optimal policies for the pricing and replenishment of

Research ArticleOptimal Policies for the Pricing and Replenishment of FashionApparel considering the Effect of Fashion Level

Qi Chen Qi Xu and Wenjie Wang

Glorious Sun School of Business and Management Donghua University Shanghai 200051 China

Correspondence should be addressed to Qi Xu xuqidhueducn

Received 29 October 2018 Revised 21 January 2019 Accepted 4 February 2019 Published 17 February 2019

Academic Editor Dehua Shen

Copyright copy 2019 Qi Chen et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Fashion apparel with short product lifecycles and highly volatile demand requires careful attention during both the initial orderingperiods before the selling season and during the selling season with its decisions regarding price and replenishment UsingPontryaginrsquos maximum principle method this study investigates the problem of the dynamic pricing strategy and replenishmentcycle for fashion apparel by considering the effect of fashion level on demand First we provide a framework for fashion apparelby formulating a model that includes both price and demand at different fashion levels We then provide an algorithm to derivethe optimal dynamic pricing strategy and replenishment cycle Numerical examples and sensitivity analyses of the main systemparameters are provided to demonstrate the obtained results which form the basis for managerial insights It is shown thatthe apparel retailer has three types of optimal dynamic pricing strategies and that the optimal strategy is independent of thereplenishment cycle The apparel retailer is able to realize the profit advantage of a continuously variable price policy by adjustingthe sales price periodically

1 Introduction

Fast fashion is an industrial practice widely applied in fashionretailing For example famous international retail brandssuch as HampM Top Shop and Zara have all implementedfast fashion [1] The sales of fashion garments are not onlyinfluenced by price but also by their fashion level whichlargely depends on the elements of style color and materialFashion products are quite different from other traditionalproducts In general the higher the fashion level the highertheir market value However fashion level drops over timeresulting in a continuous decline in market value a rapidreduction of demand and product backlogs The latest datashowed that the fashion industry faces excess inventory Forexample HampM the famous fast fashion brand companyannounced at the end of March 2018 that the total value ofits unsold accumulation of clothes was $4 billion In 2016 thetotal inventory of 79 textile and apparel quoted companies inChina amounted to $81 billion

In the fashion retail industry inventory planning deci-sions are crucial Having a short product lifecycle and highlyvolatile demand fashion items require careful attention in

the initial ordering process before selling season and in thedecisions relating to price and replenishment during theselling season [2] Some apparel firms apply dynamic pricingstrategies based on adjusting prices during the selling seasonto promote demand lower inventory levels and realizegreater profit [3] Such strategies involve charging high priceswhen the fashion apparelrsquos value is relatively high and laterwhen the value decreases the items are sold at a discountprice For example Uniqlo GAP Nike and Zara use dynamicpricing strategies which affect consumersrsquo selection Uniqloa Japanese fashion retailer successfully implemented thesestrategies thus demonstrating that it is better to reduce pricesat the beginning of the year and sell its apparel at a 60discount in a season than to raise prices by 30 at the endof the year [3]

However the abovementioned pricing strategy ignorescertain attributes of the fashion apparel industry Over timethe market value of fashion apparel continuously decreasesIn recent years due to the rapid evolution of informationtechnologies it has become increasingly easier to implementdynamic pricing Therefore apparel retailers should reex-amine their pricing policies and explore different means of

HindawiComplexityVolume 2019 Article ID 9253605 12 pageshttpsdoiorg10115520199253605

2 Complexity

implementing dynamic pricing making them more respon-sive to consumer demand According to the time-varyingcharacteristics of fashion clothing how to implement a time-varying dynamic pricing strategy has obvious practical signif-icance for managing inventory replenishment and improvingnet profits In this light this study aims to investigate thejoint optimal dynamic pricing and replenishment policies ofa fashion apparel retailer by considering fashion level

Nevertheless few research efforts have explicitly consid-ered the inventory decisions of fashion apparel from theperspective of pricing and replenishment cycles Fisher et al[4] studied how replenishment order quantity may minimizethe cost of lost sales back orders and obsolete inventoryin a two-stage model Kogan and Spiegel [5] developed amodel with time and price-dependent demand and an initialinventory level Tsao [6] developed an analytical model forthe inventory control of deteriorating fashion goods underthe conditions of trade credit and partial backlogsWhile theyidentified the optimal pricing and replenishment strategyprice remained a static variable Due to the market value offashion apparel decreasing over time the abovementionedpricing models are not fit for the fashion apparel industryFashion apparel has a short lifecycle and products showchanges in market value similar to instantaneous deteriorat-ing items Here deterioration is defined as decay damagespoilage evaporation obsolescence pilferage loss of utilityor the loss of a commodityrsquos marginal value all of whichdecrease usefulness Most physical goods such as medicinevolatile liquids and fashion goods undergo a process of decayor deterioration [7] Therefore decision-making regardingthe joint dynamic pricing strategy and replenishment cyclefor deteriorating items can be used as a reference for inven-tory management in the fashion apparel industry

For firmsmanaging the inventories of deteriorating itemsrepresents an important challenge that has understandablyattracted widespread attention from both researchers andpractitioners Ghare and Schrader [8] were the first toestablish an economic order quantity model to study theoptimal ordering policy for deteriorating items Wu et al [9]formulated an inventory model with deteriorating items andprice-sensitive demand and obtained the retailerrsquos optimalselling price and the length of replenishment cycle Asdemand for deteriorating items is affected by both priceand time that remains before a productrsquos expiration it isessential to model these simultaneous effects and establisha joint optimal pricing and inventory replenishment policyHowever few researchers have explicitly considered thedependency of demand on both price and time Maihamiet al [10] established joint pricing and inventory control fordeteriorating items with demand dependent on time andprice and with partial backlogging allowed Avinadav et al[11] also used a price- and time-dependent demand functionto develop a mathematical model that calculates the optimalprice and replenishment period of perishable items Li et al[12] considered the problem of joint pricing inventories andinvestment preservation by considering the general price-dependent demand rate In the abovementioned inventorymodels price remains static over the time horizon

In recent years due to the rapid evolution of informa-tion technologies dynamic pricing has become increasinglyviable Consequently firmsrsquo optimal dynamic pricing strategyhas attracted scholarly attention Gallego and Ryzin [13]formulated an inventory model to investigate the problemof dynamic pricing in which demand was price sensitiveand stochasticThere is abundant research on pricing modelsthat explicitly address dynamic pricing [14ndash21] However theabovementioned models rarely regard pricing strategy as aprocess of dynamic change Wang et al [22] developed aninventory model to identify the optimal dynamic pricingstrategy for noninstantaneous deteriorating items Li et al[23] considered the problem of dynamic pricing and periodicordering for deteriorating items with a stochastic inventorylevel that depends on stock-dependent demand and theselling price Herbon et al [24] formulated an inventorymodel with deteriorating items and price-sensitive demandand obtained the retailerrsquos optimal dynamic selling priceand the length of replenishment cycle Tashakkor et al [25]addressed the variability of the deterioration rate in a jointdynamic pricing and replenishment cycle problem where thedeterioration rate is an increasing step function Zhang etal [26] used Pontryaginrsquos maximum principle to obtain boththe optimal dynamic pricing strategy and the replenishmentcycle for noninstantaneous deteriorating items with demanddependent on a storersquos sales price and the quantity of the itemsdisplayed

The study by Tsao [6] is the only one that analyzedthe joint pricing and replenishment cycle decision-makingproblem for fashion goods The studies that are closestto ours are [6 26] Our work mainly differs from theirsin three aspects First in their approach the quantity ofdeteriorating fashion goods continuously decreases duringthe replenishment period [6 26] In contrast in our studywhile fashion apparelrsquos market value decreases over time itsquantity does not Second in our study the fashion leveldecreases over time and exerts a negative effect on thedemand rate This factor however is not considered in theproblem of pricing fashion goods in Tsaorsquos model [6] In ourmodel of fashion apparel we assume that the demand ratedepends on both sales price and fashion level which is amorerealistic assumption To the best of our knowledge our studyis the first to operate under this assumption when analyzingthe joint optimal dynamic pricing strategy and replenishmentcycle for fashion apparel Finally in Tsaorsquos model price wasa static variable [6] In this study on fashion apparel weuse optimal control theory to identify the optimal dynamicpricing strategy and replenishment cycle that maximizesaverage profit per unit time This optimization problem canbe solved hierarchically such that at a lower level of thehierarchy we derive the dynamic pricing strategy for thegiven replenishment cycle by applying Pontryaginrsquos maxi-mum principle Then at a higher level of the hierarchy wederive the optimal replenishment cycle for the given pricingstrategy Furthermore we design an algorithm to derive theoptimal dynamic pricing strategy and replenishment cycleNumerical examples and sensitivity analyses of main systemparameters are provided to demonstrate the obtained resultsand some managerial insights are offered

Complexity 3

Table 1 Notations and descriptions

variables119879 length of replenishment cycle119901 (119905) selling price per unit119901max (119905) reservation price120596 (119905) fashion level of apparel119868(119905) inventory level at time 119905119872119877 (119905) marginal profit119876 order quantity119863 demand rate120587 average profit per unit timeparameters119879max maximum sales period119886 initial demand rate119887 price parameter119888 purchasing cost per unitℎ inventory holding cost per unit120578 deterioration rate of the fashion levelA ordering cost per order

The study includes six sections In Section 2 the notationsand assumptions of the problem are described Section 3 pro-vides the theoretical results of the dynamic pricing strategiesand Section 4 provides the theoretical results of the fixedprice strategies Numerical results and sensitivity analyses arerepresented in Section 5 Finally Section 6 offers a conclusionand discusses implications for management

