof Equation (4) comes from (2) by transforming andtaking ln operation on both sides The range is derived fromthe fact that the prices should be nonnegative Consider anyproduct new or old FAP If we charge the lowest price 119901

119905

for one product and the highest price 119901119905 for the other thenthis product should occupy the highest possiblemarket share(sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))33 Dynamic Programming Formulation Based on the de-scription aforementioned we define some other parameterswith respect to period t below

119868119905 = initial inventory level of old FAP before orderingat the beginning of period t119876119905 = order quantity of new FAP at the beginning ofperiod t119888119905 = unit procurement cost of new FAPℎ1119905 = unit holding cost for the excess inventory

ℎ2119905 = unit disposal cost of remaining inventory withzero shelf-life time119897119894119905 = backordered cost for unit shortage of new andold FAP 119894 = 1 2120574 = profit discount factor 120574 isin (0 1]

Here we do not factor dynamic demand substitution thatthe customer may buy the other product type when hispreferred one is out of stock At the beginning of period 119905 119905 isin1 2 119879 the retailer reviews the current inventory 119868119905 of oldFAPwhich comes from the unsold inventory of new productsat the previous period and places replenishment order 119876119905 fornew FAP as well as setting product prices 1199011119905amp1199012119905 for newand old FAPs respectively Obviously under zero lead timeassumption the inventory of new FAPs is instantaneouslyreplenished to be 119876119905 After demand realization and theretailer satisfies the demand to the maximum extent theremaining inventory levels of the new and old products are(119876119905 minus 1198631119905)+ and (119868119905 minus 1198632119905)+ respectively The (119876119905 minus 1198631119905)+of leftover that remains one period of shelf-life time will becarried to next period for sale as old FAP incurring holding

6 Mathematical Problems in Engineering

cost and becoming the initial inventory of nonfresh product119868119905+1 at period t+1 while the (119868119905 minus 1198632119905)+ of old products willbe disposed of on account of zero shelf-life time incurringdisposal cost Otherwise the relevant backordered amountof new FAPs and old FAPs for period t is (119876119905 minus 1198631119905)minus and(119868119905minus1198632119905)minus respectively whendemand is greater than on-handinventories where 119909+ = max(0 119909) and 119909minus = max(0 minus119909)

At period t the retailer revenue 119905((1199011119905 1199012119905)) can becalculated as follows

119905 ((1199011119905 1199012119905)) = (1199011119905 1199012119905) (1198631119905 1198632119905)119879 (6)

Substitute the price function (5) into (6) we can obtain theexpected revenue 119877119905((1205731t 1205732119905))119877119905 ((1205731t 1205732119905)) = 119864 (119905 (1199011119905 1199012119905))= sum119894=12

(120572120579119894 + ln((1 minus sum119894 120573119894119905)120573119894119905 ))120573it (1199011119905 1199012119905) 120583119905 (7)

After demand realization at period t somemore costs incursbackordered cost of 1198971119905(119876119905 minus1198631119905)minus or holding cost of ℎ1119905(119876119905 minus1198631119905)+ for new FAPs and backlogged cost of 1198972119905(119868119905 minus 1198632119905)minus ordisposal cost of ℎ2119905(119868119905 minus1198632119905)+ for old FAPs totally denoted as119905(119876119905 (1205731119905 1205732119905))119905 (119876119905 (1205731119905 1205732119905)) = (ℎ1119905 ℎ2119905) sdot ((119876119905 minus 1205731t120596119905 minus 1205761119905)

+

(119868119905 minus 1205732t120596119905 minus 1205762119905)+)+ (1198971119905 1198971119905)sdot ((119876119905 minus 1205731t120596119905 minus 1205761119905)minus(119868119905 minus 1205732t120596119905 minus 1205762119905)minus)

(8)

And the corresponding expected function is 119867119905(119876119905 (12057311199051205732119905)) = 119864(119905(119876119905 (1205731119905 1205732119905)))Let 119866119905(119876119905 (1205731119905 1205732119905)) be the profit of period t we have

119866119905 (119876119905 (1205731119905 1205732119905)) = 119905 (119876119905 (1205731119905 1205732119905))minus 119905 (119876119905 (1205731119905 1205732119905)) minus 119888119905119876119905 (9)

Accordingly the expected profit function is as below

119866119905 (119876119905 (1205731119905 1205732119905)) = 119864119866119905 (119876119905 (1205731119905 1205732119905))= 119877119905 (1205731119905 1205732119905) minus 119867119905 (119876119905 (1205731119905 1205732119905))minus 119888119905119876119905

(10)

Denote 120587119905(119868119905) to be the maximum total expected profit-to-go function from period t to T in state 119868119905 witha discount factor 120574 We have the recursive equation120587119905(119868119905) = max119876119905ge0(1205731t 1205732119905)isinΩ119905119866119905(119876119905 (1205731119905 1205732119905)) + 120574119864120587119905+1(119868119905+1)With transfer function 119868119905+1 = (119876119905 minus 1205731119905120596119905)+ by definingΦ119905(119876119905 (1205731119905 1205732119905))

Φ119905 (119876119905 (1205731119905 1205732119905)) = 119866119905 (119876119905 (1205731119905 1205732119905))+ 120574119864120587119905+1 (119868119905+1) (11)

we have the following dynamic programming model

120587119905 (119868119905) = max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731119905 1205732119905)) (12)

119904119905 120587119879+1 = minusℎ2119879+1119868119879+1 (13)

where (13) is the boundary condition since all leftovers will bedisposed of at the end of horizon planningThat is to say thereare no second markets where some unexpired inventory canbe cleared with some salvage value On the contrary the firmneeds to pay on deposing expired inventories even unexpiredleftovers Our analysis can be readily extended to address thissituation

4 Analysis of Model Structure

To analyze the structure of the optimal policy we needto know the concavity property of Φ119905(119876119905 (1205731t 1205732119905)) whichis strictly relative to revenue function 119877119905((1205731t 1205732119905)) andinventory cost function119867119905(119876119905 (1205731t 1205732119905))Lemma 2 119877119905((1205731t 1205732119905)) is continuously twice differentiableand jointly concave in (1205731t 1205732119905)Proof The continuity and twice differentiability of 119877119905((1205731t1205732119905)) is quite intuitive since it is expressed by some elementaryoperations of 1205731t and 1205732119905 To show the concavity we needto prove the negative semidefinite property of its HessianMatrix Taking the second-order derivative with respect to1205731t yields 120597211987711990512059712057321t = minus(1(1 minus 1205731119905 minus 1205732119905) + 11205731119905 + 1(1 minus1205731119905 minus 1205732119905)2)120583119905 le 0 Thus 119877119905 is concave on 1205731t Taking thesecond-order derivative with respect to 1205732t yields 120597211987711990512059712057322t =minus(1(1 minus 1205731119905 minus 1205732119905) + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 le 0which implies the concavity of 119877119905 on 1205732t And taking thecross-partial derivative with respect to 1205731t and 1205732t yields1205972119877119905(1205971205731t1205971205732t) = minus(1(1 minus 1205731119905 minus 1205732119905) + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905Pairing up the above terms leads to Hessian Matrix

119867essian (119877119905) = [[[[[

120597211987711990512059712057321t 1205972119877119905(1205971205731t1205971205732t)1205972119877119905(1205971205732t1205971205731t) 120597211987711990512059712057322t]]]]]

Mathematical Problems in Engineering 7

= [[[[[[minus( 11 minus 1205731119905 minus 1205732119905 + 11205731119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905

]]]]]]

(14)

And the corresponding determinant is as follows

1003816100381610038161003816119867 (119877119905)1003816100381610038161003816 = 120597211987711990512059712057321t sdot 120597211987711990512059712057322t minus ( 1205972119877119905(1205971205731t1205971205732t))

2

= ( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)(11205731119905 + 11205732119905)

+ 112057311199051205732119905 gt 0(15)

which implies that 119867119890119904119904119894119886119899 matrix is seminegative defi-nite On the basis of that 119877119905 is jointly concave in (1205731t1205732119905)Lemma 3 119867119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly convex in (1205731t 1205732119905) and 119876119905Proof We observe that 119905(119876119905 (1205731t 1205732119905)) is continuously andtwice differentiable for any realization of 120596119905 and 120576119894119905 And theexpectation 119867119905(119876119905 (1205731t 1205732119905)) is easily justified to be con-tinuously twice differentiable using the canonical argument(Durrett 2010)

To show the convexity we define some column vectorsexpressed by boldface lowercase letters such as h119905 =(ℎ1119905 ℎ2119905)119879 l119905 = (1198971119905 1198972119905)119879 and 120576119905 = (1205761119905 1205762119905)119879 Then thedemand and cost function can be expressed of vector formas follows

D119905 = (1198631119905 1198632119905) = 120573119905120596119905 + 120576119905 (16)119905 (119876119905 (1205731t 1205732119905)) = 119905 (120573119905 119876119905)= h119905 ((119876119905 119868119905)119879 minusD119905)++ l119905 ((119876119905 119868119905)119879 minusD119905)minus

= h119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)++ l119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)minus

(17)

Define a new vector variable 119906119905 = (119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905 andwe can get 119905(120573119905 119876119905) = 119905(u119905) = h119905(u119905)+ + l119905(u119905)minus which isconvex onu119905 (SundaranmR K 1996) Fix the randompart120596119905to its realization and consider any pair (120573119905 119876119905) and (1205731015840119905 1198761015840119905)with the convexity of 119905(u119905) we have119905 (120572 (120573119905 119876119905) + (1 minus 120572) (1205731015840119905 1198761015840119905)) = 119905 (120572120573119905+ (1 minus 120572)1205731015840119905 120572119876119905 + (1 minus 120572)1198761015840119905)= h119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)

minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)++ l119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)minus= h119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879119905 minus 1205731015840119905120596119905 minus 120576119905))++ l119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879 minus 1205731015840119905120596119905 minus 120576119905))minus = h119905 (120572u119905+ (1 minus 120572)u1015840119905)+ + l119905 (120572u119905 + (1 minus 120572) u1015840119905)minus = 119905 (120572u119905+ (1 minus 120572)u1015840119905) le 120572119905 (u119905) + (1 minus 120572) 119905 (u1015840119905)= 120572119905 (120573119905 (119876119905 119868119905)119879) + (1 minus 120572) 119905 (1205731015840119905 (1198761015840119905 119868119905)119879)

(18)

where 0 le 120572 le 1 Therefore 119905(119876119905 (1205731t 1205732119905)) = 119905(120573119905(119876119905 119868119905)119879) = 119905(u119905) is jointly convex in (1205731t 1205732119905) and 119876119905 sodoes the expectation function119867119905(119876119905 (1205731t 1205732119905))Theorem 4 At any time period t with initial inventory 119868119905the expected revenue function119866119905(119876119905 (1205731t 1205732119905)) is continuouslytwice differentiable and jointly concave in (1205731t 1205732119905) and 119876119905Proof With Lemmas 2 and 3 we can observe that119866119905(119876119905 (1205731t 1205732119905)) is continuously twice differentiable We firstobserve that for any t the three terms of 119866119905(119876119905 (1205731119905 1205732119905)) in(10) are jointly concave The first term is concave in (1205731119905 1205732119905)according to Lemma 2 The second term is jointly concavein (1205731119905 1205732119905) and 119876119905 (see Lemma 3) and the third term islinear in 119876119905 which implies the concavity With the propertythat a nonnegative weighted sum of concave functionsis concave (Stephen Boyd Lieven Vandenberghe 2004)119866119905(119876119905 (1205731119905 1205732119905)) is continuously twice differentiable andjointly concave in (1205731t 1205732119905) and 119876119905

On the basis of the concavity of 119866119905(119876119905 (1205731t 1205732119905)) we canhaveTheorem 5

Theorem 5 Consider any time period t with initial inventory119868119905 the functionΦ119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly concave in (119876119905 (1205731t 1205732119905)) and the function120587119905(119868119905) is concave and nonincreasing in 119868119905Proof By induction first we prove that Φ119879(119876119879 (1205731T 1205732119879))is continuously twice differentiable According to Lemma 2

8 Mathematical Problems in Engineering

119864(119876119879 minus 1205731119879120596119879 minus 120576119879)+ as a part of 119867119905(119876119905 (1205731t 1205732119905)) isthus continuously twice differentiable which implies thatΦ119879(119876119879 (1205731119879 1205732119879)) = 119866119879(119876119879 (1205731119879 1205732119879)) minus 120574ℎ2119879+1119864(119876119879 minus1205731119879120596119879 minus 1205762119879)+ is continuously and twice differentiableAssuming now that Φ119905(119876119905 (1205731119905 1205732119905)) is continuouslytwice differentiable for 119905 = 1 2 119879 we prove thatΦ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is also continuously twice differen-tiable according to Theorem 4 Let Λ 119879(119876119879 1205731119879) = 119864(119876119879 minus1205731119879120596119879minus1205762119879)+ = ∬1205731119879119909+119910le119876119879(119876119879minus1205731119879119909minus119910)119891120596(119909)119891120576(119910)119889119909119889119910where 119891120596(119909) and 119891120576(119910) is the pdf of random variable120596119879 and 120576119879 respectively which are generally assumed to beindependent For any realization of 1205762119879 such as 1205762119879 taking thesecond-order derivative ofΛ 119879(119876119879 1205731119879)with respective to119876119879yields 1205972Λ 119879(119876119879 1205731119879)1205971198762119879 = (11205731119879)119891((119876119879 minus 1205762119879)1205731119879) gt 0which implies that Λ 119879(119876119879 1205731119879) is convex in 119876119879 Takingthe second-order derivative with respect to 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)12059712057321119879 = (119876211987912057331119879)119891(1198761198791205731119879) gt 0 whichimplies the concavity of Λ 119879(119876119879 1205731119879) in 1205731119879 And takingthe cross-partial derivative with respect to 119876119879 and 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)1205971198761198791205971205731119879 = minus((119876119879 minus 1205762119879)12057321119879)119891((119876119879 minus1205762119879)1205731119879) Pairing up the above three derivative terms leadsto Hessian Matrix

119867(Λ 119879) = [[[[[

1205972Λ 11987912059711987621198791205972Λ 11987912059711987611987912059712057311198791205972Λ 1198791205971205731119879120597119876119879 1205972Λ 11987912059712057321119879

]]]]]

= [[[[[

11205731119879119891(119876119879 minus 12057621198791205731119879 ) minus119876119879 minus 120576211987912057321119879

119891(119876119879 minus 12057621198791205731119879 )minus119876119879 minus 120576211987912057321119879 119891(119876119879 minus 12057621198791205731119879 ) (119876119879 minus 1205762119879)212057331119879 119891(119876119879 minus 12057621198791205731119879 )

]]]]](19)

And the corresponding determinant 119867(Λ 119879) = 0 whichimplies matrix 119867119890119904119904119894119886119899 is semipositive definite On thebasis of analysis above we can conclude that Λ 119879(119876119879 1205731119879)is jointly convex in (119876119879 1205731119879) Meanwhile the last termminus120574ℎ2119879+1119864(119876119879 minus 1205731119879120596119879 minus 1205762119879)+ is obviously justified to bejointly concave in (119876119879 1205731119879) According to the concavityproperty (Stephen Boyd Lieven Vandenberghe 2004)Φ119879(119876119879 (1205731T 1205732119879)) is jointly concave in (1205731t 1205732119905) and 119876119905

120587119879(119868119879) is easily verified to be concave as well and it isobviously nonincreasing

Assume now that Φ119905(119876119905 (1205731t 1205732119905)) is jointly concave forsome 119905 = 1 2 119879 minus 1 and that 120587119905(119868119905) is also concaveand nonincreasing Then Φ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is jointlyconcave joint concavity of the first term of (10) is verified forthe case 119905 = 119879 above For any pair of points (119876119905 1205731119905) and(1198761015840119905 12057310158401119905) with given value of 120596119905 we have120587119905+1 ((120572119876119905 + (1 minus 120572)1198761015840119905)119905 minus (1205721205731119905 + (1 minus 120572) 12057310158401119905) 120596119905minus (1205721205761119905 + (1 minus 120572) 1205761119905)) = 120587119905+1 (120572 (119876119905 minus 1205731119905120596119905 minus 1205761119905)+ (1 minus 120572) (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)) ge 120572120587119905+1 (119876119905 minus 1205731119905120596119905minus 1205761119905) + (1 minus 120572) 120587119905+1 (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)

(20)

by the concavity of 120587119905+1(sdot) which implies that 119864120587119905+1(119876119905 minus1205731119905120596119905 minus 1205761119905) is jointly concave in 119876119905 and 1205731119905 With aboveargument the desired result holds