2 Modeling Assumptions

21 Problem Formulation Consider a revenue-maximizingretailer that orders and sells fashion apparel in a finitereplenishment cycle [0 119879] where the replenishment cycle 119879needs to be determined In addition to the common decisionvariables of when and how much to order we also considerselling price as a decision variable this price is dynamicallydetermined throughout the replenishment period

The dynamic pricing strategy and the fixed price strategyare both commonly followed in the fashion apparel industryWe make a performance comparison (including averageprofit and price trajectory) of these two pricing strategies byundertaking a numerical analysis The related notations usedthroughout this study are detailed in Table 1

22 Demand Function In this study we formulate thedemand function based on the following assumptions

Assumption 1 The demand function 119863(119901(119905)) explicitlydepends separately on both the apparelrsquos selling price andfashion level According to the additive combination offunctions 119889(119901(119905)) and120596(119905) respectively the demand functioncan be expressed as follows

119863(119901 (119905)) = 119889 (119901 (119905)) + 120596 (119905) (1)

The function 119889(119901(119905)) is strictly decreasing and differen-tiable with respect to 119901(119905) We note that the dependence of

demand is explicit in dynamic pricing and implicit in thetime factor Just as the fashion level of clothing deterioratesover time so too is demand implicitly affected by fashionlevel In general when 119901(119905) tends to infinity demand iszero ie lim119901(119905)997888rarrinfin119863(119901(119905)) = 0 forall119905 gt 0 and 120596(119905) =minuslim119901(119905)997888rarrinfin119889(119901(119905)) Thus 120596(119905) is a constantThis contradictsAssumption 1 which requires the existence of a time effecton fashion level A consistent definition of 119863(119901(119905)) dependson an additional assumption regarding the fashion level ofapparel which has the property of decreasing over time

Assumption 2 For fashion apparel there exists a consumerrsquosreservation price 119901max(119905) defined as the lowest price at 119905where the demand of fashion clothing is zero In other wordsif the selling price is higher than the reservation price thedemand rate must vanish

The reservation price satisfies the following conditions(i) The reservation price is higher than purchasing cost119901max(0) gt 119888 and 119901max(119905) continuously decreases over 0 le 119905 le119879max(ii) 119901max(119905) = 0 for 119905 ge 119879max and 120597119901max(119905)120597119905 lt 0 It

should be noted that the reservation price decreaseswith time119905 because the market value of fashion apparel decreases withtime

(iii) 119889(119901max(119905)) + 120596(119905) = 0 0 le 119905 le 119879max which meansthat if the clothing price is higher than the reservation priceduring the sales period consumers do not buy anythingFrom 119889(119901max(119905)) + 120596(119905) = 0 we have

119901max (119905) = 119889minus1 (120596 (119905)) (2)

The reservation price is the inverse function of fash-ion level Assumption 2 resolves the inconsistency between119863(119901(119905)) = 119889(119901(119905)) +120596(119905) and lim119901(119905)997888rarrinfin119863(119901(119905)) = 0 forall119905 gt 0

From the above assumptions the demand function of thedynamic scenario is as follows

119863(119901 (119905))= (119889 (119901 (119905)) + 120596 (119905) 0 le 119905 le 119879max 119901 (119905) le 119901max (119905)

0 119900119905ℎ119890119903119908119894119904119890(3)

Assumption 3 The demand function of the static scenario issimilar to the dynamic pricing strategy Also this demandfunction depends separately on both the apparelrsquos sellingprice and fashion level but the price does not change overtime119863(119901 119905) = 119889(119901)+120596(119905) where119889(119901) represents the demandrate under a fixed price and 120596(119905) is the fashion level that isaffected by the time factor

23 Fashion Level Function From the above assumptions thedemand function depends on the price and fashion level ofclothing To illustrate the effect of fashion level on demandwe offer the following proposition

Proposition 4 e fashion level function 120596(119905) is strictlydecreasing with 119905

From Proposition 4 we can see that the fashion levelfunction 120596(119905) is strictly decreasing with time 119905 Thereforewe can deduce that demand decreases with time 119905 which is

4 Complexity

02

03

04

05

06

07

08(t)

5 10 15 20 25 300t

(t)=08e-003t

(t)=05e-003t

Figure 1 The effect of the initial fashion level on fashion level

consistent with the actual operation of garment enterprisesso the assumption of the demand function in (3) is reason-able

To further describe the effect of fashion level on consumerdemand we assume that fashion level can be expressed asfollows

120596 (119905) = 1205960119890minus120578119905 (4)

Here 1205960 (1205960 gt 0) is the listed clothingrsquos initial fashionlevel Also 120578 (0 lt 120578 lt 1) is the fashion levelrsquos deteriorationrate If 1205960 is constant the fashion level 120596(119905) decreasesexponentially over time 119905 This model results from assumingthat the fashion level of clothing is a state variable withthe negative exponential distribution having the parameter120578 This characterizes the fashion levelrsquos attenuation propertywith time 119905

Considering the effect of the initial fashion level1205960 on thefashion level 120596(119905) as shown in Figure 1 we can see that 120596(119905)decreases as time 119905 increases The higher the initial fashionlevel 1205960 the slower the attenuation of fashion level 120596(119905) attime 119905Therefore it is obvious that the initial fashion level hasa delayed effect In practice to satisfy consumersrsquo individual-ized demand for fashion products apparel retailers often trytheir best to consider consumer psychology and enhance con-sumersrsquo feelings specifically by providing them with serviceexperiences such as trial-wear periods fashion displays andclothes matching thus raising the initial fashion level

Similarly the fashion levelrsquos deterioration rate has aneffect on fashion level 120596(119905) As shown in Figure 2 the smallerthe deterioration rate 120578 the slower the lowering of fashionlevel 120596(119905) because the fashion level largely depends on theelements of style color and material all of which influencethe deterioration rate Therefore suppliers should increasethe investment in the creative design of fashion elements tomake 120578 smaller and raise the fashion level of clothing

3 The Optimal Strategies underthe Dynamic Scenario

31 e Objective Function In the replenishment period theapparel retailerrsquos cost includes inventory holding costs and the

0

02

04

06

08

(t)

(t)=08e-009t

(t)=08e-003t

5 10 15 20 25 300

Figure 2 The effect of the deterioration rate on fashion level

ordering cost per replenishmentThus those costs are definedas 1198621 = ℎint119879

0119868(119901(119905) 119905)119889119905 and 1198622 = 119888 int119879

0119863(119901(119905) 119905)119889119905 + 119860

respectively where ℎ is the inventory holding cost per unit119888 is the purchasing cost per unit and 119860 is ordering cost perorder

The sales revenue of the apparel retailer is 119878119877 =int1198790119901(119905)119863(119901(119905) 119905)119889119905 Therefore the total profit of the retailer

per replenishment period represented by Π1 is given by

Π1 (119901 (119905) 119879) = 119878119877 minus 1198621 minus 1198622= int1198790(119901 (119905) minus 119888)119863 (119901 (119905)) 119889119905

minus ℎint1198790119868 (119901 (119905) 119905) 119889119905 minus 119860

(5)

The average profit per unit time represented by 1205871 isgiven by

1205871 (119901 (119905) 119879) = 1119879Π1 (119901 (119905) 119879) (6)

The goal of this study is to determine the optimal dynamicpricing strategy and replenishment cycle while maximizingaverage profit per unit time which can be described by thefollowing optimization problem

max119901(119905)119879

1205871 (119901 (119905) 119879)

119904119905 120597119868 (119905)120597119905 = minus119863 (119901 (119905) 119905) 0 le 119905 le 119879

119868 (119879) = 0 0 le 119901 le 119901max (119905)(7)

In this study note that the average profit per unit time isthe objective function because if the objective function is thetotal profit of a periodΠ1(119901(119905) 119879) and the decision variable119879is the upper bound of the integral which should theoreticallybe a large 119879 to maximize the objective function Howeverthis is contrary to the actual operation of garment enterpriseswhich is characterized by a shorter selling lifecycle and by

Complexity 5

having a longer replenishment cycle 119879 leading to extremelyhigh inventory costs Therefore by considering the averageprofit per replenishment period we not only compare theoperational performance of the same clothing enterpriseacross different sales periods but we also compare theoperational performance of different enterprises in the samesales period Similar studies are available [24ndash28]

The optimization problem of formula (7) can be solvedhierarchically First the optimal price 119901lowast(119905) is obtainedfor a given replenishment period 119879 Then by substitutingthe optimal price into the objective function the optimalreplenishment period 119879lowast is obtained This analysis of thehierarchical sequence covers all possible combinations of thetwodecision variables so the optimal solution of the objectivefunction can be obtained as follows

max119901(119905)119879

1205871 (119901 (119905) 119879) = max119879le119879max

max119901(119905)le119901max(119905)

1205871 (119901 (119905) 119879) (8)

32 e Optimal Dynamic Price Strategy Given the Replenish-ment Period Based on the hierarchical level discussed abovewe first consider a given replenishment period 119879 and thenidentify a dynamic price function 119901(119905) that maximizes theretailerrsquos average profit in the period The objective functionis max119901(119905)1205871(119901(119905) 119879) = (1119879)max119901(119905)Π1(119901(119905)) because119879 is aconstant and then the optimization problem presented in (7)is converted into another optimization problem as follows

max119901(119905)

Π1 (119901 (119905))

= max119901(119905)

int1198790((119901 (119905) minus 119888)119863 (119901 (119905) 119905) minus ℎ119868 (119905)) 119889119905

minus 119860

119904119905 119868 (119905) = minus119863 (119901 (119905) 119905) 0 le 119905 le 119879119868 (119879) = 0 0 le 119901 le 119901max (119905)

(9)

For the given replenishment cycle 119879 to obtain thedynamic pricing strategy we use Pontryaginrsquos maximumprinciple stated in [29] to solve the optimal controlproblem expressed in (9) The Hamiltonian function119867(119901(119905) 119868(119905) 120582(119905) 119905) is formulated as follows

119867(119901 (119905) 119868 (119905) 120582 (119905) 119905)= (119901 (119905) minus 119888)119863 (119901 (119905)) minus ℎ119868 (119905) minus 120582 (119905)119863 (119901 (119905)) (10)

where 120582(119905) is the costate variable Also 120582(119905) denotes themarginal value or shadow price of the inventory level Themaximum principle states that the necessary conditions for119901lowast(119905) with the corresponding state trajectory 119868lowast(119905) to bean optimal control are the existence of the continuous andpiecewise continuously differentiable function 120582(119905) such that

the following conditions hold By applying the necessary con-ditions of the maximum principle we obtain the following

(119905) = minus120597119867120597119868119868 (119905) = minus120597119867120597120582

120582 (0) = 0120597119867120597119901 = 0

(11)

According to (10) and (11) the optimal dynamic price canbe obtained as follows

119901lowast (119905) = 120595 (119905) 0 le 120595 (119905) le 119901max (119905)119901max (119905) 120595 (119905) ge 119901max (119905) (12)

where 120595(119905) is the implicit function that satisfies thefollowing

1198891015840 (120595 (119905)) (120595 (119905) minus 119888 minus ℎ119905) + 119863 (120595 (119905)) = 0 (13)

Thepartial derivative of the second order is obtained fromthe Hamilton function119867

12059721198671205971199012 = 21198891015840 (119901lowast (119905)) + (119901lowast (119905) minus 119888 minus ℎ119905) 11988910158401015840 (119901lowast (119905))

le 0(14)

It is shown from (14) that 119901lowast(119905) is the optimal priceAccording to the formula (12) and (13) we can see that theexpression119901lowast(119905) does not contain119879That is the optimal pricedoes not depend on the length of replenishment Given 119879the dependence of optimal price on time119905can be equivalentlyreformulated into the dependence on the time remaininguntil replenishment 119879 minus 119905 This property is intuitive and theapparel retailer can decide on the optimal price according tohow much time is left before the next replenishment period

As discussed above the optimal dynamic pricing and thereservation price vary with time The next proposition com-pares the optimal dynamic pricing 119901lowast(119905) with the reservationprice 119901max(119905) and the marginal cost denoted by 119872119862 where119872119862(119905) = 119888 + ℎ119905Proposition 5 ere exists a unique replenishment timedenoted by 1198791 such that

(i) 119872119862(119905) lt 119901lowast(119905) lt 119901max(119905) for 0 le 119905 lt 1198791(ii) 119872119862(119905) = 119901lowast(119905) = 119901max(119905) for 119905 = 1198791(iii) 119872119862(119905) gt 119901lowast(119905) = 119901max(119905) for 119905 ge 1198791Proposition 5 provides an important insight namely the

optimal dynamic price is always less than the reservationprice but the optimal dynamic pricemay be either more thanor less than the marginal cost when considering the fashionlevel of apparel and the reservation price of consumers In

6 Complexity

particular if the reservation price is equal to the marginalcost ie 119901max(1198791) = 119888 + ℎ1198791 the replenishment cycle1198791 can be obtained Note that the specific case of thefunctions described in Proposition 5 is schematically plottedin Figure 3

In the following sections we specifically investigate theoptimal dynamic price path With the influence of fashionlevel the optimal dynamic price of apparel has different pathsof change Let the demand function 119889(119901(119905)) be linear ie119889(119901(119905)) = 119886 minus 119887119901(119905) (119886 gt 0 119887 gt 0) Substituting 119889(119901(119905)) intothe 119889(119901max(119905)) + 120596(119905) = 0 in Assumption 2 we obtain thereservation price

119901max (119905) = 1119887 (119886 + 120596 (119905)) (15)

Then substituting 119889(119901(119905)) into (13) we have minus119887(119901lowast(119905) minus 119888 minusℎ119905) + 119886 minus 119887119901lowast(119905) + 120596(119905) = 0 Thus the equivalent form of (12)is obtained

119901lowast (119905) = 12 (119901max (119905) + 119888 + ℎ119905) 0 le 119905 le 1198791119901max (119905) 119905 gt 1198791

(16)

Consider that the second derivative of the Hamiltonfunction119867 to 119901 is 12059721198671205971199012 = minus2119887 lt 0 which illustrates that119901lowast(119905) is optimal Equation (16) shows that the optimal price119901lowast(119905) monotonically increases in ℎ Next we differentiateboth sides of (16) with respect to 119905 which yields the following

120597119901lowast (119905)120597119905 = 119887

2 (1205961015840 (119905) + 119887ℎ) for 0 le 119905 le 1198791 (17)

The next proposition illustrates the effect of fashion levelon the optimal dynamic price path

Proposition 6 In terms of the property of the fashion levelfunction 120596(119905) = 1205960119890minus120578119905 being convex ie 12059610158401015840(119905) gt 0 theoptimal price 119901lowast(119905) has three possible paths

(i) 119901lowast(119905)monotonically decreases in time 119905 for 0 le 119905 lt 11987911205961015840(119905) lt minus119887ℎ(ii) 119901lowast(119905) monotonically increases in time 119905 for 0 le 119905 lt 11987911205961015840(119905) gt minus119887ℎ(iii) 119901lowast(119905) first decreases and then increases for 0 le 119905 lt 1198791Note that the specific case of optimal dynamic price

described in Proposition 6 is schematically plotted in Figure 3We take case (iii) of Proposition 6 as an example because in thebeginning 1205961015840(119905) lt minus119887ℎ 119901lowast(119905) decreased until 1205961015840(119905) = minus119887ℎand 119901lowast(119905) then increased with the increase of the derivativeof the fashion level function 1205961015840(119905) the red curve shown inFigure 3 Figure 3 also shows that the optimal dynamic price119901lowast(119905) consists of two parts and the first part is 119901lowast(119905) = 120595(119905)for 0 lt 119905 lt 1198791 e second part is 119901lowast(119905) = 119901max(119905) and when119905 = 1198791 the optimal dynamic price begins to switch to anotherpath ie 119901lowast(119905) = 119901max(119905) for 119905 ge 1198791 the green curve shown inFigure 3

t0

MC

plowast(t)

T1

(t)

pＧＲ(t)

Figure 3 The optimal dynamic price paths

33 e Optimal Replenishment Period Given Dynamic PriceThe problem at the higher hierarchical level of the decompo-sition discussed above is stated as follows given the optimaldynamic price function 119901lowast(119905) maximize the average profitunder the replenishment period 119879 as a single decisionvariable Then the optimization problem presented in (9) isconverted into another optimization problem as follows

max119879

1205871 (119901lowast (119905) 119879)119904119905 119879 le 119879max

(18)

In Section 32 we obtain that there exists a unique time1198791 such that beyond that time point the optimal price equalsthe reservation price 119901max(119905) and there can be no demandTherefore the optimal replenishment time must be no laterthan 1198791 ie 119879lowast lt 1198791

In addition119872119862(119905) is the apparel retailerrsquos marginal costat time 119905 Now we can define instantaneous marginal profit119872119877 as the rate of contribution to profit at time t exclusive offixed cost

119872119877(119905) = 119901lowast (119905) minus 119872119862 (119905) (19)

According to (19) when 119905 = 119879 the retailerrsquos marginal profit is119872119877(119879) = 119901lowast(119879) minus 119872(119879)Proposition 7 Given the optimal dynamic price 119901lowast(119905) if1205871(119901lowast(119905) 1198792) gt 0 the optimal replenishment period 119879lowast =min119879max 1198792 where 1198792 is unique solution of

1205871 (119901lowast (119905) 119879) = 119863 (119901lowast (119905))119872119877 (119879) 0 le 119879 le 1198791 (20)

Otherwise (20) has no solution and the optimal replenishmentperiod 119879lowast = min119879max 1198791

On the basis of 119901lowast(119905) and 119879lowast the optimal order quantitycan be calculated as follows

119876lowast = int119879lowast

0(119886 minus 119887119901lowast (119905) + 120596 (119905)) 119889119905 (21)

Complexity 7

34 Algorithm Based on the propositions above we proposethe following algorithm to obtain the optimal solution

Step 1 Let 119889(119901(119905)) = 119886 minus 119887119901(119905) and 120596(119905) = 1205960119890minus120578119905Step 2 Determine 119901max(119905) and 120595(119905) from (2) and (13)respectively

Step 3 According to Proposition 5 determine 1198791 by solving119872119862(1198791) = 119901max(1198791) and define 119901lowast(119905) = 120595(119905) for 0 le 119905 le 119879and 119901lowast(119905) = 119901max(119905) for 119905 gt 1198791Step 4 Verify the second-order condition (14)

Step 5 Compute 119879lowast from Proposition 7

Step 6 Compute the optimal order quantity 119876lowast and theaverage profit per unit time 120587lowast1 from (21) and (6) respectively

4 The Optimal Strategies underthe Static Scenario

Thedynamic pricing strategy discussed above is one in whichdemand is affected by fashion level By means of a dynamicpricing strategy it is easy to distinguish the consumers whoare sensitive and insensitive to fashion thus maximizing theretailerrsquos profit