5 Optimal Policy

Following the structure property of model we can obtain thefollowing result

Theorem 6 Consider any period t with the initial inventory119868119905 there is a unique maximizer (119876lowast119905 (120573lowast1119905 120573lowast2119905)) to maximizethe constrained expected periodical profit function

(119876lowast119905 (120573lowast1119905 120573lowast2119905)) isin arg max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731t 1205732119905)) (21)

Proof It can be easily proved on the basis of Theorem 5

On the basis of the dynamic programming analysisincluding the concavity property we can construct an opti-mization problem for any period t with constrained condi-tions as follows

119872119894119899119894119898119894119911119890 minus Φ119905 (119876119905 (1205731t 1205732119905))

119904119905

(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905))(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) 119905) le 0(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905))(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)) le 0(1205731t + 1205732119905 minus sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905))(1205731t + 1205732119905 minus

sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)) le 0minus119876119905 le 0

(22)

Mathematical Problems in Engineering 9

The Lagrangian and the Karush-Kuhn-Tucker(KTT) condi-tions provide an efficient solution approach to this optimiza-tion problemDenote119860119905 = (exp(1205721205791minus119901119905))(1+exp(1205721205791minus119901119905))119860119905 = (exp(1205721205791 minus 119901119905))(1 + exp(1205721205791 minus 119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119862119905 = (sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))

and 119862119905 = (sum119894=12 exp(120572120579119894 minus119901119905))(1 +sum119894=12 exp(120572120579119894 minus119901119905)) Weconstruct for each period t theLagrangian of the optimizationproblem as

119871 119905 (119876119905 (1205731t 1205732119905) (1205821199051 1205821199052 1205821199053 1205821199054))= minusΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905)+ 1205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905)+ 1205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) minus 1205821199054119876119905

(23)

where the Lagrange multipliers are 120582119905119894 ge 0 119894 = 1 2 3 4 Let(119876lowast119905 (120573lowast1119905 120573lowast2119905)) be the optimizing solution we can constructthe KKT conditions as follows

minus nablaΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (0 21205731t minus 119860119905 minus 119860119905 0)+ 1205821199052 (0 0 21205732t minus 119861119905 minus 119861119905) + 1205821199053 (0 21205731t + 21205732119905minus 119862119905 minus 119862119905 21205731t + 21205732119905 minus 119862119905 minus 119862) minus 1205821199054 (1 0 0) = 0

1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905) = 01205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905) = 01205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) = 01205821199054119876119905 = 0

(24)

where the first equation represents FOC of Lagrangianand the latter four equations represent the conditions ofcomplementary slackness

With solution of theKKT conditions (119876lowast119905 (120573lowast1119905 120573lowast2119905)) and(4) we can get the optimization pricing and ordering policyof (119876lowast119905 (119901lowast1119905 119901lowast2119905)) at period t where (119901lowast1119905 119901lowast2119905) is as follows119901lowast119894119905 = 119875119894119905 (120573lowast1119905 120573lowast2119905) = 120572120579119894 + ln((1 minus sum119894 120573lowast119894119905)120573lowast119894119905 )

119894 = 1 2(25)

Note that (119876lowast119905 (120573lowast1119905 120573lowast2119905)) depends on state variable 119868119905 andparameters of (ℎ1119905 ℎ2119905) and (1198971119905 1198972119905) as well as 120583119905 which

implies that the retailer decides the optimal policy on thebasis of these parameters at every period t With back-ward induction we get the optimal strategy a sequence of[(119876lowast1 (119901lowast11 119901lowast21)) (119876lowast2 (119901lowast12 119901lowast22)) (119876lowast119879 (119901lowast1119879 119901lowast2119879))]6 Numerical Experiment

We now conduct a numerical example to illustrate our resultsand select the parameters as follows The demand followsthe multiplicative form with iid terms 120596119905 sim U(0 100)for all periods We assume (1205791 1205792)119879 = (10 5)119879 sincefresh product has a higher perceptive value than nonfreshones The cost structure is symmetric h119905 = (05 05)119879l119905 = (45 45) and 119888119905 = 10 For simplicity to compute theoptimal prices we consider a finite horizon with 119879 = 5the discount rate is 120574 = 09 and the initial inventory level119868119905 = 0 Using Matlab to program the dynamic model andget the following results with KKT condition method wecan obtain a sequence of order quantity and optimal pricessuch as [(78 (148 97)) (68 (126 86)) (92 (175 114))(43 (109 62)) (52 (115 102))] at which the maximumprofit 120587lowast = 4 2356 occurs

In Table 1 we compare the performance betweendynamic pricing policy and a general policy under oneperiod lifetime setting where all products are sold at aregular price at any period of time Similarly we assume(1205791 1205792)119879 = (10 5)119879 in dynamic pricing policy while(1205791 1205792)119879 = (10 10)119879 in general policy Note that in generalsetting the optimal pricing and ordering policy can beobtained by way of general dynamic programming methodThree examples of different finite horizon time (119879 = 51020)are conducted with the same cost structure and discount ratementioned above Accordingly the optimal solutions suchas pricing and order quantity as well as dynamic inventoryare shown in Table 1 where superscript g and d representregular policy and dynamic policy respectively

A summary of interesting observations is as follows Thedynamic policy is obviously better than general pricing andinventory policy with which the retailerrsquos profit increasesby 15-20 Further the demand is also matched to a morecertain extent by the supply since the order quantity is almostmore than that under general policy while the Based on theresults we can conclude that dynamic pricing and inventorypolicy is beneficial for retailerrsquos revenue management inselling fresh agriculture products Furthermore from theperspective of agriculture industrial development it also canmaximally reduce the operating costs such as holding costsor lost costs since the optimal match between productssupply and demand

7 Concluding Remark

In this paper we address a joint pricing and inventorycontrol problem for stochastic fresh agriproduct inventorysystems under consumer choice over a finite horizon in thebacklogging case In the model the products are perishablealong time and the customers are sensitive to the ages ofinventories which result in consumer choice behavior based

10 Mathematical Problems in Engineering

Table 1 The optimal solution under dynamic pricing and regular pricing

Periods T Order quantity Pricing Inventory 120587d 120587g (120587119889 minus 120587119892)120587119892Qd Qg (1199011198891 1199011198892) 119901119892 (1198681198891 1198681198892 ) 119868119892

5

78 72 (14897) 125 (490) 9

42356 35782 18468 65 (12686) 104 (312) 892 80 (175114) 128 (5611) 1343 45 (10962) 81 (220) 352 45 (11582) 93 (121) 5

10

81 70 (156107) 122 (520) 10

83438 69844 195

72 61 (13684) 103 (386) 783 58 (165104) 132 (569) 876 59 (11978) 85 (322) 470 73 (12592) 81 (346) 456 41 (16598) 146 (301) 968 59 (12288) 95 (312) 865 47 (154104) 138 (339) 758 56 (12473) 92 (230) 349 39 (12864) 97 (191) 2

20

90 86 (15598) 131 (540) 16

175328 149632 172

72 65 (14385) 104 (366) 883 57 (165101) 111 (409) 966 75 (13276) 86 (314) 572 54 (12995) 92 (346) 456 51 (16392) 132 (353) 768 62 (13287) 98 (313) 465 45 (151122) 136 (335) 457 69 (13879) 97 (280) 348 46 (15592) 102 (211) 281 72 (13871) 98 (430) 270 58 (13992) 104 (386) 778 50 (15468) 119 (492) 571 48 (13288) 95 (333) 061 63 (14582) 84 (356) 856 52 (16578) 146 (373) 1076 48 (11388) 99 (323) 666 64 (14594) 128 (395) 852 49 (13778) 98 (230) 350 46 (14569) 91 (220) 1

on product freshness By using MNL model to describeconsumer discrete choice we construct a stochastic anddynamic inventory model facing strategic customer behaviorwith joint ordering and discounting decisions We are ableto characterize the firmrsquos optimal policy with the use ofa transformation technique and employing the concept ofthe concavity and finally develop a Lagrangian and theKarush-Kuhn-Tucker (KTT) conditions to provide an efficientsolution approach to this optimization problem

A limitation in our model is that we assume that thelifetime of the product is only two periods Although this istrue for some fresh agriproducts in themarket there are otherproducts that are perishable but yet have longer lifetimes Toextend ourmodel to the products withmore than two periods

lifetimes the state space has to be increased more to specifythe amounts of inventory that are going to expire at differentpoints of time in the futureThis definitelymakes the problemquite hard Another issue is to incorporate an operatingcost for discounting pricing practice which however willintroduce additional technical complexity to the dynamicprogram There are several future research directions Oneis to extend the model to the environment where all thedemand can be lost or only non-FAP demand can be lostwhile the fresh FAP can be backlogged instead Another is toconsider the case where the product has a fixed finite lifetimeof exactly 119899(119899 ge 3) periods or random lifetime Finally itwould be interesting to account for stock-out based substitu-tions

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

of analysis above we denote 120573119894119905(1199011119905 1199012119905) the market share ofproduct i at a certain period t which is similar to equation (1)and expressed as follows

120573119894119905 (1199011119905 1199012119905) = exp (120572120579119894 minus 119901119894119905)1 + sum119896=12 exp (120572120579119896 minus 119901119896119905) 119894 = 1 2 (2)

Note that the total market share sum119894=12 120573119894119905(1199011119905 1199012119905) le 1Denote 120596119905 to be the amount of arriving customers to buy

the products at period t which is generally uncertain andindependently distributed with finitemean of 120583119905 and variance120591t across all time periods Note that the expected demandof new or old product is proportional to the market share120573119894119905(1199011119905 1199012119905) So provided with price (1199011119905 1199012119905) as well as ademand disturbance on product i at period t 120576119894119905 we can writeout the demand function of product i denoted as 119863119894119905 belowan additive form which is commonly used in the literature(see eg [28 32])

119863119894119905 = 120573119894119905 (1199011119905 1199012119905) 120596119905 + 120576119894119905 119894 = 1 2 (3)

where 120576119894119905 is assumed to be identically distributed over timewith zero mean and finite variance as well as cdf 119865(sdot) and

pdf 119891(sdot) Denote 119889119894119905(1199011119905 1199012119905) as the expected demand levelthen 119889119894119905(1199011119905 1199012119905) = 120573119894119905(1199011119905 1199012119905)119864(120596119905)+119864(120576119894119905) = 120573119894119905(1199011119905 1199012119905)120583119905The selling price is generally restricted to an interval [119901

119905 119901119905]

and 1199011119905 ge 1199012119905Note that the monotonicity of the expected demand

function implies a one-to-one correspondence between theselling price and the expected demand And we observe thatif we use the inverse of the market share function to expressthe price variables as a function of the market share thenthe revenue function becomes jointly concave under somecertain condition allowing us to analyze the structure ofmodel Accordingly we have the following

Proposition 1 Under the MNL market share model the pricefunction at period t is

119901119894119905 = 119875119894119905 (1205731119905 1205732119905) = 120572120579119894 + ln((1 minus sum119894 120573119894119905)120573119894119905 ) 119894 = 1 2

(4)

e corresponding market share space Ω119905 = (1205731119905 1205732119905) 119901119905 le1198752119905(1205731119905 1205732119905) le 1198751119905(1205731119905 1205732119905) le 119901119905 119894 = 1 2 can be expressed as

Ω119905 =

(1205731119905 1205732119905)

exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) le 1205731119905 leexp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905)sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905) le 1205731119905 + 1205732119905 lesum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)

exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905) le 1205732119905 leexp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)

(5)

which is convex and compact

Proof Equation (4) comes from (2) by transforming andtaking ln operation on both sides The range is derived fromthe fact that the prices should be nonnegative Consider anyproduct new or old FAP If we charge the lowest price 119901

119905

for one product and the highest price 119901119905 for the other thenthis product should occupy the highest possiblemarket share(sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))33 Dynamic Programming Formulation Based on the de-scription aforementioned we define some other parameterswith respect to period t below

119868119905 = initial inventory level of old FAP before orderingat the beginning of period t119876119905 = order quantity of new FAP at the beginning ofperiod t119888119905 = unit procurement cost of new FAPℎ1119905 = unit holding cost for the excess inventory

ℎ2119905 = unit disposal cost of remaining inventory withzero shelf-life time119897119894119905 = backordered cost for unit shortage of new andold FAP 119894 = 1 2120574 = profit discount factor 120574 isin (0 1]

Here we do not factor dynamic demand substitution thatthe customer may buy the other product type when hispreferred one is out of stock At the beginning of period 119905 119905 isin1 2 119879 the retailer reviews the current inventory 119868119905 of oldFAPwhich comes from the unsold inventory of new productsat the previous period and places replenishment order 119876119905 fornew FAP as well as setting product prices 1199011119905amp1199012119905 for newand old FAPs respectively Obviously under zero lead timeassumption the inventory of new FAPs is instantaneouslyreplenished to be 119876119905 After demand realization and theretailer satisfies the demand to the maximum extent theremaining inventory levels of the new and old products are(119876119905 minus 1198631119905)+ and (119868119905 minus 1198632119905)+ respectively The (119876119905 minus 1198631119905)+of leftover that remains one period of shelf-life time will becarried to next period for sale as old FAP incurring holding

6 Mathematical Problems in Engineering

cost and becoming the initial inventory of nonfresh product119868119905+1 at period t+1 while the (119868119905 minus 1198632119905)+ of old products willbe disposed of on account of zero shelf-life time incurringdisposal cost Otherwise the relevant backordered amountof new FAPs and old FAPs for period t is (119876119905 minus 1198631119905)minus and(119868119905minus1198632119905)minus respectively whendemand is greater than on-handinventories where 119909+ = max(0 119909) and 119909minus = max(0 minus119909)

At period t the retailer revenue 119905((1199011119905 1199012119905)) can becalculated as follows

119905 ((1199011119905 1199012119905)) = (1199011119905 1199012119905) (1198631119905 1198632119905)119879 (6)

Substitute the price function (5) into (6) we can obtain theexpected revenue 119877119905((1205731t 1205732119905))119877119905 ((1205731t 1205732119905)) = 119864 (119905 (1199011119905 1199012119905))= sum119894=12

(120572120579119894 + ln((1 minus sum119894 120573119894119905)120573119894119905 ))120573it (1199011119905 1199012119905) 120583119905 (7)

After demand realization at period t somemore costs incursbackordered cost of 1198971119905(119876119905 minus1198631119905)minus or holding cost of ℎ1119905(119876119905 minus1198631119905)+ for new FAPs and backlogged cost of 1198972119905(119868119905 minus 1198632119905)minus ordisposal cost of ℎ2119905(119868119905 minus1198632119905)+ for old FAPs totally denoted as119905(119876119905 (1205731119905 1205732119905))119905 (119876119905 (1205731119905 1205732119905)) = (ℎ1119905 ℎ2119905) sdot ((119876119905 minus 1205731t120596119905 minus 1205761119905)

+

(119868119905 minus 1205732t120596119905 minus 1205762119905)+)+ (1198971119905 1198971119905)sdot ((119876119905 minus 1205731t120596119905 minus 1205761119905)minus(119868119905 minus 1205732t120596119905 minus 1205762119905)minus)

(8)

And the corresponding expected function is 119867119905(119876119905 (12057311199051205732119905)) = 119864(119905(119876119905 (1205731119905 1205732119905)))Let 119866119905(119876119905 (1205731119905 1205732119905)) be the profit of period t we have

119866119905 (119876119905 (1205731119905 1205732119905)) = 119905 (119876119905 (1205731119905 1205732119905))minus 119905 (119876119905 (1205731119905 1205732119905)) minus 119888119905119876119905 (9)

Accordingly the expected profit function is as below

119866119905 (119876119905 (1205731119905 1205732119905)) = 119864119866119905 (119876119905 (1205731119905 1205732119905))= 119877119905 (1205731119905 1205732119905) minus 119867119905 (119876119905 (1205731119905 1205732119905))minus 119888119905119876119905

(10)

Denote 120587119905(119868119905) to be the maximum total expected profit-to-go function from period t to T in state 119868119905 witha discount factor 120574 We have the recursive equation120587119905(119868119905) = max119876119905ge0(1205731t 1205732119905)isinΩ119905119866119905(119876119905 (1205731119905 1205732119905)) + 120574119864120587119905+1(119868119905+1)With transfer function 119868119905+1 = (119876119905 minus 1205731119905120596119905)+ by definingΦ119905(119876119905 (1205731119905 1205732119905))

Φ119905 (119876119905 (1205731119905 1205732119905)) = 119866119905 (119876119905 (1205731119905 1205732119905))+ 120574119864120587119905+1 (119868119905+1) (11)

we have the following dynamic programming model

120587119905 (119868119905) = max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731119905 1205732119905)) (12)