Besides the dynamic pricing strategy retailers often fol-low the fixed price strategy because of its long-run benefitsSome famous retailers such as Macyrsquos often use a fixed pricestrategy So for fashion clothing the determination of whichparticular strategy is more beneficial dynamic pricing orfixed pricing becomes important To facilitate a comparisonbetween dynamic pricing and fixed pricing strategies thissection discusses the decision-making problem of fashionretailers under fixed price conditions

Section 22 showed that the demand function under thefixed pricing strategy is as follows

119863(119901 119905) = 119889 (119901) + 120596 (119905) = 119886 minus 119887119901 + 1205960119890minus120578119905 (22)

It is worth noting that price does not change withtime The retailerrsquos total profit per replenishment periodrepresented by Π2 is given by the following

Π2 (119901 119879) = int1198790(119901 minus 119888)119863 (119901 119905) 119889119905 minus ℎint119879

0119868 (119901 119905) 119889119905

minus 119860(23)

The goal of Section 4 is to determine the fixed pricingstrategy and replenishment cycle while maximizing averageprofit per unit time which can be described by the followingoptimization problem

max119901119879

1205872 (119901 119879) = 1119879Π2 (119901 119879) (24)

Similar to the dynamic pricing scenario the method ofthe sequential hierarchical optimization can still be used tosolve the objective function under fixed price conditionsThe

problem at the lower hierarchical level of the decompositiondiscussed above is stated as follows given the replenishmentperiod119879 identify a fixed price denoted by119901119865 thatmaximizesthe total profit for the period

max119901

1205872 (119901 119879)

= max119901

1119879 int1198790(119901 minus 119888)119863 (119901 119905) 119889119905 minus ℎint119879

0119868 (119901 119905) 119889119905 minus 119860

(25)

Proposition 8 For any given replenishment period 119879 isin[0 119879max]1205872(119901 119879) is strictly concave in119901 and the optimal priceis given by 119901lowast119865 = (12)119888 + 119886119887 + ℎ1198792 minus (1205960119887119879120578)(119890minus120578119879 minus 1)

The problem at the higher hierarchical level of thedecomposition discussed above is stated as follows given theoptimal dynamic price function 119901lowast119865 maximize the averageprofit under the replenishment period 119879 as a single decisionvariable Then the optimization problem presented in (24) isconverted into another optimization problem as follows

max119879

1205872 (119901lowast119865 119879)

= max119879

1119879 int1198790(119901lowast119865 minus 119888)119863119889119905 minus ℎint119879

0119868 119889119905 minus 119860

(26)

According to 119901lowast119865 and 119879lowast the optimal order quantity can becalculated as follows


0(119886 minus 119887119901lowast119865 + 120596 (119905)) 119889119905 (27)

Due to the mathematical complexity involved it is diffi-cult for us to propose an explicit solution for (26) Thus inthe following we present the numerical results between thefixed price model and the dynamic price model

5 Numerical Example

In this section we conduct numerical analyses to gainmanagerial insights Set 119886 = 20 119887 = 2 119888 = 2 ℎ = 05 120578 = 02119860 = 250 119879max = 20 and 1205960 = 20 In Section 51 we use thesolution algorithm described in Section 3 to find the optimalsolution Also we compare the operational performances ofthe dynamic and static scenarios focusing on the averageprofit and selling price In Section 52 we examine the effectsof the deterioration rate of fashion level and purchasing coston the optimal solution and obtain some useful insights

51 Optimal Solutions Applying the algorithm presented inSection 3 we obtain the optimal price paths and replenish-ment period under the dynamic scenario In Figure 4 thepath of optimal dynamic price 119901lowast(119905) consists of two partsThe first part is the path between the marginal cost 119872119862(119905)and reservation price 119901max(119905)The optimal price decreases for0 le 119905 lt 693 and then increases to 119905 = 1198791 = 167 Afterwardsdemand vanishes and the optimal price decreases and equalsthe reservation price ie 119901lowast(119905) = 119901max(119905)

As shown in Figure 5 the objective 1205871(119879) is plottedas a function of the replenishment period Furthermore

8 Complexity

MC(t)

plowast(t)

pmax (t)

T1

5 10 15 20 250t

0

5

10

15

20

Figure 4 The optimal dynamic pricing strategy

0 5 10 15 20

T

1(plowast(t)T)

D(plowast(t)timesMR(T)

minus200

minus100

0

100

200

Tlowast

Figure 5 The optimal replenishment period

119863(119901lowast(119905))119872119877(119879) is plotted as a function of the replenishmentperiod The optimal replenishment period determined from1205871(119901lowast(119905) 119879) = 119863(119901lowast(119905))119872119877(119879) is 119879lowast = 1198792 = 531

That is the optimal replenishment period is determinedby the point of intersection of the average profit andmarginalprofit curve

Through numerical simulation we also compare thecorresponding price trajectories as shown in Figure 6 Herethe corresponding optimal dynamic pricing strategy 119901lowast(119905) is

119901lowast (119905) = 5119890minus02119905 + 025119905 + 6 0 le 119905 le 16710119890minus02119905 + 10 119905 gt 167 (28)

and the average profit unit time is 120587lowast1 = 38726 (from (6))In contrast if the apparel retailer cannot vary the price theoptimal fixed price is 119901lowast119865 = 975 and the average profit unittime is 120587lowast2 = 38104 lt 120587lowast1

Furthermore we investigate the effects of parameterchanges on the profits obtained from the proposed methodTable 2 shows that the difference in the profits Δ120587 = 120587lowast1 minus 120587lowast2increases as the deterioration rate of fashion level increases

0 5 10 15 20 25t

PlowastF

0

2

4

6

8

10

12

p plowast(t)

Figure 6 Comparison of dynamic pricing strategy and fixed pricestrategy

because the faster the deterioration rate the less fashionablethe garments and hence consumers have less incentive tobuy Consequently the apparel retailer dynamically adjustsprices during different selling periods The retailer chargesa high price when the fashion apparelrsquos value is elevatedand later when its value decreases such items are sold at adiscount priceThus by adjusting the sales price periodicallythe apparel retailer is able to realize the profit advantage of acontinuously variable price policy

52 Sensitivity Analysis

521 Impact of the Deterioration Rate (120578) As discussedabove the deterioration rate affects the optimal dynamicprice the optimal replenishment period the order quantityand the apparel retailerrsquos profit Figure 7 shows that thecurve of the optimal dynamic price 119901lowast(119905) with different 120578 isin01 02 03 and the other parameters is kept unchangedA higher value of 120578 represents a greater attenuation ratefor the fashion level It is shown from Figure 7 that for asmaller value of 120578 the apparel retailer sets a higher sellingprice due to the greater marketing demand during the salesperiod This higher price is set because the smaller value of120578 the more fashionable the garments As a result greatermarketing demand consumes inventory items more quicklyOverall a smaller 120578 implies both greater demand and greaterorder quantity thus shortening the replenishment periodand Table 3 shows that 119879lowast decreases as 120578 increases Finallyas shown in Table 3 average profit increases as 120578 decreasesbecause fashion level depends largely on stylistic elementsConsequently taking measures to lower 120578 such as increasinginvestment in the creative design of fashion elements isbeneficial for the apparel retailer

522 Impact of the Purchasing Cost (119888) Table 4 shows thereplenishment period 119879lowast the order quantity 119876lowast and theaverage profit 120587lowast1 with different 119888 isin 1 2 3 and the otherparameters are kept unchanged Figure 8 shows the curveof the optimal dynamic price 119901lowast(119905) From the figure andtable we can see that 119901lowast(119905) and 119879lowast are increasing in 119888 while

Complexity 9

Table 2 Profit comparison under different 120578120578 006 008 010 012 014 016 018 020120587lowast1 70822 64576 59074 54163 49740 45729 42073 38726120587lowast2 70821 64548 58980 53982 49457 45336 41566 38104Δ120587 0001 0029 0094 0182 0283 0393 0507 0623

Table 3 The impact of 120578 on decisions of retailer

120578 119879lowast 119876lowast 120587lowast1020 5312 6816 3873019 5315 6892 4036018 5324 6976 4207017 5339 7067 4386016 5359 7167 4573015 5385 7277 4769014 5419 7398 4974013 5460 7532 5190012 5509 7680 5416011 5569 7844 5655

Table 4 The impact of c on decisions of retailer

119888 119879lowast 119876lowast 120587lowast11 497 7004 52192 531 6816 38733 575 6618 26554 637 6408 15765 737 6182 650

0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)

Figure 7 The optimal dynamic pricing strategy with different 120578

119876lowast and 120587lowast1 are decreasing in 119888 This implies that when thepurchasing cost 119888 increases the apparel retailer adds thiscost to the selling price 119901lowast(119905) Facing a large 119888 the apparelretailer compensates for the increased purchasing costs bylengthening the replenishment cycle reducing orders andraising the selling price

6 ConclusionsConsidering how both fashion level and changes in reserva-tion price influence the performance of apparel retailers this

0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)


study proposes two pricing strategies for such enterprisesthe dynamic pricing strategy and the fixed pricing strategyWe formulate a garment inventory model with demandbeing dependent on price and fashion level and determinethe optimal prices and the optimal replenishment cyclewith the objective of maximizing average profits in a finitetime horizon We characterize the properties of the optimalsolution and propose an algorithm Our research advancesthe following conclusions and managerial insights First arelatively rapid deterioration rate of fashion level means thatthe garment is of a relatively poor design For the fashionapparel retailer a dynamic price strategy is better than afixed price strategy This implies that the apparel retailerby periodically adjusting sales price can realize the profitadvantage of a continuously variable price policy Second theapparel retailer has three types of optimal dynamic pricingstrategies from which to choose with the optimal dynamicpricing strategy being independent of replenishment cycle119879 The optimal price strategy depends on time 119905 and isdependent on remaining time 119879 minus 119905 This is intuitive and theapparel retailer can decide on the optimal price of fashionclothing according to how much time is left before the nextreplenishment period In contrast the optimal fixed price isdependent on the replenishment period Third numericalanalysis shows that both the deterioration rate of fashionlevel and the purchasing cost exert significant effects onboth the optimal joint policy and the profit thus providingfashion apparel retailers with guidance that can improve theirprofitability

Appendix

Proof of Proposition 4 From Assumption 2 differentiating119889(119901max(119905)) + 120596(119905) = 0 with respect to 119905 yields 120597120596(119905)120597119905 =

10 Complexity

minus(120597119889(119901max(119905))120597119901max(119905))(120597119901max(119905)120597119905) From Assumption 1we have 120597119889(119901max(119905))120597119901max(119905) lt 0 and 120597119901max(119905)120597119905 lt 0 for0 le 119905 le 119879max We can obtain 120597120596(119905)120597119905 lt 0 This completes theproof

Proof of Proposition 5 From Assumption 2 the reservationprice 119901max(119905) decreases monotonically in time 119901max(0) gt 119888If 119905 gt 119879max then 119901max(119905) = 0 Therefore there exists a uniquereplenishment time denoted by 1198791 such that 119901max(119905) gt119872119862(119905) for 119905 lt 1198791 and 119901max(119905) lt 119872119862(119905) for 119905 gt 1198791 FromAssumption 1 we have 1198891015840(120595(119905)) lt 0 From (13) we canobtain (119888 + ℎ119905 minus 120595(119905)) lt 0 that is119872119862(119905) lt 119901lowast(119905) therefore119872119862(119905) lt 119901lowast(119905) lt 119901max(119905)

According to the demand function of Assumption 1 if120595(119905) gt 119901max(119905) then 119863(119901(119905)) = 0 and if it is substitutedinto 10) then 119901lowast(119905) = 119901max(119905) (Thus considering consumerreservation price and marginal cost there exists the uniquereplenishment period 1198791 that makes 119872119862(119905) lt 119901lowast(119905) lt119901max(119905) for 0 le 119905 le 1198791119872119862(119905) = 119901lowast(119905) = 119901max(119905) for 119905 = 1198791and119872119862(119905) gt 119901lowast(119905) = 119901max(119905) for 119905 ge 1198791)Proof of Proposition 6 We differentiate both sides of (16) withrespect to 119905 which yields 120597119901lowast(119905)120597119905 = (1198872)(1205961015840(119905)+119887ℎ) for 0 le119905 le 1198791 If 1205961015840(119905) lt minus119887ℎ then 120597119901lowast(119905)120597119905 lt 0 and we obtain line(i) in Proposition 6 If 1205961015840(119905) gt minus119887ℎ then 120597119901lowast(119905)120597119905 gt 0 andwe obtain line (ii) in Proposition 6This proves Proposition 6and completes the proof

Proof of Proposition 7 For notational convenience we denote1205871(119879) = 1205871(119901lowast(119905) 119879) and 119863(119905) = 119863(119901lowast(119905)) Differentiating1205871(119901lowast(119905) 119879) with respect to 119879 yields

1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879

0(119901lowast (119905) minus 119888)119863 (119901lowast (119905)) 119889119905

minus ℎint1198790119868 (119901lowast (119905) 119905) 119889119905 minus 119860

= minus 11198792 int

119879


minus ℎint1198790119868 (119901lowast (119905) 119905) 119889119905 minus 119860 + 1

119879 (119901lowast (119879) minus 119888)

sdot 119863 (119879) minus ℎ 119889119889119879 int1198790int119879119905119863(119901lowast (119904) 119904) 119889119904 119889119905

= 1119879 (119901lowast (119879) minus 119888)119863 (119879) minus ℎ 119889

119889119879sdot int1198790int119879119905119863 (119904) 119889119904 119889119905 = minus120587 (119879)119879 + 1

119879 (119901lowast (119879)minus 119888)119863 (119879) minus ℎ119879119863 (119879) = minus 1119879 120587 (119879) minus 119863 (119879)sdot (119901lowast (119879) minus 119888 minus ℎ119879)

(A1)

Thus the first derivative of 1205871(119901lowast(119905) 119879) at time 1198792 is1205971205871 (119901lowast (119905) 119879)120597119879

= minus 1119879 1205871 (119901lowast (119905) 119879) minus 119863 (119901lowast (119905))119872119877 (119879)(A2)

According to the property of the optimal dynamic pricediscussed in Section 3 119863(119901lowast(119905))119872119877(119879) is positive anddecreases for 0 le 119879 le 1198791 (see Proposition 5) until 119879 = 1198791when it becomes zero In addition the limit value of the objec-tive function is lim119879997888rarr01205871(119901lowast(119905) 119879) = minusinfin That is to say1205871(119901lowast(119905) 119879) is very small at the beginning close to negativeinfinity It can be easily shown that 1205971205871(119901lowast(119905) 119879)120597119879 gt 0 and1205871(119901lowast(119905) 119879) increases and reaches a maximum at 119879 = 1198791Thus we can obtain 119879lowast = min119879max 1198791 If 1205871(119901lowast(119905) 119879)hits the zero level and becomes positive then there existsan extreme point 1198792 satisfying 1205971205871(119901lowast(119905) 119879)120597119879|119879=1198792 = 0Hence we can obtain 1205871(119901lowast(119905) 119879) = 119863(119901lowast(119905))119872119877(119879) Nextwe prove the uniqueness of 1198792 Taking the second-orderderivative of 1205871(119901lowast(119905) 119879) with respect to 119879 yields

12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879

+ 1119879 (119901lowast (119879) minus 119888)119863 (119879) minus ℎ119879119863 (119879)10038161003816100381610038161003816100381610038161003816119879=1198792

= minus 11198792 minus119863 (1198792) [(119901lowast (1198792))1015840 minus ℎ] minus 1198891015840 (1198792)

sdot [119901lowast (1198792) minus 119888 minus ℎ1198792] = 11198792 1198891015840 (119901lowast (1198792))

sdot (119901lowast (1198792))1015840 + 1205961015840 (1198792) (119901lowast (1198792) minus 119888 minus ℎ1198792)+ 119863 (1198792) [(119901lowast (1198792))1015840 minus ℎ] = 1

1198792 1198891015840 (119901lowast (1198792))

sdot (119901lowast (1198792))1015840 (119901lowast (1198792) minus 119888 minus ℎ1198792) + 119863 (1198792)minus ℎ119863 (1198792) + 1205961015840 (1198792) (119901lowast (1198792) minus 119888 minus ℎ1198792)

(A3)

Thus the second derivative of 1205871(119901lowast(119905) 119879) at time 1198792 is12059721205871 (119879)1205971198792

100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840

sdot (119901lowast (1198792) minus 119888 minus ℎ119879) + 119863 (1198792) minus ℎ119863 (1198792)+ 1205961015840 (1198792) (119901lowast (1198792) minus 119888 minus ℎ1198792)

(A4)

From Proposition 4 the fashion level function decreaseswith time 119905 ie 1205961015840(1198792) lt 0 Therefore the last two termson the right-hand side of (A4) are less than zero From1198891015840(120595(119905))(120595(119905) minus 119888 minus ℎ119905) + 119863(120595(119905)) = 0 the first termon the right-hand side of (A4) is equal to zero Thus(12059721205871(119879)1205971198792)|119879=1198792 = (11198792)minusℎ119863(1198792) + 1205961015840(1198792)(119901lowast(1198792) minus 119888 minusℎ1198792) From (119901lowast(1198792) minus 119888 minus ℎ1198792) gt 0 and 1205961015840(1198792) lt 0 we have

Complexity 11

12059721205871(119879)1205971198792|119879=1198792 lt 0 Hence there exists a unique point1198792 satisfying 1205971205871(119901lowast(119905) 119879)120597119879|119879=1198792 = 0 This completes theproof

Proof of Proposition 8 Taking the first-order derivative ofΠ2(119901 119879) with respect to 119879 we obtain120597Π2 (119901 119879)120597119901 = 1198791205971205872 (119901 119879)120597119901

= 120597120597119901 int1198790(119901 minus 119888) (119886 minus 119887119901 + 1205960119890minus120578119905) 119889119905

minus ℎ 120597120597119901 int1198790int119879119905(119886 minus 119887119901 + 1205960119890minus120578119904) 119889119904 119889119905

= int1198790119886 minus 119887119901 + 1205960119890minus120578119905 minus 119887 (119901 minus 119888) 119889119905

minus ℎint1198790minus119887 (119879 minus 119905) 119889119905

= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922

minus 1205960120578 (119890minus120578119879 minus 1)

(A5)

It can be easily shown that 12059721205872(119901 119879)1205971199012 = minus2119887 lt0 Therefore there exists a unique price 119901lowast119865 satisfying1205971205872(119901 119879)120597119901|119875119865 = 0 and we can obtain 119901lowast119865 = (12)119888 + 119886119887 +ℎ1198792 minus (1205960119887119879120578)(119890minus120578119879 minus 1) This completes the proof

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this study

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (71572033 71832001)

References

[1] T-M Choi C-L Hui N Liu S-F Ng and Y Yu ldquoFast fashionsales forecasting with limited data and timerdquo Decision SupportSystems vol 59 no 2 pp 84ndash92 2014