119904119905 120587119879+1 = minusℎ2119879+1119868119879+1 (13)

where (13) is the boundary condition since all leftovers will bedisposed of at the end of horizon planningThat is to say thereare no second markets where some unexpired inventory canbe cleared with some salvage value On the contrary the firmneeds to pay on deposing expired inventories even unexpiredleftovers Our analysis can be readily extended to address thissituation

4 Analysis of Model Structure

To analyze the structure of the optimal policy we needto know the concavity property of Φ119905(119876119905 (1205731t 1205732119905)) whichis strictly relative to revenue function 119877119905((1205731t 1205732119905)) andinventory cost function119867119905(119876119905 (1205731t 1205732119905))Lemma 2 119877119905((1205731t 1205732119905)) is continuously twice differentiableand jointly concave in (1205731t 1205732119905)Proof The continuity and twice differentiability of 119877119905((1205731t1205732119905)) is quite intuitive since it is expressed by some elementaryoperations of 1205731t and 1205732119905 To show the concavity we needto prove the negative semidefinite property of its HessianMatrix Taking the second-order derivative with respect to1205731t yields 120597211987711990512059712057321t = minus(1(1 minus 1205731119905 minus 1205732119905) + 11205731119905 + 1(1 minus1205731119905 minus 1205732119905)2)120583119905 le 0 Thus 119877119905 is concave on 1205731t Taking thesecond-order derivative with respect to 1205732t yields 120597211987711990512059712057322t =minus(1(1 minus 1205731119905 minus 1205732119905) + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 le 0which implies the concavity of 119877119905 on 1205732t And taking thecross-partial derivative with respect to 1205731t and 1205732t yields1205972119877119905(1205971205731t1205971205732t) = minus(1(1 minus 1205731119905 minus 1205732119905) + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905Pairing up the above terms leads to Hessian Matrix

119867essian (119877119905) = [[[[[

120597211987711990512059712057321t 1205972119877119905(1205971205731t1205971205732t)1205972119877119905(1205971205732t1205971205731t) 120597211987711990512059712057322t]]]]]

Mathematical Problems in Engineering 7

= [[[[[[minus( 11 minus 1205731119905 minus 1205732119905 + 11205731119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905

]]]]]]

(14)

And the corresponding determinant is as follows

1003816100381610038161003816119867 (119877119905)1003816100381610038161003816 = 120597211987711990512059712057321t sdot 120597211987711990512059712057322t minus ( 1205972119877119905(1205971205731t1205971205732t))

2

= ( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)(11205731119905 + 11205732119905)

+ 112057311199051205732119905 gt 0(15)

which implies that 119867119890119904119904119894119886119899 matrix is seminegative defi-nite On the basis of that 119877119905 is jointly concave in (1205731t1205732119905)Lemma 3 119867119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly convex in (1205731t 1205732119905) and 119876119905Proof We observe that 119905(119876119905 (1205731t 1205732119905)) is continuously andtwice differentiable for any realization of 120596119905 and 120576119894119905 And theexpectation 119867119905(119876119905 (1205731t 1205732119905)) is easily justified to be con-tinuously twice differentiable using the canonical argument(Durrett 2010)

To show the convexity we define some column vectorsexpressed by boldface lowercase letters such as h119905 =(ℎ1119905 ℎ2119905)119879 l119905 = (1198971119905 1198972119905)119879 and 120576119905 = (1205761119905 1205762119905)119879 Then thedemand and cost function can be expressed of vector formas follows

D119905 = (1198631119905 1198632119905) = 120573119905120596119905 + 120576119905 (16)119905 (119876119905 (1205731t 1205732119905)) = 119905 (120573119905 119876119905)= h119905 ((119876119905 119868119905)119879 minusD119905)++ l119905 ((119876119905 119868119905)119879 minusD119905)minus

= h119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)++ l119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)minus

(17)

Define a new vector variable 119906119905 = (119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905 andwe can get 119905(120573119905 119876119905) = 119905(u119905) = h119905(u119905)+ + l119905(u119905)minus which isconvex onu119905 (SundaranmR K 1996) Fix the randompart120596119905to its realization and consider any pair (120573119905 119876119905) and (1205731015840119905 1198761015840119905)with the convexity of 119905(u119905) we have119905 (120572 (120573119905 119876119905) + (1 minus 120572) (1205731015840119905 1198761015840119905)) = 119905 (120572120573119905+ (1 minus 120572)1205731015840119905 120572119876119905 + (1 minus 120572)1198761015840119905)= h119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)

minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)++ l119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)minus= h119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879119905 minus 1205731015840119905120596119905 minus 120576119905))++ l119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879 minus 1205731015840119905120596119905 minus 120576119905))minus = h119905 (120572u119905+ (1 minus 120572)u1015840119905)+ + l119905 (120572u119905 + (1 minus 120572) u1015840119905)minus = 119905 (120572u119905+ (1 minus 120572)u1015840119905) le 120572119905 (u119905) + (1 minus 120572) 119905 (u1015840119905)= 120572119905 (120573119905 (119876119905 119868119905)119879) + (1 minus 120572) 119905 (1205731015840119905 (1198761015840119905 119868119905)119879)

(18)

where 0 le 120572 le 1 Therefore 119905(119876119905 (1205731t 1205732119905)) = 119905(120573119905(119876119905 119868119905)119879) = 119905(u119905) is jointly convex in (1205731t 1205732119905) and 119876119905 sodoes the expectation function119867119905(119876119905 (1205731t 1205732119905))Theorem 4 At any time period t with initial inventory 119868119905the expected revenue function119866119905(119876119905 (1205731t 1205732119905)) is continuouslytwice differentiable and jointly concave in (1205731t 1205732119905) and 119876119905Proof With Lemmas 2 and 3 we can observe that119866119905(119876119905 (1205731t 1205732119905)) is continuously twice differentiable We firstobserve that for any t the three terms of 119866119905(119876119905 (1205731119905 1205732119905)) in(10) are jointly concave The first term is concave in (1205731119905 1205732119905)according to Lemma 2 The second term is jointly concavein (1205731119905 1205732119905) and 119876119905 (see Lemma 3) and the third term islinear in 119876119905 which implies the concavity With the propertythat a nonnegative weighted sum of concave functionsis concave (Stephen Boyd Lieven Vandenberghe 2004)119866119905(119876119905 (1205731119905 1205732119905)) is continuously twice differentiable andjointly concave in (1205731t 1205732119905) and 119876119905

On the basis of the concavity of 119866119905(119876119905 (1205731t 1205732119905)) we canhaveTheorem 5

Theorem 5 Consider any time period t with initial inventory119868119905 the functionΦ119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly concave in (119876119905 (1205731t 1205732119905)) and the function120587119905(119868119905) is concave and nonincreasing in 119868119905Proof By induction first we prove that Φ119879(119876119879 (1205731T 1205732119879))is continuously twice differentiable According to Lemma 2

8 Mathematical Problems in Engineering

119864(119876119879 minus 1205731119879120596119879 minus 120576119879)+ as a part of 119867119905(119876119905 (1205731t 1205732119905)) isthus continuously twice differentiable which implies thatΦ119879(119876119879 (1205731119879 1205732119879)) = 119866119879(119876119879 (1205731119879 1205732119879)) minus 120574ℎ2119879+1119864(119876119879 minus1205731119879120596119879 minus 1205762119879)+ is continuously and twice differentiableAssuming now that Φ119905(119876119905 (1205731119905 1205732119905)) is continuouslytwice differentiable for 119905 = 1 2 119879 we prove thatΦ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is also continuously twice differen-tiable according to Theorem 4 Let Λ 119879(119876119879 1205731119879) = 119864(119876119879 minus1205731119879120596119879minus1205762119879)+ = ∬1205731119879119909+119910le119876119879(119876119879minus1205731119879119909minus119910)119891120596(119909)119891120576(119910)119889119909119889119910where 119891120596(119909) and 119891120576(119910) is the pdf of random variable120596119879 and 120576119879 respectively which are generally assumed to beindependent For any realization of 1205762119879 such as 1205762119879 taking thesecond-order derivative ofΛ 119879(119876119879 1205731119879)with respective to119876119879yields 1205972Λ 119879(119876119879 1205731119879)1205971198762119879 = (11205731119879)119891((119876119879 minus 1205762119879)1205731119879) gt 0which implies that Λ 119879(119876119879 1205731119879) is convex in 119876119879 Takingthe second-order derivative with respect to 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)12059712057321119879 = (119876211987912057331119879)119891(1198761198791205731119879) gt 0 whichimplies the concavity of Λ 119879(119876119879 1205731119879) in 1205731119879 And takingthe cross-partial derivative with respect to 119876119879 and 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)1205971198761198791205971205731119879 = minus((119876119879 minus 1205762119879)12057321119879)119891((119876119879 minus1205762119879)1205731119879) Pairing up the above three derivative terms leadsto Hessian Matrix

119867(Λ 119879) = [[[[[

1205972Λ 11987912059711987621198791205972Λ 11987912059711987611987912059712057311198791205972Λ 1198791205971205731119879120597119876119879 1205972Λ 11987912059712057321119879

]]]]]

= [[[[[

11205731119879119891(119876119879 minus 12057621198791205731119879 ) minus119876119879 minus 120576211987912057321119879

119891(119876119879 minus 12057621198791205731119879 )minus119876119879 minus 120576211987912057321119879 119891(119876119879 minus 12057621198791205731119879 ) (119876119879 minus 1205762119879)212057331119879 119891(119876119879 minus 12057621198791205731119879 )

]]]]](19)

And the corresponding determinant 119867(Λ 119879) = 0 whichimplies matrix 119867119890119904119904119894119886119899 is semipositive definite On thebasis of analysis above we can conclude that Λ 119879(119876119879 1205731119879)is jointly convex in (119876119879 1205731119879) Meanwhile the last termminus120574ℎ2119879+1119864(119876119879 minus 1205731119879120596119879 minus 1205762119879)+ is obviously justified to bejointly concave in (119876119879 1205731119879) According to the concavityproperty (Stephen Boyd Lieven Vandenberghe 2004)Φ119879(119876119879 (1205731T 1205732119879)) is jointly concave in (1205731t 1205732119905) and 119876119905

120587119879(119868119879) is easily verified to be concave as well and it isobviously nonincreasing

Assume now that Φ119905(119876119905 (1205731t 1205732119905)) is jointly concave forsome 119905 = 1 2 119879 minus 1 and that 120587119905(119868119905) is also concaveand nonincreasing Then Φ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is jointlyconcave joint concavity of the first term of (10) is verified forthe case 119905 = 119879 above For any pair of points (119876119905 1205731119905) and(1198761015840119905 12057310158401119905) with given value of 120596119905 we have120587119905+1 ((120572119876119905 + (1 minus 120572)1198761015840119905)119905 minus (1205721205731119905 + (1 minus 120572) 12057310158401119905) 120596119905minus (1205721205761119905 + (1 minus 120572) 1205761119905)) = 120587119905+1 (120572 (119876119905 minus 1205731119905120596119905 minus 1205761119905)+ (1 minus 120572) (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)) ge 120572120587119905+1 (119876119905 minus 1205731119905120596119905minus 1205761119905) + (1 minus 120572) 120587119905+1 (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)

(20)

by the concavity of 120587119905+1(sdot) which implies that 119864120587119905+1(119876119905 minus1205731119905120596119905 minus 1205761119905) is jointly concave in 119876119905 and 1205731119905 With aboveargument the desired result holds

5 Optimal Policy

Following the structure property of model we can obtain thefollowing result

Theorem 6 Consider any period t with the initial inventory119868119905 there is a unique maximizer (119876lowast119905 (120573lowast1119905 120573lowast2119905)) to maximizethe constrained expected periodical profit function

(119876lowast119905 (120573lowast1119905 120573lowast2119905)) isin arg max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731t 1205732119905)) (21)

Proof It can be easily proved on the basis of Theorem 5

On the basis of the dynamic programming analysisincluding the concavity property we can construct an opti-mization problem for any period t with constrained condi-tions as follows

119872119894119899119894119898119894119911119890 minus Φ119905 (119876119905 (1205731t 1205732119905))

119904119905

(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905))(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) 119905) le 0(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905))(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)) le 0(1205731t + 1205732119905 minus sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905))(1205731t + 1205732119905 minus

sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)) le 0minus119876119905 le 0

(22)

Mathematical Problems in Engineering 9

The Lagrangian and the Karush-Kuhn-Tucker(KTT) condi-tions provide an efficient solution approach to this optimiza-tion problemDenote119860119905 = (exp(1205721205791minus119901119905))(1+exp(1205721205791minus119901119905))119860119905 = (exp(1205721205791 minus 119901119905))(1 + exp(1205721205791 minus 119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119862119905 = (sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))

and 119862119905 = (sum119894=12 exp(120572120579119894 minus119901119905))(1 +sum119894=12 exp(120572120579119894 minus119901119905)) Weconstruct for each period t theLagrangian of the optimizationproblem as

119871 119905 (119876119905 (1205731t 1205732119905) (1205821199051 1205821199052 1205821199053 1205821199054))= minusΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905)+ 1205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905)+ 1205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) minus 1205821199054119876119905

(23)

where the Lagrange multipliers are 120582119905119894 ge 0 119894 = 1 2 3 4 Let(119876lowast119905 (120573lowast1119905 120573lowast2119905)) be the optimizing solution we can constructthe KKT conditions as follows

minus nablaΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (0 21205731t minus 119860119905 minus 119860119905 0)+ 1205821199052 (0 0 21205732t minus 119861119905 minus 119861119905) + 1205821199053 (0 21205731t + 21205732119905minus 119862119905 minus 119862119905 21205731t + 21205732119905 minus 119862119905 minus 119862) minus 1205821199054 (1 0 0) = 0

1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905) = 01205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905) = 01205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) = 01205821199054119876119905 = 0

(24)

where the first equation represents FOC of Lagrangianand the latter four equations represent the conditions ofcomplementary slackness

With solution of theKKT conditions (119876lowast119905 (120573lowast1119905 120573lowast2119905)) and(4) we can get the optimization pricing and ordering policyof (119876lowast119905 (119901lowast1119905 119901lowast2119905)) at period t where (119901lowast1119905 119901lowast2119905) is as follows119901lowast119894119905 = 119875119894119905 (120573lowast1119905 120573lowast2119905) = 120572120579119894 + ln((1 minus sum119894 120573lowast119894119905)120573lowast119894119905 )

119894 = 1 2(25)

Note that (119876lowast119905 (120573lowast1119905 120573lowast2119905)) depends on state variable 119868119905 andparameters of (ℎ1119905 ℎ2119905) and (1198971119905 1198972119905) as well as 120583119905 which

implies that the retailer decides the optimal policy on thebasis of these parameters at every period t With back-ward induction we get the optimal strategy a sequence of[(119876lowast1 (119901lowast11 119901lowast21)) (119876lowast2 (119901lowast12 119901lowast22)) (119876lowast119879 (119901lowast1119879 119901lowast2119879))]6 Numerical Experiment

We now conduct a numerical example to illustrate our resultsand select the parameters as follows The demand followsthe multiplicative form with iid terms 120596119905 sim U(0 100)for all periods We assume (1205791 1205792)119879 = (10 5)119879 sincefresh product has a higher perceptive value than nonfreshones The cost structure is symmetric h119905 = (05 05)119879l119905 = (45 45) and 119888119905 = 10 For simplicity to compute theoptimal prices we consider a finite horizon with 119879 = 5the discount rate is 120574 = 09 and the initial inventory level119868119905 = 0 Using Matlab to program the dynamic model andget the following results with KKT condition method wecan obtain a sequence of order quantity and optimal pricessuch as [(78 (148 97)) (68 (126 86)) (92 (175 114))(43 (109 62)) (52 (115 102))] at which the maximumprofit 120587lowast = 4 2356 occurs

In Table 1 we compare the performance betweendynamic pricing policy and a general policy under oneperiod lifetime setting where all products are sold at aregular price at any period of time Similarly we assume(1205791 1205792)119879 = (10 5)119879 in dynamic pricing policy while(1205791 1205792)119879 = (10 10)119879 in general policy Note that in generalsetting the optimal pricing and ordering policy can beobtained by way of general dynamic programming methodThree examples of different finite horizon time (119879 = 51020)are conducted with the same cost structure and discount ratementioned above Accordingly the optimal solutions suchas pricing and order quantity as well as dynamic inventoryare shown in Table 1 where superscript g and d representregular policy and dynamic policy respectively