[2] B Shen H-L Chan P-S Chow and K A Thoney-BarlettaldquoInventory management research for the fashion industryrdquoInternational Journal of Inventory Research vol 3 no 4 pp 297ndash317 2016

[3] J Dong and D DWu ldquoTwo-period pricing and quick responsewith strategic customersrdquo International Journal of ProductionEconomics 2017

[4] M Fisher K Rajaram and A Raman ldquoOptimizing inventoryreplenishment of retail fashion productsrdquo Manufacturing andService Operations Management vol 3 no 3 pp 230ndash241 2001

[5] K Kogan and U Spiegel ldquoOptimal policies for inventory usageproduction and pricing of fashion goods over a selling seasonrdquoJournal of the Operational Research Society vol 57 no 3 pp304ndash315 2006

[6] Y-C Tsao ldquoTwo-phase pricing and inventory management fordeteriorating and fashion goods under trade creditrdquoMathemat-ical Methods of Operations Research vol 72 no 1 pp 107ndash1272010

[7] H-M Wee ldquoEconomic production lot size model for deterio-rating items with partial back-orderingrdquo Computers amp Indus-trial Engineering vol 24 no 3 pp 449ndash458 1993

[8] P M Ghare and G F Schrader ldquoA model for exponentiallydecaying inventory systemsrdquo International Journal of Produc-tion Economics vol 21 pp 449ndash460 1963

[9] K-S Wu L-Y Ouyang and C-T Yang ldquoCoordinating replen-ishment and pricing policies for non-instantaneous deteriorat-ing items with price-sensitive demandrdquo International Journal ofSystems Science vol 40 no 12 pp 1273ndash1281 2009

[10] R Maihami and I N Kamalabadi ldquoJoint pricing and inventorycontrol for non-instantaneous deteriorating items with partialbacklogging and time and price dependent demandrdquo Interna-tional Journal of Production Economics vol 136 no 1 pp 116ndash122 2012

[11] T Avinadav A Herbon and U Spiegel ldquoOptimal inventorypolicy for a perishable item with demand function sensitive toprice and timerdquo International Journal of Production Economicsvol 144 no 2 pp 497ndash506 2013

[12] G Li X He J Zhou and H Wu ldquoPricing replenishmentand preservation technology investment decisions for non-instantaneous deteriorating itemsrdquoOmega vol 84 pp 114ndash1262019

[13] G Gallego and G Van Ryzin ldquoOptimal dynamic pricingof inventories with stochastic demand over finite horizonsrdquoManagement Science vol 40 no 8 pp 999ndash1020 1994

[14] W Zhao and Y-S Zheng ldquoOptimal dynamic pricing for per-ishable assets with nonhomogeneous demandrdquo ManagementScience vol 46 no 3 pp 375ndash388 2000

[15] F Caro and J Gallien ldquoDynamic assortment with demandlearning for seasonal consumer goodsrdquo Management Sciencevol 53 no 2 pp 276ndash292 2007

[16] E P Chew C Lee R Liu K-S Hong and A Zhang ldquoOptimaldynamic pricing and ordering decisions for perishable prod-uctsrdquo International Journal of Production Economics vol 157 no1 pp 39ndash48 2014

[17] G Gallego and M Hu ldquoDynamic pricing of perishable assetsunder competitionrdquo Management Science vol 60 no 5 pp1241ndash1259 2014

[18] M Rabbani N P Zia and H Rafiei ldquoJoint optimal dynamicpricing and replenishment policies for items with simultaneousquality and physical quantity deteriorationrdquoAppliedMathemat-ics and Computation vol 287 pp 149ndash160 2016

[19] C-Y Dye and C-T Yang ldquoOptimal dynamic pricing andpreservation technology investment for deteriorating productswith reference price effectsrdquo Omega vol 62 pp 52ndash67 2016

[20] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing for dete-riorating items with reference price effects when inventoriesstimulate demandrdquo European Journal of Operational Researchvol 262 no 1 pp 136ndash150 2017

12 Complexity

[21] G Liu J Zhang and W Tang ldquoJoint dynamic pricing andinvestment strategy for perishable foods with price-qualitydependent demandrdquoAnnals ofOperations Research vol 226 no1 pp 397ndash416 2015

[22] Y Wang J Zhang and W Tang ldquoDynamic pricing for non-instantaneous deteriorating itemsrdquo Journal of Intelligent Man-ufacturing vol 26 no 4 pp 629ndash640 2015

[23] Y Li S Zhang and J Han ldquoDynamic pricing and periodicordering for a stochastic inventory system with deterioratingitemsrdquo Automatica vol 76 pp 200ndash213 2017

[24] A Herbon and E Khmelnitsky ldquoOptimal dynamic pricingand ordering of a perishable product under additive effects ofprice and time on demandrdquo European Journal of OperationalResearch vol 260 no 2 pp 546ndash556 2017

[25] N Tashakkor S H Mirmohammadi and M Iranpoor ldquoJointoptimization of dynamic pricing and replenishment cycle con-sidering variable non-instantaneous deterioration and stock-dependent demandrdquo Computers amp Industrial Engineering vol123 pp 232ndash241 2018

[26] J Zhang Y Wang L Lu and W Tang ldquoOptimal dynamicpricing and replenishment cycle for non-instantaneous deteri-oration items with inventory-level-dependent demandrdquo Inter-national Journal of Production Economics vol 170 pp 136ndash1452015

[27] R Maihami and B Karimi ldquoOptimizing the pricing andreplenishment policy for non-instantaneous deteriorating itemswith stochastic demand and promotional effortsrdquo Computers ampOperations Research vol 51 pp 302ndash312 2014

[28] A Rajan Rakesh and R Steinberg ldquoDynamic pricing andordering decisions by a monopolistrdquo Management Science vol38 no 2 pp 240ndash262 1992

[29] S P Sethi and G L Thompson Optimal Control eoryApplications to Management Science and Economics SpringerHeidelberg Germany 2000

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2 Complexity

implementing dynamic pricing making them more respon-sive to consumer demand According to the time-varyingcharacteristics of fashion clothing how to implement a time-varying dynamic pricing strategy has obvious practical signif-icance for managing inventory replenishment and improvingnet profits In this light this study aims to investigate thejoint optimal dynamic pricing and replenishment policies ofa fashion apparel retailer by considering fashion level

Nevertheless few research efforts have explicitly consid-ered the inventory decisions of fashion apparel from theperspective of pricing and replenishment cycles Fisher et al[4] studied how replenishment order quantity may minimizethe cost of lost sales back orders and obsolete inventoryin a two-stage model Kogan and Spiegel [5] developed amodel with time and price-dependent demand and an initialinventory level Tsao [6] developed an analytical model forthe inventory control of deteriorating fashion goods underthe conditions of trade credit and partial backlogsWhile theyidentified the optimal pricing and replenishment strategyprice remained a static variable Due to the market value offashion apparel decreasing over time the abovementionedpricing models are not fit for the fashion apparel industryFashion apparel has a short lifecycle and products showchanges in market value similar to instantaneous deteriorat-ing items Here deterioration is defined as decay damagespoilage evaporation obsolescence pilferage loss of utilityor the loss of a commodityrsquos marginal value all of whichdecrease usefulness Most physical goods such as medicinevolatile liquids and fashion goods undergo a process of decayor deterioration [7] Therefore decision-making regardingthe joint dynamic pricing strategy and replenishment cyclefor deteriorating items can be used as a reference for inven-tory management in the fashion apparel industry

For firmsmanaging the inventories of deteriorating itemsrepresents an important challenge that has understandablyattracted widespread attention from both researchers andpractitioners Ghare and Schrader [8] were the first toestablish an economic order quantity model to study theoptimal ordering policy for deteriorating items Wu et al [9]formulated an inventory model with deteriorating items andprice-sensitive demand and obtained the retailerrsquos optimalselling price and the length of replenishment cycle Asdemand for deteriorating items is affected by both priceand time that remains before a productrsquos expiration it isessential to model these simultaneous effects and establisha joint optimal pricing and inventory replenishment policyHowever few researchers have explicitly considered thedependency of demand on both price and time Maihamiet al [10] established joint pricing and inventory control fordeteriorating items with demand dependent on time andprice and with partial backlogging allowed Avinadav et al[11] also used a price- and time-dependent demand functionto develop a mathematical model that calculates the optimalprice and replenishment period of perishable items Li et al[12] considered the problem of joint pricing inventories andinvestment preservation by considering the general price-dependent demand rate In the abovementioned inventorymodels price remains static over the time horizon

In recent years due to the rapid evolution of informa-tion technologies dynamic pricing has become increasinglyviable Consequently firmsrsquo optimal dynamic pricing strategyhas attracted scholarly attention Gallego and Ryzin [13]formulated an inventory model to investigate the problemof dynamic pricing in which demand was price sensitiveand stochasticThere is abundant research on pricing modelsthat explicitly address dynamic pricing [14ndash21] However theabovementioned models rarely regard pricing strategy as aprocess of dynamic change Wang et al [22] developed aninventory model to identify the optimal dynamic pricingstrategy for noninstantaneous deteriorating items Li et al[23] considered the problem of dynamic pricing and periodicordering for deteriorating items with a stochastic inventorylevel that depends on stock-dependent demand and theselling price Herbon et al [24] formulated an inventorymodel with deteriorating items and price-sensitive demandand obtained the retailerrsquos optimal dynamic selling priceand the length of replenishment cycle Tashakkor et al [25]addressed the variability of the deterioration rate in a jointdynamic pricing and replenishment cycle problem where thedeterioration rate is an increasing step function Zhang etal [26] used Pontryaginrsquos maximum principle to obtain boththe optimal dynamic pricing strategy and the replenishmentcycle for noninstantaneous deteriorating items with demanddependent on a storersquos sales price and the quantity of the itemsdisplayed

The study by Tsao [6] is the only one that analyzedthe joint pricing and replenishment cycle decision-makingproblem for fashion goods The studies that are closestto ours are [6 26] Our work mainly differs from theirsin three aspects First in their approach the quantity ofdeteriorating fashion goods continuously decreases duringthe replenishment period [6 26] In contrast in our studywhile fashion apparelrsquos market value decreases over time itsquantity does not Second in our study the fashion leveldecreases over time and exerts a negative effect on thedemand rate This factor however is not considered in theproblem of pricing fashion goods in Tsaorsquos model [6] In ourmodel of fashion apparel we assume that the demand ratedepends on both sales price and fashion level which is amorerealistic assumption To the best of our knowledge our studyis the first to operate under this assumption when analyzingthe joint optimal dynamic pricing strategy and replenishmentcycle for fashion apparel Finally in Tsaorsquos model price wasa static variable [6] In this study on fashion apparel weuse optimal control theory to identify the optimal dynamicpricing strategy and replenishment cycle that maximizesaverage profit per unit time This optimization problem canbe solved hierarchically such that at a lower level of thehierarchy we derive the dynamic pricing strategy for thegiven replenishment cycle by applying Pontryaginrsquos maxi-mum principle Then at a higher level of the hierarchy wederive the optimal replenishment cycle for the given pricingstrategy Furthermore we design an algorithm to derive theoptimal dynamic pricing strategy and replenishment cycleNumerical examples and sensitivity analyses of main systemparameters are provided to demonstrate the obtained resultsand some managerial insights are offered

Complexity 3









119863(119901 (119905)) = 119889 (119901 (119905)) + 120596 (119905) (1)







119901max (119905) = 119889minus1 (120596 (119905)) (2)



119863(119901 (119905))= (119889 (119901 (119905)) + 120596 (119905) 0 le 119905 le 119879max 119901 (119905) le 119901max (119905)

0 119900119905ℎ119890119903119908119894119904119890(3)





4 Complexity

02

03

04

05

06

07

08(t)

5 10 15 20 25 300t

(t)=08e-003t

(t)=05e-003t




120596 (119905) = 1205960119890minus120578119905 (4)






0

02

04

06

08

(t)

(t)=08e-009t

(t)=08e-003t

5 10 15 20 25 300



0119868(119901(119905) 119905)119889119905 and 1198622 = 119888 int119879

0119863(119901(119905) 119905)119889119905 + 119860





minus ℎint1198790119868 (119901 (119905) 119905) 119889119905 minus 119860

(5)


1205871 (119901 (119905) 119879) = 1119879Π1 (119901 (119905) 119879) (6)


max119901(119905)119879

1205871 (119901 (119905) 119879)

119904119905 120597119868 (119905)120597119905 = minus119863 (119901 (119905) 119905) 0 le 119905 le 119879

119868 (119879) = 0 0 le 119901 le 119901max (119905)(7)


Complexity 5



max119901(119905)119879

1205871 (119901 (119905) 119879) = max119879le119879max

max119901(119905)le119901max(119905)

1205871 (119901 (119905) 119879) (8)


max119901(119905)

Π1 (119901 (119905))

= max119901(119905)

int1198790((119901 (119905) minus 119888)119863 (119901 (119905) 119905) minus ℎ119868 (119905)) 119889119905

minus 119860


(9)





(119905) = minus120597119867120597119868119868 (119905) = minus120597119867120597120582

120582 (0) = 0120597119867120597119901 = 0

(11)




1198891015840 (120595 (119905)) (120595 (119905) minus 119888 minus ℎ119905) + 119863 (120595 (119905)) = 0 (13)



le 0(14)





6 Complexity



119901max (119905) = 1119887 (119886 + 120596 (119905)) (15)



(16)


120597119901lowast (119905)120597119905 = 119887

2 (1205961015840 (119905) + 119887ℎ) for 0 le 119905 le 1198791 (17)





t0

MC

plowast(t)

T1

(t)

pＧＲ(t)



max119879

1205871 (119901lowast (119905) 119879)119904119905 119879 le 119879max

(18)



119872119877(119905) = 119901lowast (119905) minus 119872119862 (119905) (19)






0(119886 minus 119887119901lowast (119905) + 120596 (119905)) 119889119905 (21)

Complexity 7










119863(119901 119905) = 119889 (119901) + 120596 (119905) = 119886 minus 119887119901 + 1205960119890minus120578119905 (22)



0119868 (119901 119905) 119889119905

minus 119860(23)


max119901119879

1205872 (119901 119879) = 1119879Π2 (119901 119879) (24)



max119901

1205872 (119901 119879)

= max119901


0119868 (119901 119905) 119889119905 minus 119860

(25)



max119879

1205872 (119901lowast119865 119879)

= max119879


0119868 119889119905 minus 119860

(26)



0(119886 minus 119887119901lowast119865 + 120596 (119905)) 119889119905 (27)


5 Numerical Example




8 Complexity

MC(t)

plowast(t)

pmax (t)

T1

5 10 15 20 250t

0

5

10

15

20


0 5 10 15 20

T

1(plowast(t)T)


minus200

minus100

0

100

200

Tlowast








0 5 10 15 20 25t

PlowastF

0

2

4

6

8

10

12

p plowast(t)






Complexity 9






0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)




0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)



Appendix


10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3









119863(119901 (119905)) = 119889 (119901 (119905)) + 120596 (119905) (1)







119901max (119905) = 119889minus1 (120596 (119905)) (2)



119863(119901 (119905))= (119889 (119901 (119905)) + 120596 (119905) 0 le 119905 le 119879max 119901 (119905) le 119901max (119905)

0 119900119905ℎ119890119903119908119894119904119890(3)





4 Complexity

02

03

04

05

06

07

08(t)

5 10 15 20 25 300t

(t)=08e-003t

(t)=05e-003t




120596 (119905) = 1205960119890minus120578119905 (4)






0

02

04

06

08

(t)

(t)=08e-009t

(t)=08e-003t

5 10 15 20 25 300



0119868(119901(119905) 119905)119889119905 and 1198622 = 119888 int119879

0119863(119901(119905) 119905)119889119905 + 119860





minus ℎint1198790119868 (119901 (119905) 119905) 119889119905 minus 119860

(5)


1205871 (119901 (119905) 119879) = 1119879Π1 (119901 (119905) 119879) (6)


max119901(119905)119879

1205871 (119901 (119905) 119879)

119904119905 120597119868 (119905)120597119905 = minus119863 (119901 (119905) 119905) 0 le 119905 le 119879

119868 (119879) = 0 0 le 119901 le 119901max (119905)(7)


Complexity 5



max119901(119905)119879

1205871 (119901 (119905) 119879) = max119879le119879max

max119901(119905)le119901max(119905)

1205871 (119901 (119905) 119879) (8)


max119901(119905)

Π1 (119901 (119905))

= max119901(119905)

int1198790((119901 (119905) minus 119888)119863 (119901 (119905) 119905) minus ℎ119868 (119905)) 119889119905

minus 119860


(9)





(119905) = minus120597119867120597119868119868 (119905) = minus120597119867120597120582

120582 (0) = 0120597119867120597119901 = 0

(11)




1198891015840 (120595 (119905)) (120595 (119905) minus 119888 minus ℎ119905) + 119863 (120595 (119905)) = 0 (13)



le 0(14)





6 Complexity



119901max (119905) = 1119887 (119886 + 120596 (119905)) (15)



(16)


120597119901lowast (119905)120597119905 = 119887

2 (1205961015840 (119905) + 119887ℎ) for 0 le 119905 le 1198791 (17)





t0

MC

plowast(t)

T1

(t)

pＧＲ(t)



max119879

1205871 (119901lowast (119905) 119879)119904119905 119879 le 119879max

(18)



119872119877(119905) = 119901lowast (119905) minus 119872119862 (119905) (19)






0(119886 minus 119887119901lowast (119905) + 120596 (119905)) 119889119905 (21)

Complexity 7










119863(119901 119905) = 119889 (119901) + 120596 (119905) = 119886 minus 119887119901 + 1205960119890minus120578119905 (22)



0119868 (119901 119905) 119889119905

minus 119860(23)


max119901119879

1205872 (119901 119879) = 1119879Π2 (119901 119879) (24)



max119901

1205872 (119901 119879)

= max119901


0119868 (119901 119905) 119889119905 minus 119860

(25)



max119879

1205872 (119901lowast119865 119879)

= max119879


0119868 119889119905 minus 119860

(26)



0(119886 minus 119887119901lowast119865 + 120596 (119905)) 119889119905 (27)


5 Numerical Example




8 Complexity

MC(t)

plowast(t)

pmax (t)

T1

5 10 15 20 250t

0

5

10

15

20


0 5 10 15 20

T

1(plowast(t)T)


minus200

minus100

0

100

200

Tlowast








0 5 10 15 20 25t

PlowastF

0

2

4

6

8

10

12

p plowast(t)






Complexity 9






0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)




0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)



Appendix


10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity

02

03

04

05

06

07

08(t)

5 10 15 20 25 300t

(t)=08e-003t

(t)=05e-003t




120596 (119905) = 1205960119890minus120578119905 (4)






0

02

04

06

08

(t)