A summary of interesting observations is as follows Thedynamic policy is obviously better than general pricing andinventory policy with which the retailerrsquos profit increasesby 15-20 Further the demand is also matched to a morecertain extent by the supply since the order quantity is almostmore than that under general policy while the Based on theresults we can conclude that dynamic pricing and inventorypolicy is beneficial for retailerrsquos revenue management inselling fresh agriculture products Furthermore from theperspective of agriculture industrial development it also canmaximally reduce the operating costs such as holding costsor lost costs since the optimal match between productssupply and demand

7 Concluding Remark

In this paper we address a joint pricing and inventorycontrol problem for stochastic fresh agriproduct inventorysystems under consumer choice over a finite horizon in thebacklogging case In the model the products are perishablealong time and the customers are sensitive to the ages ofinventories which result in consumer choice behavior based

10 Mathematical Problems in Engineering

Table 1 The optimal solution under dynamic pricing and regular pricing

Periods T Order quantity Pricing Inventory 120587d 120587g (120587119889 minus 120587119892)120587119892Qd Qg (1199011198891 1199011198892) 119901119892 (1198681198891 1198681198892 ) 119868119892

5

78 72 (14897) 125 (490) 9

42356 35782 18468 65 (12686) 104 (312) 892 80 (175114) 128 (5611) 1343 45 (10962) 81 (220) 352 45 (11582) 93 (121) 5

10

81 70 (156107) 122 (520) 10

83438 69844 195

72 61 (13684) 103 (386) 783 58 (165104) 132 (569) 876 59 (11978) 85 (322) 470 73 (12592) 81 (346) 456 41 (16598) 146 (301) 968 59 (12288) 95 (312) 865 47 (154104) 138 (339) 758 56 (12473) 92 (230) 349 39 (12864) 97 (191) 2

20

90 86 (15598) 131 (540) 16

175328 149632 172

72 65 (14385) 104 (366) 883 57 (165101) 111 (409) 966 75 (13276) 86 (314) 572 54 (12995) 92 (346) 456 51 (16392) 132 (353) 768 62 (13287) 98 (313) 465 45 (151122) 136 (335) 457 69 (13879) 97 (280) 348 46 (15592) 102 (211) 281 72 (13871) 98 (430) 270 58 (13992) 104 (386) 778 50 (15468) 119 (492) 571 48 (13288) 95 (333) 061 63 (14582) 84 (356) 856 52 (16578) 146 (373) 1076 48 (11388) 99 (323) 666 64 (14594) 128 (395) 852 49 (13778) 98 (230) 350 46 (14569) 91 (220) 1

on product freshness By using MNL model to describeconsumer discrete choice we construct a stochastic anddynamic inventory model facing strategic customer behaviorwith joint ordering and discounting decisions We are ableto characterize the firmrsquos optimal policy with the use ofa transformation technique and employing the concept ofthe concavity and finally develop a Lagrangian and theKarush-Kuhn-Tucker (KTT) conditions to provide an efficientsolution approach to this optimization problem

A limitation in our model is that we assume that thelifetime of the product is only two periods Although this istrue for some fresh agriproducts in themarket there are otherproducts that are perishable but yet have longer lifetimes Toextend ourmodel to the products withmore than two periods

lifetimes the state space has to be increased more to specifythe amounts of inventory that are going to expire at differentpoints of time in the futureThis definitelymakes the problemquite hard Another issue is to incorporate an operatingcost for discounting pricing practice which however willintroduce additional technical complexity to the dynamicprogram There are several future research directions Oneis to extend the model to the environment where all thedemand can be lost or only non-FAP demand can be lostwhile the fresh FAP can be backlogged instead Another is toconsider the case where the product has a fixed finite lifetimeof exactly 119899(119899 ge 3) periods or random lifetime Finally itwould be interesting to account for stock-out based substitu-tions

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

cost and becoming the initial inventory of nonfresh product119868119905+1 at period t+1 while the (119868119905 minus 1198632119905)+ of old products willbe disposed of on account of zero shelf-life time incurringdisposal cost Otherwise the relevant backordered amountof new FAPs and old FAPs for period t is (119876119905 minus 1198631119905)minus and(119868119905minus1198632119905)minus respectively whendemand is greater than on-handinventories where 119909+ = max(0 119909) and 119909minus = max(0 minus119909)

At period t the retailer revenue 119905((1199011119905 1199012119905)) can becalculated as follows

119905 ((1199011119905 1199012119905)) = (1199011119905 1199012119905) (1198631119905 1198632119905)119879 (6)

Substitute the price function (5) into (6) we can obtain theexpected revenue 119877119905((1205731t 1205732119905))119877119905 ((1205731t 1205732119905)) = 119864 (119905 (1199011119905 1199012119905))= sum119894=12

(120572120579119894 + ln((1 minus sum119894 120573119894119905)120573119894119905 ))120573it (1199011119905 1199012119905) 120583119905 (7)

After demand realization at period t somemore costs incursbackordered cost of 1198971119905(119876119905 minus1198631119905)minus or holding cost of ℎ1119905(119876119905 minus1198631119905)+ for new FAPs and backlogged cost of 1198972119905(119868119905 minus 1198632119905)minus ordisposal cost of ℎ2119905(119868119905 minus1198632119905)+ for old FAPs totally denoted as119905(119876119905 (1205731119905 1205732119905))119905 (119876119905 (1205731119905 1205732119905)) = (ℎ1119905 ℎ2119905) sdot ((119876119905 minus 1205731t120596119905 minus 1205761119905)

+

(119868119905 minus 1205732t120596119905 minus 1205762119905)+)+ (1198971119905 1198971119905)sdot ((119876119905 minus 1205731t120596119905 minus 1205761119905)minus(119868119905 minus 1205732t120596119905 minus 1205762119905)minus)

(8)

And the corresponding expected function is 119867119905(119876119905 (12057311199051205732119905)) = 119864(119905(119876119905 (1205731119905 1205732119905)))Let 119866119905(119876119905 (1205731119905 1205732119905)) be the profit of period t we have

119866119905 (119876119905 (1205731119905 1205732119905)) = 119905 (119876119905 (1205731119905 1205732119905))minus 119905 (119876119905 (1205731119905 1205732119905)) minus 119888119905119876119905 (9)

Accordingly the expected profit function is as below

119866119905 (119876119905 (1205731119905 1205732119905)) = 119864119866119905 (119876119905 (1205731119905 1205732119905))= 119877119905 (1205731119905 1205732119905) minus 119867119905 (119876119905 (1205731119905 1205732119905))minus 119888119905119876119905

(10)

Denote 120587119905(119868119905) to be the maximum total expected profit-to-go function from period t to T in state 119868119905 witha discount factor 120574 We have the recursive equation120587119905(119868119905) = max119876119905ge0(1205731t 1205732119905)isinΩ119905119866119905(119876119905 (1205731119905 1205732119905)) + 120574119864120587119905+1(119868119905+1)With transfer function 119868119905+1 = (119876119905 minus 1205731119905120596119905)+ by definingΦ119905(119876119905 (1205731119905 1205732119905))

Φ119905 (119876119905 (1205731119905 1205732119905)) = 119866119905 (119876119905 (1205731119905 1205732119905))+ 120574119864120587119905+1 (119868119905+1) (11)

we have the following dynamic programming model

120587119905 (119868119905) = max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731119905 1205732119905)) (12)

119904119905 120587119879+1 = minusℎ2119879+1119868119879+1 (13)

where (13) is the boundary condition since all leftovers will bedisposed of at the end of horizon planningThat is to say thereare no second markets where some unexpired inventory canbe cleared with some salvage value On the contrary the firmneeds to pay on deposing expired inventories even unexpiredleftovers Our analysis can be readily extended to address thissituation

4 Analysis of Model Structure

To analyze the structure of the optimal policy we needto know the concavity property of Φ119905(119876119905 (1205731t 1205732119905)) whichis strictly relative to revenue function 119877119905((1205731t 1205732119905)) andinventory cost function119867119905(119876119905 (1205731t 1205732119905))Lemma 2 119877119905((1205731t 1205732119905)) is continuously twice differentiableand jointly concave in (1205731t 1205732119905)Proof The continuity and twice differentiability of 119877119905((1205731t1205732119905)) is quite intuitive since it is expressed by some elementaryoperations of 1205731t and 1205732119905 To show the concavity we needto prove the negative semidefinite property of its HessianMatrix Taking the second-order derivative with respect to1205731t yields 120597211987711990512059712057321t = minus(1(1 minus 1205731119905 minus 1205732119905) + 11205731119905 + 1(1 minus1205731119905 minus 1205732119905)2)120583119905 le 0 Thus 119877119905 is concave on 1205731t Taking thesecond-order derivative with respect to 1205732t yields 120597211987711990512059712057322t =minus(1(1 minus 1205731119905 minus 1205732119905) + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 le 0which implies the concavity of 119877119905 on 1205732t And taking thecross-partial derivative with respect to 1205731t and 1205732t yields1205972119877119905(1205971205731t1205971205732t) = minus(1(1 minus 1205731119905 minus 1205732119905) + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905Pairing up the above terms leads to Hessian Matrix

119867essian (119877119905) = [[[[[

120597211987711990512059712057321t 1205972119877119905(1205971205731t1205971205732t)1205972119877119905(1205971205732t1205971205731t) 120597211987711990512059712057322t]]]]]

Mathematical Problems in Engineering 7

= [[[[[[minus( 11 minus 1205731119905 minus 1205732119905 + 11205731119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905

]]]]]]

(14)

And the corresponding determinant is as follows

1003816100381610038161003816119867 (119877119905)1003816100381610038161003816 = 120597211987711990512059712057321t sdot 120597211987711990512059712057322t minus ( 1205972119877119905(1205971205731t1205971205732t))

2

= ( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)(11205731119905 + 11205732119905)

+ 112057311199051205732119905 gt 0(15)

which implies that 119867119890119904119904119894119886119899 matrix is seminegative defi-nite On the basis of that 119877119905 is jointly concave in (1205731t1205732119905)Lemma 3 119867119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly convex in (1205731t 1205732119905) and 119876119905Proof We observe that 119905(119876119905 (1205731t 1205732119905)) is continuously andtwice differentiable for any realization of 120596119905 and 120576119894119905 And theexpectation 119867119905(119876119905 (1205731t 1205732119905)) is easily justified to be con-tinuously twice differentiable using the canonical argument(Durrett 2010)

To show the convexity we define some column vectorsexpressed by boldface lowercase letters such as h119905 =(ℎ1119905 ℎ2119905)119879 l119905 = (1198971119905 1198972119905)119879 and 120576119905 = (1205761119905 1205762119905)119879 Then thedemand and cost function can be expressed of vector formas follows

D119905 = (1198631119905 1198632119905) = 120573119905120596119905 + 120576119905 (16)119905 (119876119905 (1205731t 1205732119905)) = 119905 (120573119905 119876119905)= h119905 ((119876119905 119868119905)119879 minusD119905)++ l119905 ((119876119905 119868119905)119879 minusD119905)minus

= h119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)++ l119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)minus

(17)

Define a new vector variable 119906119905 = (119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905 andwe can get 119905(120573119905 119876119905) = 119905(u119905) = h119905(u119905)+ + l119905(u119905)minus which isconvex onu119905 (SundaranmR K 1996) Fix the randompart120596119905to its realization and consider any pair (120573119905 119876119905) and (1205731015840119905 1198761015840119905)with the convexity of 119905(u119905) we have119905 (120572 (120573119905 119876119905) + (1 minus 120572) (1205731015840119905 1198761015840119905)) = 119905 (120572120573119905+ (1 minus 120572)1205731015840119905 120572119876119905 + (1 minus 120572)1198761015840119905)= h119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)

minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)++ l119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)minus= h119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879119905 minus 1205731015840119905120596119905 minus 120576119905))++ l119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879 minus 1205731015840119905120596119905 minus 120576119905))minus = h119905 (120572u119905+ (1 minus 120572)u1015840119905)+ + l119905 (120572u119905 + (1 minus 120572) u1015840119905)minus = 119905 (120572u119905+ (1 minus 120572)u1015840119905) le 120572119905 (u119905) + (1 minus 120572) 119905 (u1015840119905)= 120572119905 (120573119905 (119876119905 119868119905)119879) + (1 minus 120572) 119905 (1205731015840119905 (1198761015840119905 119868119905)119879)

(18)

where 0 le 120572 le 1 Therefore 119905(119876119905 (1205731t 1205732119905)) = 119905(120573119905(119876119905 119868119905)119879) = 119905(u119905) is jointly convex in (1205731t 1205732119905) and 119876119905 sodoes the expectation function119867119905(119876119905 (1205731t 1205732119905))Theorem 4 At any time period t with initial inventory 119868119905the expected revenue function119866119905(119876119905 (1205731t 1205732119905)) is continuouslytwice differentiable and jointly concave in (1205731t 1205732119905) and 119876119905Proof With Lemmas 2 and 3 we can observe that119866119905(119876119905 (1205731t 1205732119905)) is continuously twice differentiable We firstobserve that for any t the three terms of 119866119905(119876119905 (1205731119905 1205732119905)) in(10) are jointly concave The first term is concave in (1205731119905 1205732119905)according to Lemma 2 The second term is jointly concavein (1205731119905 1205732119905) and 119876119905 (see Lemma 3) and the third term islinear in 119876119905 which implies the concavity With the propertythat a nonnegative weighted sum of concave functionsis concave (Stephen Boyd Lieven Vandenberghe 2004)119866119905(119876119905 (1205731119905 1205732119905)) is continuously twice differentiable andjointly concave in (1205731t 1205732119905) and 119876119905

On the basis of the concavity of 119866119905(119876119905 (1205731t 1205732119905)) we canhaveTheorem 5

Theorem 5 Consider any time period t with initial inventory119868119905 the functionΦ119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly concave in (119876119905 (1205731t 1205732119905)) and the function120587119905(119868119905) is concave and nonincreasing in 119868119905Proof By induction first we prove that Φ119879(119876119879 (1205731T 1205732119879))is continuously twice differentiable According to Lemma 2

8 Mathematical Problems in Engineering

119864(119876119879 minus 1205731119879120596119879 minus 120576119879)+ as a part of 119867119905(119876119905 (1205731t 1205732119905)) isthus continuously twice differentiable which implies thatΦ119879(119876119879 (1205731119879 1205732119879)) = 119866119879(119876119879 (1205731119879 1205732119879)) minus 120574ℎ2119879+1119864(119876119879 minus1205731119879120596119879 minus 1205762119879)+ is continuously and twice differentiableAssuming now that Φ119905(119876119905 (1205731119905 1205732119905)) is continuouslytwice differentiable for 119905 = 1 2 119879 we prove thatΦ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is also continuously twice differen-tiable according to Theorem 4 Let Λ 119879(119876119879 1205731119879) = 119864(119876119879 minus1205731119879120596119879minus1205762119879)+ = ∬1205731119879119909+119910le119876119879(119876119879minus1205731119879119909minus119910)119891120596(119909)119891120576(119910)119889119909119889119910where 119891120596(119909) and 119891120576(119910) is the pdf of random variable120596119879 and 120576119879 respectively which are generally assumed to beindependent For any realization of 1205762119879 such as 1205762119879 taking thesecond-order derivative ofΛ 119879(119876119879 1205731119879)with respective to119876119879yields 1205972Λ 119879(119876119879 1205731119879)1205971198762119879 = (11205731119879)119891((119876119879 minus 1205762119879)1205731119879) gt 0which implies that Λ 119879(119876119879 1205731119879) is convex in 119876119879 Takingthe second-order derivative with respect to 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)12059712057321119879 = (119876211987912057331119879)119891(1198761198791205731119879) gt 0 whichimplies the concavity of Λ 119879(119876119879 1205731119879) in 1205731119879 And takingthe cross-partial derivative with respect to 119876119879 and 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)1205971198761198791205971205731119879 = minus((119876119879 minus 1205762119879)12057321119879)119891((119876119879 minus1205762119879)1205731119879) Pairing up the above three derivative terms leadsto Hessian Matrix

119867(Λ 119879) = [[[[[

1205972Λ 11987912059711987621198791205972Λ 11987912059711987611987912059712057311198791205972Λ 1198791205971205731119879120597119876119879 1205972Λ 11987912059712057321119879

]]]]]

= [[[[[

11205731119879119891(119876119879 minus 12057621198791205731119879 ) minus119876119879 minus 120576211987912057321119879

119891(119876119879 minus 12057621198791205731119879 )minus119876119879 minus 120576211987912057321119879 119891(119876119879 minus 12057621198791205731119879 ) (119876119879 minus 1205762119879)212057331119879 119891(119876119879 minus 12057621198791205731119879 )

]]]]](19)

And the corresponding determinant 119867(Λ 119879) = 0 whichimplies matrix 119867119890119904119904119894119886119899 is semipositive definite On thebasis of analysis above we can conclude that Λ 119879(119876119879 1205731119879)is jointly convex in (119876119879 1205731119879) Meanwhile the last termminus120574ℎ2119879+1119864(119876119879 minus 1205731119879120596119879 minus 1205762119879)+ is obviously justified to bejointly concave in (119876119879 1205731119879) According to the concavityproperty (Stephen Boyd Lieven Vandenberghe 2004)Φ119879(119876119879 (1205731T 1205732119879)) is jointly concave in (1205731t 1205732119905) and 119876119905

120587119879(119868119879) is easily verified to be concave as well and it isobviously nonincreasing

Assume now that Φ119905(119876119905 (1205731t 1205732119905)) is jointly concave forsome 119905 = 1 2 119879 minus 1 and that 120587119905(119868119905) is also concaveand nonincreasing Then Φ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is jointlyconcave joint concavity of the first term of (10) is verified forthe case 119905 = 119879 above For any pair of points (119876119905 1205731119905) and(1198761015840119905 12057310158401119905) with given value of 120596119905 we have120587119905+1 ((120572119876119905 + (1 minus 120572)1198761015840119905)119905 minus (1205721205731119905 + (1 minus 120572) 12057310158401119905) 120596119905minus (1205721205761119905 + (1 minus 120572) 1205761119905)) = 120587119905+1 (120572 (119876119905 minus 1205731119905120596119905 minus 1205761119905)+ (1 minus 120572) (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)) ge 120572120587119905+1 (119876119905 minus 1205731119905120596119905minus 1205761119905) + (1 minus 120572) 120587119905+1 (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)

(20)

by the concavity of 120587119905+1(sdot) which implies that 119864120587119905+1(119876119905 minus1205731119905120596119905 minus 1205761119905) is jointly concave in 119876119905 and 1205731119905 With aboveargument the desired result holds

5 Optimal Policy

Following the structure property of model we can obtain thefollowing result

Theorem 6 Consider any period t with the initial inventory119868119905 there is a unique maximizer (119876lowast119905 (120573lowast1119905 120573lowast2119905)) to maximizethe constrained expected periodical profit function

(119876lowast119905 (120573lowast1119905 120573lowast2119905)) isin arg max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731t 1205732119905)) (21)

Proof It can be easily proved on the basis of Theorem 5

On the basis of the dynamic programming analysisincluding the concavity property we can construct an opti-mization problem for any period t with constrained condi-tions as follows

119872119894119899119894119898119894119911119890 minus Φ119905 (119876119905 (1205731t 1205732119905))

119904119905

(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905))(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) 119905) le 0(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905))(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)) le 0(1205731t + 1205732119905 minus sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905))(1205731t + 1205732119905 minus

sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)) le 0minus119876119905 le 0

(22)

Mathematical Problems in Engineering 9

The Lagrangian and the Karush-Kuhn-Tucker(KTT) condi-tions provide an efficient solution approach to this optimiza-tion problemDenote119860119905 = (exp(1205721205791minus119901119905))(1+exp(1205721205791minus119901119905))119860119905 = (exp(1205721205791 minus 119901119905))(1 + exp(1205721205791 minus 119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119862119905 = (sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))

and 119862119905 = (sum119894=12 exp(120572120579119894 minus119901119905))(1 +sum119894=12 exp(120572120579119894 minus119901119905)) Weconstruct for each period t theLagrangian of the optimizationproblem as

119871 119905 (119876119905 (1205731t 1205732119905) (1205821199051 1205821199052 1205821199053 1205821199054))= minusΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905)+ 1205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905)+ 1205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) minus 1205821199054119876119905

(23)

where the Lagrange multipliers are 120582119905119894 ge 0 119894 = 1 2 3 4 Let(119876lowast119905 (120573lowast1119905 120573lowast2119905)) be the optimizing solution we can constructthe KKT conditions as follows

minus nablaΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (0 21205731t minus 119860119905 minus 119860119905 0)+ 1205821199052 (0 0 21205732t minus 119861119905 minus 119861119905) + 1205821199053 (0 21205731t + 21205732119905minus 119862119905 minus 119862119905 21205731t + 21205732119905 minus 119862119905 minus 119862) minus 1205821199054 (1 0 0) = 0

1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905) = 01205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905) = 01205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) = 01205821199054119876119905 = 0

(24)

where the first equation represents FOC of Lagrangianand the latter four equations represent the conditions ofcomplementary slackness

With solution of theKKT conditions (119876lowast119905 (120573lowast1119905 120573lowast2119905)) and(4) we can get the optimization pricing and ordering policyof (119876lowast119905 (119901lowast1119905 119901lowast2119905)) at period t where (119901lowast1119905 119901lowast2119905) is as follows119901lowast119894119905 = 119875119894119905 (120573lowast1119905 120573lowast2119905) = 120572120579119894 + ln((1 minus sum119894 120573lowast119894119905)120573lowast119894119905 )

119894 = 1 2(25)

Note that (119876lowast119905 (120573lowast1119905 120573lowast2119905)) depends on state variable 119868119905 andparameters of (ℎ1119905 ℎ2119905) and (1198971119905 1198972119905) as well as 120583119905 which

implies that the retailer decides the optimal policy on thebasis of these parameters at every period t With back-ward induction we get the optimal strategy a sequence of[(119876lowast1 (119901lowast11 119901lowast21)) (119876lowast2 (119901lowast12 119901lowast22)) (119876lowast119879 (119901lowast1119879 119901lowast2119879))]6 Numerical Experiment

We now conduct a numerical example to illustrate our resultsand select the parameters as follows The demand followsthe multiplicative form with iid terms 120596119905 sim U(0 100)for all periods We assume (1205791 1205792)119879 = (10 5)119879 sincefresh product has a higher perceptive value than nonfreshones The cost structure is symmetric h119905 = (05 05)119879l119905 = (45 45) and 119888119905 = 10 For simplicity to compute theoptimal prices we consider a finite horizon with 119879 = 5the discount rate is 120574 = 09 and the initial inventory level119868119905 = 0 Using Matlab to program the dynamic model andget the following results with KKT condition method wecan obtain a sequence of order quantity and optimal pricessuch as [(78 (148 97)) (68 (126 86)) (92 (175 114))(43 (109 62)) (52 (115 102))] at which the maximumprofit 120587lowast = 4 2356 occurs

In Table 1 we compare the performance betweendynamic pricing policy and a general policy under oneperiod lifetime setting where all products are sold at aregular price at any period of time Similarly we assume(1205791 1205792)119879 = (10 5)119879 in dynamic pricing policy while(1205791 1205792)119879 = (10 10)119879 in general policy Note that in generalsetting the optimal pricing and ordering policy can beobtained by way of general dynamic programming methodThree examples of different finite horizon time (119879 = 51020)are conducted with the same cost structure and discount ratementioned above Accordingly the optimal solutions suchas pricing and order quantity as well as dynamic inventoryare shown in Table 1 where superscript g and d representregular policy and dynamic policy respectively

A summary of interesting observations is as follows Thedynamic policy is obviously better than general pricing andinventory policy with which the retailerrsquos profit increasesby 15-20 Further the demand is also matched to a morecertain extent by the supply since the order quantity is almostmore than that under general policy while the Based on theresults we can conclude that dynamic pricing and inventorypolicy is beneficial for retailerrsquos revenue management inselling fresh agriculture products Furthermore from theperspective of agriculture industrial development it also canmaximally reduce the operating costs such as holding costsor lost costs since the optimal match between productssupply and demand

7 Concluding Remark

In this paper we address a joint pricing and inventorycontrol problem for stochastic fresh agriproduct inventorysystems under consumer choice over a finite horizon in thebacklogging case In the model the products are perishablealong time and the customers are sensitive to the ages ofinventories which result in consumer choice behavior based

10 Mathematical Problems in Engineering

Table 1 The optimal solution under dynamic pricing and regular pricing

Periods T Order quantity Pricing Inventory 120587d 120587g (120587119889 minus 120587119892)120587119892Qd Qg (1199011198891 1199011198892) 119901119892 (1198681198891 1198681198892 ) 119868119892

5

78 72 (14897) 125 (490) 9

42356 35782 18468 65 (12686) 104 (312) 892 80 (175114) 128 (5611) 1343 45 (10962) 81 (220) 352 45 (11582) 93 (121) 5

10

81 70 (156107) 122 (520) 10

83438 69844 195

72 61 (13684) 103 (386) 783 58 (165104) 132 (569) 876 59 (11978) 85 (322) 470 73 (12592) 81 (346) 456 41 (16598) 146 (301) 968 59 (12288) 95 (312) 865 47 (154104) 138 (339) 758 56 (12473) 92 (230) 349 39 (12864) 97 (191) 2

20

90 86 (15598) 131 (540) 16

175328 149632 172

72 65 (14385) 104 (366) 883 57 (165101) 111 (409) 966 75 (13276) 86 (314) 572 54 (12995) 92 (346) 456 51 (16392) 132 (353) 768 62 (13287) 98 (313) 465 45 (151122) 136 (335) 457 69 (13879) 97 (280) 348 46 (15592) 102 (211) 281 72 (13871) 98 (430) 270 58 (13992) 104 (386) 778 50 (15468) 119 (492) 571 48 (13288) 95 (333) 061 63 (14582) 84 (356) 856 52 (16578) 146 (373) 1076 48 (11388) 99 (323) 666 64 (14594) 128 (395) 852 49 (13778) 98 (230) 350 46 (14569) 91 (220) 1

on product freshness By using MNL model to describeconsumer discrete choice we construct a stochastic anddynamic inventory model facing strategic customer behaviorwith joint ordering and discounting decisions We are ableto characterize the firmrsquos optimal policy with the use ofa transformation technique and employing the concept ofthe concavity and finally develop a Lagrangian and theKarush-Kuhn-Tucker (KTT) conditions to provide an efficientsolution approach to this optimization problem

A limitation in our model is that we assume that thelifetime of the product is only two periods Although this istrue for some fresh agriproducts in themarket there are otherproducts that are perishable but yet have longer lifetimes Toextend ourmodel to the products withmore than two periods

lifetimes the state space has to be increased more to specifythe amounts of inventory that are going to expire at differentpoints of time in the futureThis definitelymakes the problemquite hard Another issue is to incorporate an operatingcost for discounting pricing practice which however willintroduce additional technical complexity to the dynamicprogram There are several future research directions Oneis to extend the model to the environment where all thedemand can be lost or only non-FAP demand can be lostwhile the fresh FAP can be backlogged instead Another is toconsider the case where the product has a fixed finite lifetimeof exactly 119899(119899 ge 3) periods or random lifetime Finally itwould be interesting to account for stock-out based substitu-tions

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

= [[[[[[minus( 11 minus 1205731119905 minus 1205732119905 + 11205731119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905minus( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905 minus( 11 minus 1205731119905 minus 1205732119905 + 11205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)120583119905

]]]]]]

(14)

And the corresponding determinant is as follows

1003816100381610038161003816119867 (119877119905)1003816100381610038161003816 = 120597211987711990512059712057321t sdot 120597211987711990512059712057322t minus ( 1205972119877119905(1205971205731t1205971205732t))

2

= ( 11 minus 1205731119905 minus 1205732119905 + 1(1 minus 1205731119905 minus 1205732119905)2)(11205731119905 + 11205732119905)

+ 112057311199051205732119905 gt 0(15)

which implies that 119867119890119904119904119894119886119899 matrix is seminegative defi-nite On the basis of that 119877119905 is jointly concave in (1205731t1205732119905)Lemma 3 119867119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly convex in (1205731t 1205732119905) and 119876119905Proof We observe that 119905(119876119905 (1205731t 1205732119905)) is continuously andtwice differentiable for any realization of 120596119905 and 120576119894119905 And theexpectation 119867119905(119876119905 (1205731t 1205732119905)) is easily justified to be con-tinuously twice differentiable using the canonical argument(Durrett 2010)

To show the convexity we define some column vectorsexpressed by boldface lowercase letters such as h119905 =(ℎ1119905 ℎ2119905)119879 l119905 = (1198971119905 1198972119905)119879 and 120576119905 = (1205761119905 1205762119905)119879 Then thedemand and cost function can be expressed of vector formas follows

D119905 = (1198631119905 1198632119905) = 120573119905120596119905 + 120576119905 (16)119905 (119876119905 (1205731t 1205732119905)) = 119905 (120573119905 119876119905)= h119905 ((119876119905 119868119905)119879 minusD119905)++ l119905 ((119876119905 119868119905)119879 minusD119905)minus

= h119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)++ l119905 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)minus

(17)

Define a new vector variable 119906119905 = (119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905 andwe can get 119905(120573119905 119876119905) = 119905(u119905) = h119905(u119905)+ + l119905(u119905)minus which isconvex onu119905 (SundaranmR K 1996) Fix the randompart120596119905to its realization and consider any pair (120573119905 119876119905) and (1205731015840119905 1198761015840119905)with the convexity of 119905(u119905) we have119905 (120572 (120573119905 119876119905) + (1 minus 120572) (1205731015840119905 1198761015840119905)) = 119905 (120572120573119905+ (1 minus 120572)1205731015840119905 120572119876119905 + (1 minus 120572)1198761015840119905)= h119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)

minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)++ l119905 ((120572 (119876119905 119868119905)119879 + (1 minus 120572) (1198761015840119905 119868119905)119879)minus (120572120573119905 + (1 minus 120572)1205731015840119905) 120596119905 minus (120572 + (1 minus 120572)) 120576119905)minus= h119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879119905 minus 1205731015840119905120596119905 minus 120576119905))++ l119905 (120572 ((119876119905 119868119905)119879 minus 120573119905120596119905 minus 120576119905)+ (1 minus 120572) ((1198761015840 119868119905)119879 minus 1205731015840119905120596119905 minus 120576119905))minus = h119905 (120572u119905+ (1 minus 120572)u1015840119905)+ + l119905 (120572u119905 + (1 minus 120572) u1015840119905)minus = 119905 (120572u119905+ (1 minus 120572)u1015840119905) le 120572119905 (u119905) + (1 minus 120572) 119905 (u1015840119905)= 120572119905 (120573119905 (119876119905 119868119905)119879) + (1 minus 120572) 119905 (1205731015840119905 (1198761015840119905 119868119905)119879)

(18)

where 0 le 120572 le 1 Therefore 119905(119876119905 (1205731t 1205732119905)) = 119905(120573119905(119876119905 119868119905)119879) = 119905(u119905) is jointly convex in (1205731t 1205732119905) and 119876119905 sodoes the expectation function119867119905(119876119905 (1205731t 1205732119905))Theorem 4 At any time period t with initial inventory 119868119905the expected revenue function119866119905(119876119905 (1205731t 1205732119905)) is continuouslytwice differentiable and jointly concave in (1205731t 1205732119905) and 119876119905Proof With Lemmas 2 and 3 we can observe that119866119905(119876119905 (1205731t 1205732119905)) is continuously twice differentiable We firstobserve that for any t the three terms of 119866119905(119876119905 (1205731119905 1205732119905)) in(10) are jointly concave The first term is concave in (1205731119905 1205732119905)according to Lemma 2 The second term is jointly concavein (1205731119905 1205732119905) and 119876119905 (see Lemma 3) and the third term islinear in 119876119905 which implies the concavity With the propertythat a nonnegative weighted sum of concave functionsis concave (Stephen Boyd Lieven Vandenberghe 2004)119866119905(119876119905 (1205731119905 1205732119905)) is continuously twice differentiable andjointly concave in (1205731t 1205732119905) and 119876119905

On the basis of the concavity of 119866119905(119876119905 (1205731t 1205732119905)) we canhaveTheorem 5

Theorem 5 Consider any time period t with initial inventory119868119905 the functionΦ119905(119876119905 (1205731t 1205732119905)) is continuously twice differen-tiable and jointly concave in (119876119905 (1205731t 1205732119905)) and the function120587119905(119868119905) is concave and nonincreasing in 119868119905Proof By induction first we prove that Φ119879(119876119879 (1205731T 1205732119879))is continuously twice differentiable According to Lemma 2