(t)=08e-009t

(t)=08e-003t

5 10 15 20 25 300



0119868(119901(119905) 119905)119889119905 and 1198622 = 119888 int119879

0119863(119901(119905) 119905)119889119905 + 119860





minus ℎint1198790119868 (119901 (119905) 119905) 119889119905 minus 119860

(5)


1205871 (119901 (119905) 119879) = 1119879Π1 (119901 (119905) 119879) (6)


max119901(119905)119879

1205871 (119901 (119905) 119879)

119904119905 120597119868 (119905)120597119905 = minus119863 (119901 (119905) 119905) 0 le 119905 le 119879

119868 (119879) = 0 0 le 119901 le 119901max (119905)(7)


Complexity 5



max119901(119905)119879

1205871 (119901 (119905) 119879) = max119879le119879max

max119901(119905)le119901max(119905)

1205871 (119901 (119905) 119879) (8)


max119901(119905)

Π1 (119901 (119905))

= max119901(119905)

int1198790((119901 (119905) minus 119888)119863 (119901 (119905) 119905) minus ℎ119868 (119905)) 119889119905

minus 119860


(9)





(119905) = minus120597119867120597119868119868 (119905) = minus120597119867120597120582

120582 (0) = 0120597119867120597119901 = 0

(11)




1198891015840 (120595 (119905)) (120595 (119905) minus 119888 minus ℎ119905) + 119863 (120595 (119905)) = 0 (13)



le 0(14)





6 Complexity



119901max (119905) = 1119887 (119886 + 120596 (119905)) (15)



(16)


120597119901lowast (119905)120597119905 = 119887

2 (1205961015840 (119905) + 119887ℎ) for 0 le 119905 le 1198791 (17)





t0

MC

plowast(t)

T1

(t)

pＧＲ(t)



max119879

1205871 (119901lowast (119905) 119879)119904119905 119879 le 119879max

(18)



119872119877(119905) = 119901lowast (119905) minus 119872119862 (119905) (19)






0(119886 minus 119887119901lowast (119905) + 120596 (119905)) 119889119905 (21)

Complexity 7










119863(119901 119905) = 119889 (119901) + 120596 (119905) = 119886 minus 119887119901 + 1205960119890minus120578119905 (22)



0119868 (119901 119905) 119889119905

minus 119860(23)


max119901119879

1205872 (119901 119879) = 1119879Π2 (119901 119879) (24)



max119901

1205872 (119901 119879)

= max119901


0119868 (119901 119905) 119889119905 minus 119860

(25)



max119879

1205872 (119901lowast119865 119879)

= max119879


0119868 119889119905 minus 119860

(26)



0(119886 minus 119887119901lowast119865 + 120596 (119905)) 119889119905 (27)


5 Numerical Example




8 Complexity

MC(t)

plowast(t)

pmax (t)

T1

5 10 15 20 250t

0

5

10

15

20


0 5 10 15 20

T

1(plowast(t)T)


minus200

minus100

0

100

200

Tlowast








0 5 10 15 20 25t

PlowastF

0

2

4

6

8

10

12

p plowast(t)






Complexity 9






0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)




0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)



Appendix


10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5



max119901(119905)119879

1205871 (119901 (119905) 119879) = max119879le119879max

max119901(119905)le119901max(119905)

1205871 (119901 (119905) 119879) (8)


max119901(119905)

Π1 (119901 (119905))

= max119901(119905)

int1198790((119901 (119905) minus 119888)119863 (119901 (119905) 119905) minus ℎ119868 (119905)) 119889119905

minus 119860


(9)





(119905) = minus120597119867120597119868119868 (119905) = minus120597119867120597120582

120582 (0) = 0120597119867120597119901 = 0

(11)




1198891015840 (120595 (119905)) (120595 (119905) minus 119888 minus ℎ119905) + 119863 (120595 (119905)) = 0 (13)



le 0(14)





6 Complexity



119901max (119905) = 1119887 (119886 + 120596 (119905)) (15)



(16)


120597119901lowast (119905)120597119905 = 119887

2 (1205961015840 (119905) + 119887ℎ) for 0 le 119905 le 1198791 (17)





t0

MC

plowast(t)

T1

(t)

pＧＲ(t)



max119879

1205871 (119901lowast (119905) 119879)119904119905 119879 le 119879max

(18)



119872119877(119905) = 119901lowast (119905) minus 119872119862 (119905) (19)






0(119886 minus 119887119901lowast (119905) + 120596 (119905)) 119889119905 (21)

Complexity 7










119863(119901 119905) = 119889 (119901) + 120596 (119905) = 119886 minus 119887119901 + 1205960119890minus120578119905 (22)



0119868 (119901 119905) 119889119905

minus 119860(23)


max119901119879

1205872 (119901 119879) = 1119879Π2 (119901 119879) (24)



max119901

1205872 (119901 119879)

= max119901


0119868 (119901 119905) 119889119905 minus 119860

(25)



max119879

1205872 (119901lowast119865 119879)

= max119879


0119868 119889119905 minus 119860

(26)



0(119886 minus 119887119901lowast119865 + 120596 (119905)) 119889119905 (27)


5 Numerical Example




8 Complexity

MC(t)

plowast(t)

pmax (t)

T1

5 10 15 20 250t

0

5

10

15

20


0 5 10 15 20

T

1(plowast(t)T)


minus200

minus100

0

100

200

Tlowast








0 5 10 15 20 25t

PlowastF

0

2

4

6

8

10

12

p plowast(t)






Complexity 9






0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)




0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)



Appendix


10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity



119901max (119905) = 1119887 (119886 + 120596 (119905)) (15)



(16)


120597119901lowast (119905)120597119905 = 119887

2 (1205961015840 (119905) + 119887ℎ) for 0 le 119905 le 1198791 (17)





t0

MC

plowast(t)

T1

(t)

pＧＲ(t)



max119879

1205871 (119901lowast (119905) 119879)119904119905 119879 le 119879max

(18)



119872119877(119905) = 119901lowast (119905) minus 119872119862 (119905) (19)






0(119886 minus 119887119901lowast (119905) + 120596 (119905)) 119889119905 (21)

Complexity 7










119863(119901 119905) = 119889 (119901) + 120596 (119905) = 119886 minus 119887119901 + 1205960119890minus120578119905 (22)



0119868 (119901 119905) 119889119905

minus 119860(23)


max119901119879

1205872 (119901 119879) = 1119879Π2 (119901 119879) (24)



max119901

1205872 (119901 119879)

= max119901


0119868 (119901 119905) 119889119905 minus 119860

(25)



max119879

1205872 (119901lowast119865 119879)

= max119879


0119868 119889119905 minus 119860

(26)



0(119886 minus 119887119901lowast119865 + 120596 (119905)) 119889119905 (27)


5 Numerical Example




8 Complexity

MC(t)

plowast(t)

pmax (t)

T1

5 10 15 20 250t

0

5

10

15

20


0 5 10 15 20

T

1(plowast(t)T)


minus200

minus100

0

100

200

Tlowast








0 5 10 15 20 25t

PlowastF

0

2

4

6

8

10

12

p plowast(t)






Complexity 9






0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)




0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)



Appendix


10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7










119863(119901 119905) = 119889 (119901) + 120596 (119905) = 119886 minus 119887119901 + 1205960119890minus120578119905 (22)



0119868 (119901 119905) 119889119905

minus 119860(23)


max119901119879

1205872 (119901 119879) = 1119879Π2 (119901 119879) (24)



max119901

1205872 (119901 119879)

= max119901


0119868 (119901 119905) 119889119905 minus 119860

(25)



max119879

1205872 (119901lowast119865 119879)

= max119879


0119868 119889119905 minus 119860

(26)



0(119886 minus 119887119901lowast119865 + 120596 (119905)) 119889119905 (27)


5 Numerical Example




8 Complexity

MC(t)

plowast(t)

pmax (t)

T1

5 10 15 20 250t

0

5

10

15

20


0 5 10 15 20

T

1(plowast(t)T)


minus200

minus100

0

100

200

Tlowast








0 5 10 15 20 25t

PlowastF

0

2

4

6

8

10

12

p plowast(t)






Complexity 9






0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)




0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)



Appendix


10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

MC(t)

plowast(t)

pmax (t)

T1

5 10 15 20 250t

0

5

10

15

20


0 5 10 15 20

T

1(plowast(t)T)


minus200

minus100

0

100

200

Tlowast








0 5 10 15 20 25t

PlowastF

0

2

4

6

8

10

12

p plowast(t)






Complexity 9






0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)




0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)



Appendix


10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9






0 5 10 15 20 25t

8

85

9

95

10

105

11

115

=03

=02

=01

plowast(t)




0 5 10 15 20 25t

8

85

9

95

10

105

11

115

c=3

c=2

c=1

pmax(t)

plowast(t)



Appendix


10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity





1205971205871 (119879)120597119879 = 120597120597119879 1119879Π1 (119901 (119905) 119879)

= 120597120597119879 1119879 int119879




119879



119879 (119901lowast (119879) minus 119888)





(A1)




12059721205871 (119879)1205971198792100381610038161003816100381610038161003816100381610038161003816119879=1198792 =

120597120597119879 minus120587 (119879)119879





1198792 1198891015840 (119901lowast (1198792))


(A3)


100381610038161003816100381610038161003816100381610038161003816119879=1198792 =11198792 1198891015840 ((119901lowast (1198792))) (119901lowast (1198792))1015840


(A4)


Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11







= 119886119879 minus 2119887119901119879 + 119887119888119879 + 119887ℎ11987922


(A5)


Data Availability




Acknowledgments


References





















12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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