8 Mathematical Problems in Engineering

119864(119876119879 minus 1205731119879120596119879 minus 120576119879)+ as a part of 119867119905(119876119905 (1205731t 1205732119905)) isthus continuously twice differentiable which implies thatΦ119879(119876119879 (1205731119879 1205732119879)) = 119866119879(119876119879 (1205731119879 1205732119879)) minus 120574ℎ2119879+1119864(119876119879 minus1205731119879120596119879 minus 1205762119879)+ is continuously and twice differentiableAssuming now that Φ119905(119876119905 (1205731119905 1205732119905)) is continuouslytwice differentiable for 119905 = 1 2 119879 we prove thatΦ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is also continuously twice differen-tiable according to Theorem 4 Let Λ 119879(119876119879 1205731119879) = 119864(119876119879 minus1205731119879120596119879minus1205762119879)+ = ∬1205731119879119909+119910le119876119879(119876119879minus1205731119879119909minus119910)119891120596(119909)119891120576(119910)119889119909119889119910where 119891120596(119909) and 119891120576(119910) is the pdf of random variable120596119879 and 120576119879 respectively which are generally assumed to beindependent For any realization of 1205762119879 such as 1205762119879 taking thesecond-order derivative ofΛ 119879(119876119879 1205731119879)with respective to119876119879yields 1205972Λ 119879(119876119879 1205731119879)1205971198762119879 = (11205731119879)119891((119876119879 minus 1205762119879)1205731119879) gt 0which implies that Λ 119879(119876119879 1205731119879) is convex in 119876119879 Takingthe second-order derivative with respect to 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)12059712057321119879 = (119876211987912057331119879)119891(1198761198791205731119879) gt 0 whichimplies the concavity of Λ 119879(119876119879 1205731119879) in 1205731119879 And takingthe cross-partial derivative with respect to 119876119879 and 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)1205971198761198791205971205731119879 = minus((119876119879 minus 1205762119879)12057321119879)119891((119876119879 minus1205762119879)1205731119879) Pairing up the above three derivative terms leadsto Hessian Matrix

119867(Λ 119879) = [[[[[

1205972Λ 11987912059711987621198791205972Λ 11987912059711987611987912059712057311198791205972Λ 1198791205971205731119879120597119876119879 1205972Λ 11987912059712057321119879

]]]]]

= [[[[[

11205731119879119891(119876119879 minus 12057621198791205731119879 ) minus119876119879 minus 120576211987912057321119879

119891(119876119879 minus 12057621198791205731119879 )minus119876119879 minus 120576211987912057321119879 119891(119876119879 minus 12057621198791205731119879 ) (119876119879 minus 1205762119879)212057331119879 119891(119876119879 minus 12057621198791205731119879 )

]]]]](19)

And the corresponding determinant 119867(Λ 119879) = 0 whichimplies matrix 119867119890119904119904119894119886119899 is semipositive definite On thebasis of analysis above we can conclude that Λ 119879(119876119879 1205731119879)is jointly convex in (119876119879 1205731119879) Meanwhile the last termminus120574ℎ2119879+1119864(119876119879 minus 1205731119879120596119879 minus 1205762119879)+ is obviously justified to bejointly concave in (119876119879 1205731119879) According to the concavityproperty (Stephen Boyd Lieven Vandenberghe 2004)Φ119879(119876119879 (1205731T 1205732119879)) is jointly concave in (1205731t 1205732119905) and 119876119905

120587119879(119868119879) is easily verified to be concave as well and it isobviously nonincreasing

Assume now that Φ119905(119876119905 (1205731t 1205732119905)) is jointly concave forsome 119905 = 1 2 119879 minus 1 and that 120587119905(119868119905) is also concaveand nonincreasing Then Φ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is jointlyconcave joint concavity of the first term of (10) is verified forthe case 119905 = 119879 above For any pair of points (119876119905 1205731119905) and(1198761015840119905 12057310158401119905) with given value of 120596119905 we have120587119905+1 ((120572119876119905 + (1 minus 120572)1198761015840119905)119905 minus (1205721205731119905 + (1 minus 120572) 12057310158401119905) 120596119905minus (1205721205761119905 + (1 minus 120572) 1205761119905)) = 120587119905+1 (120572 (119876119905 minus 1205731119905120596119905 minus 1205761119905)+ (1 minus 120572) (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)) ge 120572120587119905+1 (119876119905 minus 1205731119905120596119905minus 1205761119905) + (1 minus 120572) 120587119905+1 (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)

(20)

by the concavity of 120587119905+1(sdot) which implies that 119864120587119905+1(119876119905 minus1205731119905120596119905 minus 1205761119905) is jointly concave in 119876119905 and 1205731119905 With aboveargument the desired result holds

5 Optimal Policy

Following the structure property of model we can obtain thefollowing result

Theorem 6 Consider any period t with the initial inventory119868119905 there is a unique maximizer (119876lowast119905 (120573lowast1119905 120573lowast2119905)) to maximizethe constrained expected periodical profit function

(119876lowast119905 (120573lowast1119905 120573lowast2119905)) isin arg max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731t 1205732119905)) (21)

Proof It can be easily proved on the basis of Theorem 5

On the basis of the dynamic programming analysisincluding the concavity property we can construct an opti-mization problem for any period t with constrained condi-tions as follows

119872119894119899119894119898119894119911119890 minus Φ119905 (119876119905 (1205731t 1205732119905))

119904119905

(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905))(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) 119905) le 0(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905))(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)) le 0(1205731t + 1205732119905 minus sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905))(1205731t + 1205732119905 minus

sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)) le 0minus119876119905 le 0

(22)

Mathematical Problems in Engineering 9

The Lagrangian and the Karush-Kuhn-Tucker(KTT) condi-tions provide an efficient solution approach to this optimiza-tion problemDenote119860119905 = (exp(1205721205791minus119901119905))(1+exp(1205721205791minus119901119905))119860119905 = (exp(1205721205791 minus 119901119905))(1 + exp(1205721205791 minus 119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119862119905 = (sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))

and 119862119905 = (sum119894=12 exp(120572120579119894 minus119901119905))(1 +sum119894=12 exp(120572120579119894 minus119901119905)) Weconstruct for each period t theLagrangian of the optimizationproblem as

119871 119905 (119876119905 (1205731t 1205732119905) (1205821199051 1205821199052 1205821199053 1205821199054))= minusΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905)+ 1205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905)+ 1205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) minus 1205821199054119876119905

(23)

where the Lagrange multipliers are 120582119905119894 ge 0 119894 = 1 2 3 4 Let(119876lowast119905 (120573lowast1119905 120573lowast2119905)) be the optimizing solution we can constructthe KKT conditions as follows

minus nablaΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (0 21205731t minus 119860119905 minus 119860119905 0)+ 1205821199052 (0 0 21205732t minus 119861119905 minus 119861119905) + 1205821199053 (0 21205731t + 21205732119905minus 119862119905 minus 119862119905 21205731t + 21205732119905 minus 119862119905 minus 119862) minus 1205821199054 (1 0 0) = 0

1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905) = 01205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905) = 01205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) = 01205821199054119876119905 = 0

(24)

where the first equation represents FOC of Lagrangianand the latter four equations represent the conditions ofcomplementary slackness

With solution of theKKT conditions (119876lowast119905 (120573lowast1119905 120573lowast2119905)) and(4) we can get the optimization pricing and ordering policyof (119876lowast119905 (119901lowast1119905 119901lowast2119905)) at period t where (119901lowast1119905 119901lowast2119905) is as follows119901lowast119894119905 = 119875119894119905 (120573lowast1119905 120573lowast2119905) = 120572120579119894 + ln((1 minus sum119894 120573lowast119894119905)120573lowast119894119905 )

119894 = 1 2(25)

Note that (119876lowast119905 (120573lowast1119905 120573lowast2119905)) depends on state variable 119868119905 andparameters of (ℎ1119905 ℎ2119905) and (1198971119905 1198972119905) as well as 120583119905 which

implies that the retailer decides the optimal policy on thebasis of these parameters at every period t With back-ward induction we get the optimal strategy a sequence of[(119876lowast1 (119901lowast11 119901lowast21)) (119876lowast2 (119901lowast12 119901lowast22)) (119876lowast119879 (119901lowast1119879 119901lowast2119879))]6 Numerical Experiment

We now conduct a numerical example to illustrate our resultsand select the parameters as follows The demand followsthe multiplicative form with iid terms 120596119905 sim U(0 100)for all periods We assume (1205791 1205792)119879 = (10 5)119879 sincefresh product has a higher perceptive value than nonfreshones The cost structure is symmetric h119905 = (05 05)119879l119905 = (45 45) and 119888119905 = 10 For simplicity to compute theoptimal prices we consider a finite horizon with 119879 = 5the discount rate is 120574 = 09 and the initial inventory level119868119905 = 0 Using Matlab to program the dynamic model andget the following results with KKT condition method wecan obtain a sequence of order quantity and optimal pricessuch as [(78 (148 97)) (68 (126 86)) (92 (175 114))(43 (109 62)) (52 (115 102))] at which the maximumprofit 120587lowast = 4 2356 occurs

In Table 1 we compare the performance betweendynamic pricing policy and a general policy under oneperiod lifetime setting where all products are sold at aregular price at any period of time Similarly we assume(1205791 1205792)119879 = (10 5)119879 in dynamic pricing policy while(1205791 1205792)119879 = (10 10)119879 in general policy Note that in generalsetting the optimal pricing and ordering policy can beobtained by way of general dynamic programming methodThree examples of different finite horizon time (119879 = 51020)are conducted with the same cost structure and discount ratementioned above Accordingly the optimal solutions suchas pricing and order quantity as well as dynamic inventoryare shown in Table 1 where superscript g and d representregular policy and dynamic policy respectively

A summary of interesting observations is as follows Thedynamic policy is obviously better than general pricing andinventory policy with which the retailerrsquos profit increasesby 15-20 Further the demand is also matched to a morecertain extent by the supply since the order quantity is almostmore than that under general policy while the Based on theresults we can conclude that dynamic pricing and inventorypolicy is beneficial for retailerrsquos revenue management inselling fresh agriculture products Furthermore from theperspective of agriculture industrial development it also canmaximally reduce the operating costs such as holding costsor lost costs since the optimal match between productssupply and demand

7 Concluding Remark

In this paper we address a joint pricing and inventorycontrol problem for stochastic fresh agriproduct inventorysystems under consumer choice over a finite horizon in thebacklogging case In the model the products are perishablealong time and the customers are sensitive to the ages ofinventories which result in consumer choice behavior based

10 Mathematical Problems in Engineering

Table 1 The optimal solution under dynamic pricing and regular pricing

Periods T Order quantity Pricing Inventory 120587d 120587g (120587119889 minus 120587119892)120587119892Qd Qg (1199011198891 1199011198892) 119901119892 (1198681198891 1198681198892 ) 119868119892

5

78 72 (14897) 125 (490) 9

42356 35782 18468 65 (12686) 104 (312) 892 80 (175114) 128 (5611) 1343 45 (10962) 81 (220) 352 45 (11582) 93 (121) 5

10

81 70 (156107) 122 (520) 10

83438 69844 195

72 61 (13684) 103 (386) 783 58 (165104) 132 (569) 876 59 (11978) 85 (322) 470 73 (12592) 81 (346) 456 41 (16598) 146 (301) 968 59 (12288) 95 (312) 865 47 (154104) 138 (339) 758 56 (12473) 92 (230) 349 39 (12864) 97 (191) 2

20

90 86 (15598) 131 (540) 16

175328 149632 172

72 65 (14385) 104 (366) 883 57 (165101) 111 (409) 966 75 (13276) 86 (314) 572 54 (12995) 92 (346) 456 51 (16392) 132 (353) 768 62 (13287) 98 (313) 465 45 (151122) 136 (335) 457 69 (13879) 97 (280) 348 46 (15592) 102 (211) 281 72 (13871) 98 (430) 270 58 (13992) 104 (386) 778 50 (15468) 119 (492) 571 48 (13288) 95 (333) 061 63 (14582) 84 (356) 856 52 (16578) 146 (373) 1076 48 (11388) 99 (323) 666 64 (14594) 128 (395) 852 49 (13778) 98 (230) 350 46 (14569) 91 (220) 1

on product freshness By using MNL model to describeconsumer discrete choice we construct a stochastic anddynamic inventory model facing strategic customer behaviorwith joint ordering and discounting decisions We are ableto characterize the firmrsquos optimal policy with the use ofa transformation technique and employing the concept ofthe concavity and finally develop a Lagrangian and theKarush-Kuhn-Tucker (KTT) conditions to provide an efficientsolution approach to this optimization problem

A limitation in our model is that we assume that thelifetime of the product is only two periods Although this istrue for some fresh agriproducts in themarket there are otherproducts that are perishable but yet have longer lifetimes Toextend ourmodel to the products withmore than two periods

lifetimes the state space has to be increased more to specifythe amounts of inventory that are going to expire at differentpoints of time in the futureThis definitelymakes the problemquite hard Another issue is to incorporate an operatingcost for discounting pricing practice which however willintroduce additional technical complexity to the dynamicprogram There are several future research directions Oneis to extend the model to the environment where all thedemand can be lost or only non-FAP demand can be lostwhile the fresh FAP can be backlogged instead Another is toconsider the case where the product has a fixed finite lifetimeof exactly 119899(119899 ge 3) periods or random lifetime Finally itwould be interesting to account for stock-out based substitu-tions

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

119864(119876119879 minus 1205731119879120596119879 minus 120576119879)+ as a part of 119867119905(119876119905 (1205731t 1205732119905)) isthus continuously twice differentiable which implies thatΦ119879(119876119879 (1205731119879 1205732119879)) = 119866119879(119876119879 (1205731119879 1205732119879)) minus 120574ℎ2119879+1119864(119876119879 minus1205731119879120596119879 minus 1205762119879)+ is continuously and twice differentiableAssuming now that Φ119905(119876119905 (1205731119905 1205732119905)) is continuouslytwice differentiable for 119905 = 1 2 119879 we prove thatΦ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is also continuously twice differen-tiable according to Theorem 4 Let Λ 119879(119876119879 1205731119879) = 119864(119876119879 minus1205731119879120596119879minus1205762119879)+ = ∬1205731119879119909+119910le119876119879(119876119879minus1205731119879119909minus119910)119891120596(119909)119891120576(119910)119889119909119889119910where 119891120596(119909) and 119891120576(119910) is the pdf of random variable120596119879 and 120576119879 respectively which are generally assumed to beindependent For any realization of 1205762119879 such as 1205762119879 taking thesecond-order derivative ofΛ 119879(119876119879 1205731119879)with respective to119876119879yields 1205972Λ 119879(119876119879 1205731119879)1205971198762119879 = (11205731119879)119891((119876119879 minus 1205762119879)1205731119879) gt 0which implies that Λ 119879(119876119879 1205731119879) is convex in 119876119879 Takingthe second-order derivative with respect to 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)12059712057321119879 = (119876211987912057331119879)119891(1198761198791205731119879) gt 0 whichimplies the concavity of Λ 119879(119876119879 1205731119879) in 1205731119879 And takingthe cross-partial derivative with respect to 119876119879 and 1205731119879 yields1205972Λ 119879(119876119879 1205731119879)1205971198761198791205971205731119879 = minus((119876119879 minus 1205762119879)12057321119879)119891((119876119879 minus1205762119879)1205731119879) Pairing up the above three derivative terms leadsto Hessian Matrix

119867(Λ 119879) = [[[[[

1205972Λ 11987912059711987621198791205972Λ 11987912059711987611987912059712057311198791205972Λ 1198791205971205731119879120597119876119879 1205972Λ 11987912059712057321119879

]]]]]

= [[[[[

11205731119879119891(119876119879 minus 12057621198791205731119879 ) minus119876119879 minus 120576211987912057321119879

119891(119876119879 minus 12057621198791205731119879 )minus119876119879 minus 120576211987912057321119879 119891(119876119879 minus 12057621198791205731119879 ) (119876119879 minus 1205762119879)212057331119879 119891(119876119879 minus 12057621198791205731119879 )

]]]]](19)

And the corresponding determinant 119867(Λ 119879) = 0 whichimplies matrix 119867119890119904119904119894119886119899 is semipositive definite On thebasis of analysis above we can conclude that Λ 119879(119876119879 1205731119879)is jointly convex in (119876119879 1205731119879) Meanwhile the last termminus120574ℎ2119879+1119864(119876119879 minus 1205731119879120596119879 minus 1205762119879)+ is obviously justified to bejointly concave in (119876119879 1205731119879) According to the concavityproperty (Stephen Boyd Lieven Vandenberghe 2004)Φ119879(119876119879 (1205731T 1205732119879)) is jointly concave in (1205731t 1205732119905) and 119876119905

120587119879(119868119879) is easily verified to be concave as well and it isobviously nonincreasing

Assume now that Φ119905(119876119905 (1205731t 1205732119905)) is jointly concave forsome 119905 = 1 2 119879 minus 1 and that 120587119905(119868119905) is also concaveand nonincreasing Then Φ119905minus1(119876119905minus1 (1205731tminus1 1205732119905minus1)) is jointlyconcave joint concavity of the first term of (10) is verified forthe case 119905 = 119879 above For any pair of points (119876119905 1205731119905) and(1198761015840119905 12057310158401119905) with given value of 120596119905 we have120587119905+1 ((120572119876119905 + (1 minus 120572)1198761015840119905)119905 minus (1205721205731119905 + (1 minus 120572) 12057310158401119905) 120596119905minus (1205721205761119905 + (1 minus 120572) 1205761119905)) = 120587119905+1 (120572 (119876119905 minus 1205731119905120596119905 minus 1205761119905)+ (1 minus 120572) (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)) ge 120572120587119905+1 (119876119905 minus 1205731119905120596119905minus 1205761119905) + (1 minus 120572) 120587119905+1 (1198761015840119905 minus 12057310158401119905120596119905 minus 1205761119905)

(20)

by the concavity of 120587119905+1(sdot) which implies that 119864120587119905+1(119876119905 minus1205731119905120596119905 minus 1205761119905) is jointly concave in 119876119905 and 1205731119905 With aboveargument the desired result holds

5 Optimal Policy

Following the structure property of model we can obtain thefollowing result

Theorem 6 Consider any period t with the initial inventory119868119905 there is a unique maximizer (119876lowast119905 (120573lowast1119905 120573lowast2119905)) to maximizethe constrained expected periodical profit function

(119876lowast119905 (120573lowast1119905 120573lowast2119905)) isin arg max119876119905ge0(1205731t 1205732119905)isinΩ119905

Φ119905 (119876119905 (1205731t 1205732119905)) (21)

Proof It can be easily proved on the basis of Theorem 5

On the basis of the dynamic programming analysisincluding the concavity property we can construct an opti-mization problem for any period t with constrained condi-tions as follows

119872119894119899119894119898119894119911119890 minus Φ119905 (119876119905 (1205731t 1205732119905))

119904119905

(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905))(1205731t minus exp (1205721205791 minus 119901119905)1 + exp (1205721205791 minus 119901119905) 119905) le 0(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905))(1205732t minus exp (1205721205792 minus 119901119905)1 + exp (1205721205792 minus 119901119905)) le 0(1205731t + 1205732119905 minus sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905))(1205731t + 1205732119905 minus

sum119894=12 exp (120572120579119894 minus 119901119905)1 + sum119894=12 exp (120572120579119894 minus 119901119905)) le 0minus119876119905 le 0

(22)

Mathematical Problems in Engineering 9

The Lagrangian and the Karush-Kuhn-Tucker(KTT) condi-tions provide an efficient solution approach to this optimiza-tion problemDenote119860119905 = (exp(1205721205791minus119901119905))(1+exp(1205721205791minus119901119905))119860119905 = (exp(1205721205791 minus 119901119905))(1 + exp(1205721205791 minus 119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119862119905 = (sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))

and 119862119905 = (sum119894=12 exp(120572120579119894 minus119901119905))(1 +sum119894=12 exp(120572120579119894 minus119901119905)) Weconstruct for each period t theLagrangian of the optimizationproblem as

119871 119905 (119876119905 (1205731t 1205732119905) (1205821199051 1205821199052 1205821199053 1205821199054))= minusΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905)+ 1205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905)+ 1205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) minus 1205821199054119876119905

(23)

where the Lagrange multipliers are 120582119905119894 ge 0 119894 = 1 2 3 4 Let(119876lowast119905 (120573lowast1119905 120573lowast2119905)) be the optimizing solution we can constructthe KKT conditions as follows

minus nablaΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (0 21205731t minus 119860119905 minus 119860119905 0)+ 1205821199052 (0 0 21205732t minus 119861119905 minus 119861119905) + 1205821199053 (0 21205731t + 21205732119905minus 119862119905 minus 119862119905 21205731t + 21205732119905 minus 119862119905 minus 119862) minus 1205821199054 (1 0 0) = 0

1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905) = 01205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905) = 01205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) = 01205821199054119876119905 = 0

(24)

where the first equation represents FOC of Lagrangianand the latter four equations represent the conditions ofcomplementary slackness

With solution of theKKT conditions (119876lowast119905 (120573lowast1119905 120573lowast2119905)) and(4) we can get the optimization pricing and ordering policyof (119876lowast119905 (119901lowast1119905 119901lowast2119905)) at period t where (119901lowast1119905 119901lowast2119905) is as follows119901lowast119894119905 = 119875119894119905 (120573lowast1119905 120573lowast2119905) = 120572120579119894 + ln((1 minus sum119894 120573lowast119894119905)120573lowast119894119905 )

119894 = 1 2(25)

Note that (119876lowast119905 (120573lowast1119905 120573lowast2119905)) depends on state variable 119868119905 andparameters of (ℎ1119905 ℎ2119905) and (1198971119905 1198972119905) as well as 120583119905 which

implies that the retailer decides the optimal policy on thebasis of these parameters at every period t With back-ward induction we get the optimal strategy a sequence of[(119876lowast1 (119901lowast11 119901lowast21)) (119876lowast2 (119901lowast12 119901lowast22)) (119876lowast119879 (119901lowast1119879 119901lowast2119879))]6 Numerical Experiment

We now conduct a numerical example to illustrate our resultsand select the parameters as follows The demand followsthe multiplicative form with iid terms 120596119905 sim U(0 100)for all periods We assume (1205791 1205792)119879 = (10 5)119879 sincefresh product has a higher perceptive value than nonfreshones The cost structure is symmetric h119905 = (05 05)119879l119905 = (45 45) and 119888119905 = 10 For simplicity to compute theoptimal prices we consider a finite horizon with 119879 = 5the discount rate is 120574 = 09 and the initial inventory level119868119905 = 0 Using Matlab to program the dynamic model andget the following results with KKT condition method wecan obtain a sequence of order quantity and optimal pricessuch as [(78 (148 97)) (68 (126 86)) (92 (175 114))(43 (109 62)) (52 (115 102))] at which the maximumprofit 120587lowast = 4 2356 occurs

In Table 1 we compare the performance betweendynamic pricing policy and a general policy under oneperiod lifetime setting where all products are sold at aregular price at any period of time Similarly we assume(1205791 1205792)119879 = (10 5)119879 in dynamic pricing policy while(1205791 1205792)119879 = (10 10)119879 in general policy Note that in generalsetting the optimal pricing and ordering policy can beobtained by way of general dynamic programming methodThree examples of different finite horizon time (119879 = 51020)are conducted with the same cost structure and discount ratementioned above Accordingly the optimal solutions suchas pricing and order quantity as well as dynamic inventoryare shown in Table 1 where superscript g and d representregular policy and dynamic policy respectively

A summary of interesting observations is as follows Thedynamic policy is obviously better than general pricing andinventory policy with which the retailerrsquos profit increasesby 15-20 Further the demand is also matched to a morecertain extent by the supply since the order quantity is almostmore than that under general policy while the Based on theresults we can conclude that dynamic pricing and inventorypolicy is beneficial for retailerrsquos revenue management inselling fresh agriculture products Furthermore from theperspective of agriculture industrial development it also canmaximally reduce the operating costs such as holding costsor lost costs since the optimal match between productssupply and demand

7 Concluding Remark

In this paper we address a joint pricing and inventorycontrol problem for stochastic fresh agriproduct inventorysystems under consumer choice over a finite horizon in thebacklogging case In the model the products are perishablealong time and the customers are sensitive to the ages ofinventories which result in consumer choice behavior based

10 Mathematical Problems in Engineering

Table 1 The optimal solution under dynamic pricing and regular pricing

Periods T Order quantity Pricing Inventory 120587d 120587g (120587119889 minus 120587119892)120587119892Qd Qg (1199011198891 1199011198892) 119901119892 (1198681198891 1198681198892 ) 119868119892

5

78 72 (14897) 125 (490) 9

42356 35782 18468 65 (12686) 104 (312) 892 80 (175114) 128 (5611) 1343 45 (10962) 81 (220) 352 45 (11582) 93 (121) 5

10

81 70 (156107) 122 (520) 10

83438 69844 195

72 61 (13684) 103 (386) 783 58 (165104) 132 (569) 876 59 (11978) 85 (322) 470 73 (12592) 81 (346) 456 41 (16598) 146 (301) 968 59 (12288) 95 (312) 865 47 (154104) 138 (339) 758 56 (12473) 92 (230) 349 39 (12864) 97 (191) 2

20

90 86 (15598) 131 (540) 16

175328 149632 172

72 65 (14385) 104 (366) 883 57 (165101) 111 (409) 966 75 (13276) 86 (314) 572 54 (12995) 92 (346) 456 51 (16392) 132 (353) 768 62 (13287) 98 (313) 465 45 (151122) 136 (335) 457 69 (13879) 97 (280) 348 46 (15592) 102 (211) 281 72 (13871) 98 (430) 270 58 (13992) 104 (386) 778 50 (15468) 119 (492) 571 48 (13288) 95 (333) 061 63 (14582) 84 (356) 856 52 (16578) 146 (373) 1076 48 (11388) 99 (323) 666 64 (14594) 128 (395) 852 49 (13778) 98 (230) 350 46 (14569) 91 (220) 1

on product freshness By using MNL model to describeconsumer discrete choice we construct a stochastic anddynamic inventory model facing strategic customer behaviorwith joint ordering and discounting decisions We are ableto characterize the firmrsquos optimal policy with the use ofa transformation technique and employing the concept ofthe concavity and finally develop a Lagrangian and theKarush-Kuhn-Tucker (KTT) conditions to provide an efficientsolution approach to this optimization problem

A limitation in our model is that we assume that thelifetime of the product is only two periods Although this istrue for some fresh agriproducts in themarket there are otherproducts that are perishable but yet have longer lifetimes Toextend ourmodel to the products withmore than two periods

lifetimes the state space has to be increased more to specifythe amounts of inventory that are going to expire at differentpoints of time in the futureThis definitelymakes the problemquite hard Another issue is to incorporate an operatingcost for discounting pricing practice which however willintroduce additional technical complexity to the dynamicprogram There are several future research directions Oneis to extend the model to the environment where all thedemand can be lost or only non-FAP demand can be lostwhile the fresh FAP can be backlogged instead Another is toconsider the case where the product has a fixed finite lifetimeof exactly 119899(119899 ge 3) periods or random lifetime Finally itwould be interesting to account for stock-out based substitu-tions

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

The Lagrangian and the Karush-Kuhn-Tucker(KTT) condi-tions provide an efficient solution approach to this optimiza-tion problemDenote119860119905 = (exp(1205721205791minus119901119905))(1+exp(1205721205791minus119901119905))119860119905 = (exp(1205721205791 minus 119901119905))(1 + exp(1205721205791 minus 119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119861119905 = (exp(1205721205792 minus119901119905))(1 + exp(1205721205792 minus119901119905)) 119862119905 = (sum119894=12 exp(120572120579119894 minus 119901119905))(1 + sum119894=12 exp(120572120579119894 minus 119901119905))

and 119862119905 = (sum119894=12 exp(120572120579119894 minus119901119905))(1 +sum119894=12 exp(120572120579119894 minus119901119905)) Weconstruct for each period t theLagrangian of the optimizationproblem as

119871 119905 (119876119905 (1205731t 1205732119905) (1205821199051 1205821199052 1205821199053 1205821199054))= minusΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905)+ 1205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905)+ 1205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) minus 1205821199054119876119905

(23)

where the Lagrange multipliers are 120582119905119894 ge 0 119894 = 1 2 3 4 Let(119876lowast119905 (120573lowast1119905 120573lowast2119905)) be the optimizing solution we can constructthe KKT conditions as follows

minus nablaΦ119905 (119876119905 (1205731t 1205732119905)) + 1205821199051 (0 21205731t minus 119860119905 minus 119860119905 0)+ 1205821199052 (0 0 21205732t minus 119861119905 minus 119861119905) + 1205821199053 (0 21205731t + 21205732119905minus 119862119905 minus 119862119905 21205731t + 21205732119905 minus 119862119905 minus 119862) minus 1205821199054 (1 0 0) = 0

1205821199051 (1205731t minus 119860119905) (1205731t minus 119860119905) = 01205821199052 (1205732t minus 119861119905) (1205732t minus 119861119905) = 01205821199053 (1205731t + 1205732119905 minus 119862119905) (1205731t + 1205732119905 minus 119862119905) = 01205821199054119876119905 = 0

(24)

where the first equation represents FOC of Lagrangianand the latter four equations represent the conditions ofcomplementary slackness

With solution of theKKT conditions (119876lowast119905 (120573lowast1119905 120573lowast2119905)) and(4) we can get the optimization pricing and ordering policyof (119876lowast119905 (119901lowast1119905 119901lowast2119905)) at period t where (119901lowast1119905 119901lowast2119905) is as follows119901lowast119894119905 = 119875119894119905 (120573lowast1119905 120573lowast2119905) = 120572120579119894 + ln((1 minus sum119894 120573lowast119894119905)120573lowast119894119905 )

119894 = 1 2(25)

Note that (119876lowast119905 (120573lowast1119905 120573lowast2119905)) depends on state variable 119868119905 andparameters of (ℎ1119905 ℎ2119905) and (1198971119905 1198972119905) as well as 120583119905 which

implies that the retailer decides the optimal policy on thebasis of these parameters at every period t With back-ward induction we get the optimal strategy a sequence of[(119876lowast1 (119901lowast11 119901lowast21)) (119876lowast2 (119901lowast12 119901lowast22)) (119876lowast119879 (119901lowast1119879 119901lowast2119879))]6 Numerical Experiment

We now conduct a numerical example to illustrate our resultsand select the parameters as follows The demand followsthe multiplicative form with iid terms 120596119905 sim U(0 100)for all periods We assume (1205791 1205792)119879 = (10 5)119879 sincefresh product has a higher perceptive value than nonfreshones The cost structure is symmetric h119905 = (05 05)119879l119905 = (45 45) and 119888119905 = 10 For simplicity to compute theoptimal prices we consider a finite horizon with 119879 = 5the discount rate is 120574 = 09 and the initial inventory level119868119905 = 0 Using Matlab to program the dynamic model andget the following results with KKT condition method wecan obtain a sequence of order quantity and optimal pricessuch as [(78 (148 97)) (68 (126 86)) (92 (175 114))(43 (109 62)) (52 (115 102))] at which the maximumprofit 120587lowast = 4 2356 occurs

In Table 1 we compare the performance betweendynamic pricing policy and a general policy under oneperiod lifetime setting where all products are sold at aregular price at any period of time Similarly we assume(1205791 1205792)119879 = (10 5)119879 in dynamic pricing policy while(1205791 1205792)119879 = (10 10)119879 in general policy Note that in generalsetting the optimal pricing and ordering policy can beobtained by way of general dynamic programming methodThree examples of different finite horizon time (119879 = 51020)are conducted with the same cost structure and discount ratementioned above Accordingly the optimal solutions suchas pricing and order quantity as well as dynamic inventoryare shown in Table 1 where superscript g and d representregular policy and dynamic policy respectively

A summary of interesting observations is as follows Thedynamic policy is obviously better than general pricing andinventory policy with which the retailerrsquos profit increasesby 15-20 Further the demand is also matched to a morecertain extent by the supply since the order quantity is almostmore than that under general policy while the Based on theresults we can conclude that dynamic pricing and inventorypolicy is beneficial for retailerrsquos revenue management inselling fresh agriculture products Furthermore from theperspective of agriculture industrial development it also canmaximally reduce the operating costs such as holding costsor lost costs since the optimal match between productssupply and demand

7 Concluding Remark

In this paper we address a joint pricing and inventorycontrol problem for stochastic fresh agriproduct inventorysystems under consumer choice over a finite horizon in thebacklogging case In the model the products are perishablealong time and the customers are sensitive to the ages ofinventories which result in consumer choice behavior based

10 Mathematical Problems in Engineering

Table 1 The optimal solution under dynamic pricing and regular pricing

Periods T Order quantity Pricing Inventory 120587d 120587g (120587119889 minus 120587119892)120587119892Qd Qg (1199011198891 1199011198892) 119901119892 (1198681198891 1198681198892 ) 119868119892

5

78 72 (14897) 125 (490) 9

42356 35782 18468 65 (12686) 104 (312) 892 80 (175114) 128 (5611) 1343 45 (10962) 81 (220) 352 45 (11582) 93 (121) 5

10

81 70 (156107) 122 (520) 10

83438 69844 195

72 61 (13684) 103 (386) 783 58 (165104) 132 (569) 876 59 (11978) 85 (322) 470 73 (12592) 81 (346) 456 41 (16598) 146 (301) 968 59 (12288) 95 (312) 865 47 (154104) 138 (339) 758 56 (12473) 92 (230) 349 39 (12864) 97 (191) 2

20

90 86 (15598) 131 (540) 16

175328 149632 172

72 65 (14385) 104 (366) 883 57 (165101) 111 (409) 966 75 (13276) 86 (314) 572 54 (12995) 92 (346) 456 51 (16392) 132 (353) 768 62 (13287) 98 (313) 465 45 (151122) 136 (335) 457 69 (13879) 97 (280) 348 46 (15592) 102 (211) 281 72 (13871) 98 (430) 270 58 (13992) 104 (386) 778 50 (15468) 119 (492) 571 48 (13288) 95 (333) 061 63 (14582) 84 (356) 856 52 (16578) 146 (373) 1076 48 (11388) 99 (323) 666 64 (14594) 128 (395) 852 49 (13778) 98 (230) 350 46 (14569) 91 (220) 1

on product freshness By using MNL model to describeconsumer discrete choice we construct a stochastic anddynamic inventory model facing strategic customer behaviorwith joint ordering and discounting decisions We are ableto characterize the firmrsquos optimal policy with the use ofa transformation technique and employing the concept ofthe concavity and finally develop a Lagrangian and theKarush-Kuhn-Tucker (KTT) conditions to provide an efficientsolution approach to this optimization problem

A limitation in our model is that we assume that thelifetime of the product is only two periods Although this istrue for some fresh agriproducts in themarket there are otherproducts that are perishable but yet have longer lifetimes Toextend ourmodel to the products withmore than two periods

lifetimes the state space has to be increased more to specifythe amounts of inventory that are going to expire at differentpoints of time in the futureThis definitelymakes the problemquite hard Another issue is to incorporate an operatingcost for discounting pricing practice which however willintroduce additional technical complexity to the dynamicprogram There are several future research directions Oneis to extend the model to the environment where all thedemand can be lost or only non-FAP demand can be lostwhile the fresh FAP can be backlogged instead Another is toconsider the case where the product has a fixed finite lifetimeof exactly 119899(119899 ge 3) periods or random lifetime Finally itwould be interesting to account for stock-out based substitu-tions

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

Table 1 The optimal solution under dynamic pricing and regular pricing

Periods T Order quantity Pricing Inventory 120587d 120587g (120587119889 minus 120587119892)120587119892Qd Qg (1199011198891 1199011198892) 119901119892 (1198681198891 1198681198892 ) 119868119892

5

78 72 (14897) 125 (490) 9

42356 35782 18468 65 (12686) 104 (312) 892 80 (175114) 128 (5611) 1343 45 (10962) 81 (220) 352 45 (11582) 93 (121) 5

10

81 70 (156107) 122 (520) 10

83438 69844 195

72 61 (13684) 103 (386) 783 58 (165104) 132 (569) 876 59 (11978) 85 (322) 470 73 (12592) 81 (346) 456 41 (16598) 146 (301) 968 59 (12288) 95 (312) 865 47 (154104) 138 (339) 758 56 (12473) 92 (230) 349 39 (12864) 97 (191) 2

20

90 86 (15598) 131 (540) 16

175328 149632 172

72 65 (14385) 104 (366) 883 57 (165101) 111 (409) 966 75 (13276) 86 (314) 572 54 (12995) 92 (346) 456 51 (16392) 132 (353) 768 62 (13287) 98 (313) 465 45 (151122) 136 (335) 457 69 (13879) 97 (280) 348 46 (15592) 102 (211) 281 72 (13871) 98 (430) 270 58 (13992) 104 (386) 778 50 (15468) 119 (492) 571 48 (13288) 95 (333) 061 63 (14582) 84 (356) 856 52 (16578) 146 (373) 1076 48 (11388) 99 (323) 666 64 (14594) 128 (395) 852 49 (13778) 98 (230) 350 46 (14569) 91 (220) 1

on product freshness By using MNL model to describeconsumer discrete choice we construct a stochastic anddynamic inventory model facing strategic customer behaviorwith joint ordering and discounting decisions We are ableto characterize the firmrsquos optimal policy with the use ofa transformation technique and employing the concept ofthe concavity and finally develop a Lagrangian and theKarush-Kuhn-Tucker (KTT) conditions to provide an efficientsolution approach to this optimization problem

A limitation in our model is that we assume that thelifetime of the product is only two periods Although this istrue for some fresh agriproducts in themarket there are otherproducts that are perishable but yet have longer lifetimes Toextend ourmodel to the products withmore than two periods

lifetimes the state space has to be increased more to specifythe amounts of inventory that are going to expire at differentpoints of time in the futureThis definitelymakes the problemquite hard Another issue is to incorporate an operatingcost for discounting pricing practice which however willintroduce additional technical complexity to the dynamicprogram There are several future research directions Oneis to extend the model to the environment where all thedemand can be lost or only non-FAP demand can be lostwhile the fresh FAP can be backlogged instead Another is toconsider the case where the product has a fixed finite lifetimeof exactly 119899(119899 ge 3) periods or random lifetime Finally itwould be interesting to account for stock-out based substitu-tions

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

This paper was previously presented as an abstract in2016 International Symposium for Facing New Conditionsand Challenges the Agricultural Development in USA andChina

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by fund of Guandong ProvinceHumanities and Social Science Base for Colleges and Univer-sities Foundation Grant 11JDXM63004 ldquoCollaborative Pro-posal Agri-Business Supply Chain Contract and Coordi-nation Mechanismrdquo It is also funded by National NaturalScience Foundation of China (71501051) and Philosophyand Social Science Foundation of Guangdong Province(GD15XGL42) I should like to acknowledge the helpfulcomments on an earlier draft

References

[1] P Ghare M and G F Schrader ldquoAn inventory model for dete-riorating item for exponentially deteriorating itemsrdquo Journal ofIndustrial Engineering vol 14 pp 238ndash243 1963

[2] R P Covert and G C Philip ldquoAn EOQ model for items withWeibull distribution deteriorationrdquo IIE Transactions vol 5 no4 pp 323ndash326 1973

[3] P R Tadikamalla ldquoAn eoq inventory model for items withgamma distributed deteriorationrdquo IIE Transactions vol 10 no1 pp 100ndash103 1978

[4] A Rajan Rakesh and R Steinberg ldquoDynamic Pricing andOrdering Decisions by aMonopolistrdquoManagement Science vol38 no 2 pp 240ndash262 1992

[5] S Nahmias and W P Pierskalla ldquoOptimal ordering policiesfor a product that perishes in two periods subject to stochasticdemandrdquo Naval Research Logistics Quarterly vol 20 pp 207ndash229 1973

[6] S Nahmias ldquoOptimal Ordering Policies for Perishable Inven-torymdashIIrdquo Operations Research vol 23 no 4 pp 735ndash749 1975

[7] B E Fries ldquoOptimal ordering policy for a perishable commod-ity with fixed lifetimerdquo Operations Research vol 23 no 1 pp46ndash61 1975

[8] S Nahmias ldquoMyopic approximations for the perishable inven-tory problemrdquo Management Science vol 22 no 9 pp 1002ndash1008 1975

[9] M A Cohen ldquoAnalysis of single critical number orderingpolicies for perishable inventoriesrdquoOperations Research vol 24no 4 pp 726ndash741 1976

[10] P Nadakumar and T E Morton ldquoNear myopic heuristics forthe fixed-life perishability problemrdquo Management Science vol39 no 12 pp 1490ndash1498 1993

[11] L Liu andZ Lian ldquo(s S) continuous reviewmodels for productswith fixed lifetimesrdquoOperations Research vol 47 no 1 pp 150ndash158 1999

[12] Z Lian and L Liu ldquoContinuous review perishable inventorysystemsModels and heuristicsrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 33 no 9 pp 809ndash822 2001

[13] U Gurler and Yuksel O B ldquoA note on ldquoContinuous reviewperishable inventory systems models and heuristicsrdquordquo IIETransactions vol 35 no 3 pp 321ndash323 2003

[14] E Berk and U Gurler ldquoAnalysis of the (Q r) inventorymodel for perishables with positive lead times and lost salesrdquoOperations Research vol 56 no 5 pp 1238ndash1246 2008

[15] S Nahmias ldquoPerishable inventory theory a reviewrdquoOperationsResearch vol 30 no 4 pp 680ndash708 1982

[16] S Nahmias Perishable inventory systems (International series inoperations research and management science) 2011

[17] I Z Karaesmen A Scheller-Wolf and B Deniz ldquoManagingperishable and aging inventories Review and future researchdirectionsrdquo International Series in Operations Research andManagement Science vol 151 pp 393ndash436 2011

[18] Z Xue M Ettl and D Yao Managing freshness inventoryOptimal policy bounds and heuristics Working paper IEORDepartment Columbia University NY 2012

[19] Q Li and P Yu ldquoMultimodularity and Its Applications inThree Stochastic Dynamic Inventory ProblemsrdquoManufacturingamp Service Operations Management vol 16 no 3 pp 455ndash4632014

[20] X Chao X Gong C Shi and H Zhang ldquoApproximation algo-rithms for perishable inventory systemsrdquo Operations Researchvol 63 no 3 pp 585ndash601 2015

[21] H ZhangC Shi andXChao ldquoTechnical notemdashapproximationalgorithms for perishable inventory systems with setup costsrdquoOperations Research vol 64 no 2 pp 432ndash440 2016

[22] X Chao X Gong C Shi C Yang H Zhang and S XZhou ldquoApproximation Algorithms for Capacitated PerishableInventory Systems with Positive Lead Timesrdquo ManagementScience vol 64 no 11 pp 5038ndash5061 2018

[23] H M Wagner and T M Whitin ldquoDynamic version of theeconomic lot size modelrdquoManagement Science vol 5 no 1 pp89ndash96 1958

[24] H Kunreuther and L Schrage ldquoJoint Pricing and InventoryDecisions for Constant Priced ItemsrdquoManagement Science vol19 no 7 pp 732ndash738 1973

[25] E Zabel ldquoMultiperiod monopoly under uncertaintyrdquo Journal ofEconomic eory vol 5 no 3 pp 524ndash536 1972

[26] J Thomas ldquoPrice and Production Decisions with RandomDemandrdquo Operations Research vol 22 no 3 pp 513ndash518 1974

[27] S M Gilbert ldquoCoordination of pricing and multi-period pro-duction for constant priced goodsrdquo European Journal of Opera-tional Research vol 114 no 2 pp 330ndash337 1999

[28] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[29] A Federgruen and A Heching ldquoCombined pricing and inven-tory control under uncertaintyrdquoOperations Research vol 47 no3 pp 454ndash475 1999

[30] Y Feng and F Y Chen ldquoJoint pricing and inventory control withsetup costs and demand uncertaintyrdquoe Chinese University ofHong Kong 2003

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

[31] J Geunes H E Romeijn and K Taaffe ldquoRequirements plan-ning with pricing and order selection flexibilityrdquo OperationsResearch vol 54 no 2 pp 394ndash401 2006

[32] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies with random demand and fixed orderingcost the finite horizon caserdquoOperations Research vol 52 no 6pp 887ndash896 2004

[33] X Chen and D Simchi-Levi ldquoCoordinating inventory controland pricing strategies the continuous review modelrdquo Opera-tions Research Letters vol 34 no 3 pp 323ndash332 2006

[34] G Raz and E L Porteus ldquoA fractiles perspective to the jointpricequantity newsvendor modelrdquo Management Science vol52 no 11 pp 1764ndash1777 2006

[35] W T Huh and G Janakiraman ldquo(sS) optimality in jointinventory-pricing control an alternate approachrdquo OperationsResearch vol 56 no 3 pp 783ndash790 2008

[36] X Chen ldquoInventory centralization games with price-dependentdemand and quantity discountrdquoOperations Research vol 57 no6 pp 1394ndash1406 2009

[37] Y Song S Ray and T Boyaci ldquoOptimal dynamic jointinventory-pricing control for multiplicative demand with fixedorder costs and lost salesrdquoOperations Research vol 57 no 1 pp245ndash250 2009

[38] D Simchi-Levi X Chen and J Bramel e Logic of Logisticseory Algorithms and Applications for Logistics ManagementSpringer New York 3rd edition 2014

[39] J Eliashberg and R Steinberg ldquoCompetitive Strategies forTwo Firms with Asymmetric Production Cost StructuresrdquoManagement Science vol 37 no 11 pp 1452ndash1473 1991

[40] W Elmaghraby and P Keskinocak ldquoDynamic pricing in thepresence of inventory considerations research overview cur-rent practices and future directionsrdquoManagement Science vol49 no 10 pp 1287ndash1309 2003

[41] A Yano and S M Gilbert ldquoCoordinated Pricing and Produc-tionProcurement Decisions A Reviewrdquo in Managing BusinessInterfaces Marketing Engineering and Manufacturing Perspec-tives J Eliashberg and A Chakravarty Eds Kluwer NorwellMA 2003

[42] L M Chan Z J Shen D Simchi-Levi and J L SwannldquoCoordination of Pricing and Inventory Decisions A Surveyand Classificationrdquo in Handbook of Quantitative Supply ChainAnalysis vol 74 of International Series in Operations ResearchampManagement Science pp 335ndash392 Springer US Boston MA2004

[43] X Chen and D Simchi-Levi ldquoPricing and Inventory Manage-mentrdquo e Oxford Handbook of Pricing Management pp 784ndash822 2012

[44] P L Abad ldquoOptimal pricing and lot-sizing under conditionsof perishability and partial backorderingrdquoManagement Sciencevol 42 no 8 pp 1093ndash1104 1996

[45] M E Ferguson and O Koenigsberg ldquoHow should a firm man-age deteriorating inventoryrdquo Production Engineering Researchand Development vol 16 no 3 pp 306ndash321 2007

[46] Y Li A Lim and B Rodrigues ldquoPricing and inventory controlfor a perishable productrdquoManufacturing and Service OperationsManagement vol 11 no 3 pp 538ndash542 2009

[47] Y Li B Cheang and A Lim ldquoGrocery perishables manage-mentrdquo Production Engineering Research and Development vol21 no 3 pp 504ndash517 2012

[48] A Sainathan ldquoPricing and replenishment of competing per-ishable product variants under dynamic demand substitutionrdquo

Production Engineering Research and Development vol 22 no5 pp 1157ndash1181 2013

[49] X Chen Z Pang and L Pan ldquoCoordinating inventory controland pricing strategies for perishable productsrdquo OperationsResearch vol 62 no 2 pp 284ndash300 2014

[50] S Wu Q Liu and R Q Zhang ldquoThe reference effects on aretailerrsquos dynamic pricing and inventory strategies with strategicconsumersrdquo Operations Research vol 63 no 6 pp 1320ndash13352015

[51] Y Chen and L Y Chu Synchronizing pricing and replenishmentto serve forward-looking customers with lost sales WorkingPaper University of Southern California Los Angeles CA2016

[52] W M Kincaid and D A Darling ldquoAn inventory pricingproblemrdquo Journal of Mathematical Analysis and Applicationsvol 7 pp 183ndash208 1963

[53] G R Bitran and S VMondschein ldquoPeriodic pricing of seasonalproducts in retailingrdquo Management Science vol 43 no 1 pp64ndash79 1997

[54] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999

[55] A Goker and K Jennifer ldquoProduct line selection and pricingunder themultinomial logit choice modelrdquo in Proceedings of the2000 MSOM conference 2000

[56] D Zhang andW L Cooper ldquoRevenue management for parallelflights with customer-choice behaviorrdquo Operations Researchvol 53 no 3 pp 415ndash431 2005

[57] XXu andW JHopp ldquoAmonopolistic and oligopolistic stochas-tic flow revenue management modelrdquo Operations Research vol54 no 6 pp 1098ndash1109 2006

[58] G Perakis and A Sood ldquoCompetitive multi-period pricing forperishable products a robust optimization approachrdquo Mathe-matical Programming vol 107 no 1-2 Ser B pp 295ndash335 2006

[59] K Y Lin and S Y Sibdari ldquoDynamic price competition withdiscrete customer choicesrdquo European Journal of OperationalResearch vol 197 no 3 pp 969ndash980 2009

[60] S Jing-Sheng and X Zhengliang Demand management andinventory control for substitutable products Working paper2007

[61] VMartınez-De-Albeniz and K Talluri ldquoDynamic price compe-tition with fixed capacitiesrdquoManagement Science vol 57 no 6pp 1078ndash1093 2011

[62] P M Guadagni and J D Little ldquoA Logit Model of Brand ChoiceCalibrated on Scanner DatardquoMarketing Science vol 2 no 3 pp203ndash238 1983

[63] S P Anderson A de Palma and J-F Thisse Discrete Choiceeory of Product Differentiation MIT Press Cambridge MassUSA 1992

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom