optimal ordering and pricing policies for seasonal

Optimal Ordering and Pricing Policies for Seasonal ProductsImpacts of Demand Uncertainty and Capital ConstraintShi Jinzhao Fung Richard Y K Guo JuE

Published inDiscrete Dynamics in Nature and Society

Published 01012016

Document VersionFinal Published version also known as Publisherrsquos PDF Publisherrsquos Final version or Version of Record

LicenseCC BY

Publication record in CityU ScholarsGo to record

Published version (DOI)10115520161801658

Publication detailsShi J Fung R Y K amp Guo JE (2016) Optimal Ordering and Pricing Policies for Seasonal Products Impactsof Demand Uncertainty and Capital Constraint Discrete Dynamics in Nature and Society 2016 [1801658]httpsdoiorg10115520161801658

Citing this paperPlease note that where the full-text provided on CityU Scholars is the Post-print version (also known as Accepted AuthorManuscript Peer-reviewed or Author Final version) it may differ from the Final Published version When citing ensure thatyou check and use the publishers definitive version for pagination and other details

General rightsCopyright for the publications made accessible via the CityU Scholars portal is retained by the author(s) andor othercopyright owners and it is a condition of accessing these publications that users recognise and abide by the legalrequirements associated with these rights Users may not further distribute the material or use it for any profit-making activityor commercial gainPublisher permissionPermission for previously published items are in accordance with publishers copyright policies sourced from the SHERPARoMEO database Links to full text versions (either Published or Post-print) are only available if corresponding publishersallow open access

Take down policyContact lbscholarscityueduhk if you believe that this document breaches copyright and provide us with details We willremove access to the work immediately and investigate your claim

Download date 11042022

Research ArticleOptimal Ordering and Pricing Policies for Seasonal ProductsImpacts of Demand Uncertainty and Capital Constraint

Jinzhao Shi12 Richard Y K Fung2 and Jursquoe Guo1

1School of Management Xirsquoan Jiaotong University No 28 Xianning West Road Xirsquoan Shaanxi 710049 China2Department of Systems Engineering amp Engineering Management City University of Hong Kong Tat Chee AvenueKowloon Hong Kong

Correspondence should be addressed to Jursquoe Guo guojuemailxjtueducn

Received 31 May 2016 Accepted 25 September 2016

Academic Editor Jean J Loiseau

Copyright copy 2016 Jinzhao Shi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With a stochastic price-dependent market demand this paper investigates how demand uncertainty and capital constraint affectretailerrsquos integrated ordering and pricing policies towards seasonal products The retailer with capital constraint is normalized tobe with zero capital endowment while it can be financed by an external bank The problems are studied under a low and highdemand uncertainty scenario respectively Results show that when demand uncertainty level is relatively low the retailer facedwith demand uncertainty always sets a lower price than the riskless one while its order quantity may be smaller or larger thanthe riskless retailerrsquos which depends on the level of market size When adding a capital constraint the retailer will strictly prefera higher price but smaller quantity policy However in a high demand uncertainty scenario the impacts are more intricate Theretailer faced with demand uncertainty will always order a larger quantity than the riskless one if demand uncertainty level is highenough (above a critical value) while the capital-constrained retailer is likely to set a lower price than the well-funded one whendemand uncertainty level falls within a specific interval Therefore it can be further concluded that the impact of capital constrainton the retailerrsquos pricing decision can be influenced by different demand uncertainty levels

1 Introduction

With rapid development of science and technology moreand more products become fashionable or seasonal goodswith short sales cycle low salvage value and high demanduncertainty [1] For instance cut flowers clothing and evensmartphones are all common products with fashionableor seasonal attributes and inventory management of theseproducts is crucial aswell as complicated [2ndash4]What ismorethe vast of enterprises in the industry are small- andmedium-sized enterprises (SMEs) often with shortage of liquid funds[5] Especially under the influence of global financial crisismassive enterprises all over the world suffer from capitalshortage even bankruptcy throughout manufacturers dis-tributors and retailers Therefore in the latest decade inter-faces of operations and financial decisions receive substantialinterest and the so called ldquocapital-constrained newsvendorrdquoproblem is widely concerned [6]

Towards ldquocapital-constrained newsvendorrdquo problemmany scholars study the retailerrsquos integrated decisionon operations and financing in the presence of capitalconstraint including studies on optimal order quantity [7]purchase timing [8 9] and financing mode selection [10ndash12]These researches are all based on classical newsvendormodels where market prices and demand distributions areexogenously given However in most practical cases marketdemands are price-dependent thus characterizing priceeffect into the newsvendor problem is necessary [2 13] Forinstance demand uncertainty and demand-price elasticityof smartphones are both relatively high so it is crucial forsmartphone companies (eg Samsung Huawei and Xiaomi)to make an integrated decision on capacity and pricingbefore releasing a new smartphone to the market With arandom price-dependent market demand the current papercombines pricing decision into the ldquocapital-constrainednewsvendorrdquo problem and investigates retailerrsquos integrated

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 1801658 13 pageshttpdxdoiorg10115520161801658

2 Discrete Dynamics in Nature and Society

decisions on ordering pricing and financing which is notobserved in existing literature The main contributions andconclusions of this paper are as follows

(1) Studying Retailerrsquos Integrated Decisions on Ordering Pric-ing and Financing With introducing pricing decision intothe ldquocapital-constrained newsvendorrdquo problem this paperinvestigates the retailerrsquos integrated ordering and pricingdecisions in the presence of capital constraint Results showthat when market size is extremely small the retailer will notborrow from the bank to order any quantity Otherwise it willborrow to order and its optimal order quantity and sellingprice can be uniquely determined

(2) Investigating the Impacts of Demand Uncertainty andCapital Constraint on Retailerrsquos Optimal Ordering and PricingPolicies Three models (ie the riskless model uncertaintymodel and uncertainty-underfunded model) are developedand in-depth comparisons of optimal solutions in threemodels are carried out to reveal how demand uncertaintyand capital constraint affect retailerrsquos integrated orderingand pricing policies The problems are studied in a lowand high demand uncertainty scenario respectively whichis differentiated by an ingenious method and plenty ofconclusions are obtained through both theoretical analysesand numerical studies

The remainder of the paper is organized as followsSection 2 presents the review of related literature Section 3describes the problem and provides the model notations andassumptions Section 4 formulates three models and derivesthe optimal solutions Section 5 compares optimal solutionsin three models Section 6 carries out numerical experimentsto validate theoretic results Section 7 explores the case withhigh demand uncertainty Section 8 concludes and presentsfuture research directions

2 Literature Review

This paper is related to literature on ldquointerface of operationsand financing decisionsrdquo and ldquointegrated decisions on oper-ations and marketingrdquo In this section we review the mostrelated literature and highlight the innovations of the currentpaper

21 Interface of Operations and Financing Decisions Theinterface of operations and financing decisions has attractedwidely attention in recent years The operation-financeinterface contains a lot of categories and dimensions [14]Among them the ldquocapital-constrained newsvendor problem(CCNP)rdquo has been widely concerned Xu and Birge [15] arethe first to bring capital constraints into a simple newsvendormodel and try to understand the impact of capital structureon the retailerrsquos operational decisions Buzacott and Zhang[16] firstly incorporate asset-based financing into productiondecisions

Many researches focus on the impacts of financialconstraint on retailerrsquos orderinginventory policy Researchresults show that retailerrsquos optimal ordering policy will notbe influenced by financial constraint in the presence of

a competitively priced lending market [6 11 12 17] but it willbe affected by retailerrsquos internal capital level as well as loaninterest rate when the retailer is faced with a noncompeti-tively priced lending market [7 8]

For a retailer aside from bank credit financing (BCF)trade credit financing (TCF) (eg a delay payment schemeprovided by the supplier) is another commonly used financ-ing mode to deal with short-term capital shortage problem(see the review paper of Molamohamadi et al [18]) Jinget al [10] and Jing and Seidmann [11] further compare theefficiency between BCF and TCF for a capital-constrainedretailer and it is found that when production cost is relativelylow TCF will be more effective than BCF in mitigatingdouble marginalization Cai et al [19] prove that BCF andTCF are either complementary or substitutable for a capital-constrained retailer through both theoretical and empiricalstudies

Yan et al [20] study the CCNP problem from a per-spective of supply chain finance (SCF) system The partialtrade credit guarantee contract proposed by them can beregarded as the combination of BCF and TCF under whichthe manufacturer shares a portion of the retailerrsquos defaultrisk for the bank Other related researches extend the CCNPproblem to various directions For instance Zhang et al[21] consider a loss averse newsboy model with capitalconstraint Katehakis et al [22] tackle a dynamic multiperiodinventory management problem for a firm financed by bankloans Wang et al [23] investigate the optimal decisions andfinancing strategies of a capital-constrained manufacturerin the supply chain and Li et al [24] study the optimalfinancing decisions of two capital-constrained supply chainswith complementary products

All of the above researches are based on classicalnewsvendor models where market prices and demand dis-tributions are exogenously given However with a stochas-tic price-dependent demand the current paper combinesmarketing decision (ie pricing decision) into the ldquocapital-constrained newsvendorrdquo problem and investigates howfinancial constraint affects retailerrsquos integrated ordering andpricing decisions which is not observed in existing literature

22 Integrated Decisions on Operations andMarketing In thecontext of integrated decisions on operations and marketingmany researches focus on the retailerrsquos integrated orderingand pricing decisions under uncertain market demandsWhitin [25] is the first to formulate classical newsvendormodel with price effect where selling price and stockingquantity are set simultaneously Mills [26] refines the for-mulation by explicitly specifying demand as a function ofselling price Petruzzi andDada [13] present an overall reviewon incorporating pricing into the newsvendor model andtwo main random price-dependent demand formulationsthat is additive demand case and multiplicative demandcase are further extended to fit multiple period problemsVan Mieghem and Dada [27] further enrich the modelsby considering postponement of decisions on ordering orpricing

Based on these basic models integrated decisions onordering and pricing are investigated under different settings

Discrete Dynamics in Nature and Society 3

and perspectives such as product return [2 28] supplyuncertainty [29] service level constraint [30 31] multipleprice markdowns [32] supply chain contracts [33 34] dualsourcing channel [35 36] and multiperiod planning [337 38] Other creative works include the research on jointordering and pricing decisions considering repeat-purchasebased on the Bass model [39] retailerrsquos ordering and pricingdecisions of responding to the supplierrsquos temporary pricediscounts [40] ordering and pricing model consideringtransshipment between two independent retailers [41] andretailerrsquos joint ordering pricing and advertising decisions[42]

These studies examine retailerrsquos integrated ordering andpricing policies with different model settings However acapital-constrained retailer case is not observed in thisstream of research The current paper introduces the capitalconstraint into the ldquointegrated ordering and pricing decisionrdquoproblem and investigates retailerrsquos decisions on orderingpricing and financing simultaneously

23 Most Related Works and the Differences Our paper ismost related to the works of Petruzzi and Dada [13] andLi et al [2] Petruzzi and Dada [13] study how demanduncertainty affects retailerrsquos pricing policy but the effectof demand uncertainty on ordering policy is not analyzedHowever with a more general model setting this papergives an insight into the effects of demand uncertainty onretailerrsquos ordering and pricing policies Li et al [2] studyretailerrsquos optimal ordering and pricing policies with productreturns In their paper the retailer is assumed to be withsufficient capital to support its purchase decision whichis a conventional assumption in the ldquointegrated orderingand pricing decisionsrdquo literature However the current paperstudies retailerrsquos integrated decisions on ordering and pricingin the presence of capital constraint presents how capitalconstraint affects retailerrsquos integrated ordering and pricingpolicies and reveals how these effects influenced by differentlevels of demand uncertainty

3 Problem Description and Assumptions

This paper considers a supply chain comprising onemanufac-turer and one retailer The retailer purchases single seasonalproducts from the manufacturer before the selling seasonand then sells them to the customers The market demand israndomly price-dependent and is formulated as the followinglinear form 119863(119901 120576) = 119886 minus 119887119901 + 120576 where 120576 isin [minus119860119860] is arandom term with a uniform distribution whose PDF andCDF are 119891(120576) and 119865(120576) respectively and 119886 119887 and 119860 areall positive [2 13 34] (Normally 120576 isin [119860 119861] is adopted inrelated literature Here we adopt 120576 isin [minus119860 119860] to reduce theamount of parameters meanwhile measuring the demanduncertainty level by the absolute value of 119860 which representsthe range of the random fluctuation) The retailer is likelyto face two different kinds of financial statuses that is well-funded and capital constraint For convenience a capital-constrained retailerrsquos internal capital level is normalized tozero without loss of generality [11 12] and it can borrow

from external banks at interest rate 119903 (0 le 119903 le 1) Withrisk neutral assumption a well-funded retailerrsquos objectiveis to determine optimal order quantity and selling price tomaximize expected profit while a capital-constrained retailershould firstly make a financing decision and then make theordering and pricing decisions A rational retailer should seta selling price larger than the wholesale price announced bythe manufacturer that is 119901 gt 119908 Furthermore salvage valueof leftovers and penalty cost of shortages are all assumed to bezero [6 11 12] The main model notations are summarized inthe Notations Two crucial assumptions are made as follows

Assumption 1 119886 minus 119908119887 minus 119860 gt 0 which means that the marketdemand is always positive supposing the product is sold atwholesale price 119908 A similar assumption is made by Li et al[2] to ensure market size 119886 is not too small

Assumption 2 119860 le 119908119887 which means that random term120576 alone cannot bring a positive demand even though theproduct is sold at wholesale price 119908 It must be pointedout that this assumption is made to ensure the demanduncertainty level is not too high and it is critical for yieldingregular conclusions in this paper Towards the high demanduncertainty case where119860 gt 119908119887 results will be more irregularand cannot be derived by strict mathematical proofs hence itis further studied using numerical experiments in Section 7

This paper aims to reveal the impacts of demand uncer-tainty and capital constraint on retailerrsquos integrated orderingand pricing decisions Three models are established M0 isa riskless model where market demand is certain with therandom term 120576 fixed at its mean value M1 is an uncertaintymodel where the random term 120576 follows uniform distributionas introduced M2 is an uncertainty-underfunded modelwhere capital constraint is considered on the basis of M1Through comparing optimal solutions in models M0 andM1 the impacts of demand uncertainty on retailerrsquos optimalordering and pricing policies can be revealed Similarly theimpacts of capital constraint on retailerrsquos optimal decisionsare investigated by comparing the optimal solutions inmodels M1 and M2 All these impacts are examined undera low demand uncertainty scenario (ie 119860 le 119908119887) and a highdemand uncertainty scenario (ie119860 gt 119908119887) respectively andthen the differences of the conclusions will be pointed out

4 Model Formulations and Solutions

41 Model M0 In riskless model M0 random term 120576 equalsits mean value zero thusM0 is a deterministic model and themarket demand is119863(119901) = 119886minus119887119901 In thismodel selling price119901is the unique decision variable and the optimization problemis described as

max119901

120587M0 = (119901 minus 119908) (119886 minus 119887119901) (1)

The optimal selling price can be solved as

1199010 = 119886 + 1199081198872119887 (2)


Then the optimal order quantity can be determined as

1198760 = 119886 minus 1199081198872 (3)

42 Model M1 In uncertainty model M1 random term 120576 isin[minus119860119860] follows uniform distribution The retailerrsquos objectiveis to determine the optimal order quantity 119876lowast and optimalselling price 119901lowast to maximize expected profit So the problemis formulated as follows

max119876119901

120587M1 = 119901119864min (119876 and 119863) minus 119908119876 (4)

Based on min(119876 and 119863) = 119876 minus 119864(119876 minus 119863)+ where 119864(119876 minus119863)+ = max(119876minus119863 0) (4) can be converted into the followingformulation

max119876119901

120587M1 = (119901 minus 119908)119876 minus 119901119864 (119876 minus 119863)+ (5)

Define a substitution variable 119911 named stocking factor[2 13] 119911 = 119876 minus (119886 minus 119887119901) where 119911 isin [minus119860119860] If real-ized value of random term 120576 is larger than 119911 then marketdemand exceeds order quantity and shortages occur Oth-erwise market demand is smaller than order quantity andleftovers occur Then the problem of determining optimalquantity 119876lowast and price 119901lowast is converted into the problem ofdetermining optimal stocking factor 119911lowast and price 119901lowast and (5)can be further transformed into

max119911119901

120587M1 = (119901 minus 119908) (119911 + 119886 minus 119887119901) minus 119901Δ (119911) (6)

whereΔ(119911) = int119911minus119860(119911minus120576)119891(120576)119889120576measures the expected amount

of the leftovers The second-order derivative of 120587M1 withrespect to 119901 can be obtained as follows

1205972120587M11205971199012 = minus2119887 (7)

Obviously (7) is strictly negative thus 120587M1 is always con-cave in 119901 for any given 119911 Therefore a two-step optimizationmethod can be used to solve the problem Firstly supposing119911 is given the unique optimal 119901lowast(119911) can be obtained fromthe first-order derivative of 120587M1 concerning 119901 Substituting119901lowast(119911) into (6) then the objective function will contain onlyone decision variable 119911 Once the optimal stocking factor119911lowast is solved the optimal price can be obtained as 119901lowast(119911lowast)and the optimal order quantity will be determined by 119876lowast =119886 minus 119887119901lowast(119911lowast) + 119911lowastTheorem 3 In uncertainty model M1 the optimal stockingfactor 119911lowast is uniquely determined by

119865 (119911lowast) = 119867 (119911lowast) minus 119908119887119867 (119911lowast) + 119908119887 (8)

the optimal price 119901lowast is determined by

119901lowast (119911lowast) = 119867 (119911lowast) + 1199081198872119887 (9)

and the optimal order quantity 119876lowast is thus obtained as119876lowast = 119911lowast + 119886 minus 119887119901lowast (119911lowast) (10)

where119867(119911lowast) = 119886+119911lowastminusΔ(119911lowast) andΔ(119911lowast) = int119911lowastminus119860(119911lowastminus120576)119891(120576)119889120576

Proof See the Appendix

The retailerrsquos optimal expected profit can be obtained bysubstituting 119901lowast and 119876lowast into (5) or by substituting 119901lowast and 119911lowastinto (6)

43 Model M2 In uncertainty-underfunded model M2aside from demand uncertainty the retailer is faced with cap-ital constraint Following the convention in existing ldquocapital-constrained newsvendorrdquo literature the retailerrsquos internalcapital endowment is normalized to zero without loss ofgenerality [11 12] The retailer has access to financing fromexternal banks at loan interest rate 119903 which is exogenouslygiven and 119903 isin [0 1] The retailer should first make a decisiononwhether to opt for financing or not and then determine theoptimal order quantity 119876lowastlowast (ie equivalent to the decisionon the optimal financing amount 119861lowast) and the optimal sellingprice 119901lowastlowast At the end of the selling season the retailer repaysthe loanwith the sales revenueTheoptimization problem canbe described as follows

max119876119901

120587M2 = 119901119864min (119876 and 119863) minus 119908 (1 + 119903)119876 (11)

Equation (11) can be converted into the following formu-lation

max119876119901

120587M2 = (119901 minus 119908 (1 + 119903))119876 minus 119901119864 (119876 minus 119863)+ (12)

Then by applying the same stocking factor method pre-sented in Section 42 (12) can be further transformed as

max119911119901

120587M2 = (119901 minus 119908 (1 + 119903)) (119911 + 119886 minus 119887119901) minus 119901Δ (119911) (13)

where Δ(119911) = int119911minus119860(119911 minus 120576)119891(120576)119889120576 For any given 119911 120587M2 is

also concave in 119901 So the same two-step optimizationmethodpresented in Section 42 can be applied to solve (13) ThenTheorem 4 is concluded

Theorem 4 In uncertainty-underfunded model M2 the fol-lowing happens

(a) If market size 119886 satisfies 119886 gt 119860+119908119887(1 + 119903) the optimalstocking factor 119911lowastlowast is uniquely determined by

119865 (119911lowastlowast) = 119867 (119911lowastlowast) minus 119908119887 (1 + 119903)119867 (119911lowastlowast) + 119908119887 (1 + 119903) (14)

the optimal price 119901lowastlowast is determined by

119901lowastlowast (119911lowastlowast) = 119867 (119911lowastlowast) + 119908119887 (1 + 119903)2119887 (15)

and the optimal order quantity 119876lowastlowast is thus obtained as119876lowastlowast = 119911lowastlowast + 119886 minus 119887119901lowastlowast (119911lowastlowast) (16)


where 119867(119911lowastlowast) = 119886 + 119911lowastlowast minus Δ(119911lowastlowast) and Δ(119911lowastlowast) = int119911lowastlowastminus119860(119911lowastlowast minus120576)119891(120576)119889120576

(b) If market size 119886 satisfies 119860 + 119908119887 lt 119886 le 119860 + 119908119887(1 + 119903)the retailer will not borrow capital to order any quantity


In the case 119886 gt 119860 + 119908119887(1 + 119903) the retailerrsquos optimalexpected profit can be obtained by substituting 119901lowastlowast and 119876lowastlowastinto (12) or by substituting 119901lowastlowast and 119911lowastlowast into (13) and theoptimal borrowing amount is 119861lowast = 119908119876lowastlowast In the case 119860 +119908119887 lt 119886 le 119860 + 119908119887(1 + 119903) the retailerrsquos optimal borrowingamount and expected profit are both zero

5 Model Comparative Analyses

In this section comparative analyses of optimal solutionsin three models will be carried out to reveal how demanduncertainty and capital constraint affect retailerrsquos optimalordering and pricing policies towards seasonal products

51 Model M0 versus Model M1 From (2) and (3) theoptimal solutions of model M0 are intuitional Based on (8)in Theorem 3 the following properties concerning optimalstocking factor 119911lowast in model M1 can be concluded

Proposition 5 In model M1

(a) the optimal stocking factor 119911lowast is strictly increasing inthe market size 119886 that is 120597119911lowast120597119886 gt 0

(b) lim119886rarr119908119887+119860119911lowast = minus119860 and lim119886rarr+infin119911lowast = 119860Proof See the Appendix

Proposition 5 means that when other parameters aregiven the optimal stocking factor 119911lowast in model M1 isdetermined by the market size 119886 There exists a one-to-one correspondence relationship between 119911lowast and 119886 Basedon Proposition 5 the following proposition can be furtherobtained

Proposition 6 In model M1 the optimal order quantity 119876lowastis strictly increasing in the market size 119886 that is 120597119876lowast120597119886 gt 0and lim119886rarr119908119887+119860119876lowast = 0Proof See the Appendix

Proposition 6 reveals a one-to-one correspondence rela-tionship between 119876lowast and 119886 From (3) it is clear that theoptimal order quantity 1198760 in riskless model M0 is alsoincreasing in 119886 Then 119876lowast and 1198760 can be compared based onthe conclusions in Propositions 5 and 6The optimal prices inmodels M0 and M1 are compared as well and all the resultsare summarized inTheorem 7

Theorem 7 (a) There always exists a critical value greaterthan119860+119908119887 andwhenmarket size 119886 ismore than the optimalorder quantity in model M1 will be larger than that in modelM0 that is 119876lowast gt 1198760 Otherwise when 119886 is less than 119876lowast lt

1198760 holds Further is uniquely determined by the followingequation set where 119911 satisfies minus119860 le 119911 le 1198601199112 + 1198602 + 6119860119911 = 01199113 minus 31198601199112 + (31198602 minus 4119860 minus 4119860119908119887) 119911 + 41198602 ( minus 3119908119887)minus 1198603 = 0

(17)

(b)The optimal price inmodelM1 is nomore than that inmodelM0 that is 119901lowast le 1199010 and the case 119901lowast = 1199010 occurs only whenmarket size 119886 approaches infinityProof See the Appendix

Theorem 7 shows that when demand uncertainty level isrelatively low (ie 119860 le 119908119887) the retailer faced with demanduncertainty always sets a lower price than the riskless retailerHowever its order quantity may be smaller or larger than theriskless retailerrsquos which depends on the level of market sizeIn summary when market size is relatively small the retailerfaced with demand uncertainty tends to adopt a ldquolower priceand quantityrdquo policy to stimulate the market demand as wellas control the expected leftovers to wrestle with the demanduncertainty But when market size is relatively large it willadopt a ldquolower price but larger quantityrdquo policy to deal withthe demand uncertainty

52 Model M1 versus Model M2 When market size isextremely small that is 119860 + 119908119887 lt 119886 le 119860 + 119908119887(1 + 119903) theoptimal order quantity in model M2 is zero thus pricing ismeaningless So comparative analyses between the optimalsolutions in models M1 and M2 are performed only underthe case of 119886 gt 119860 + 119908119887(1 + 119903) which is the most majority ofthe situation

It is obvious that aside from the existing parameters inmodel M1 the optimal order quantity and selling price inmodel M2 are also affected by loan interest rate 119903 Based onTheorem 4 the following proposition concerning the optimalstocking factor 119911lowastlowast and loan interest rate 119903 can be concluded

Proposition 8 InmodelM2 the optimal stocking factor 119911lowastlowast isstrictly decreasing in the loan interest rate 119903 that is 120597119911lowastlowast120597119903 lt0Proof See the Appendix

Based on Proposition 8 the optimal order quantities andselling prices inmodelsM1 andM2 can be compared and theresults are summarized inTheorem 9

Theorem 9 When market size satisfies 119886 gt 119860 + 119908119887(1 + 119903)(a) the optimal order quantity inmodelM2 is nomore than

that in model M1 that is 119876lowastlowast le 119876lowast(b) the optimal price in model M2 is no less than that in

model M1 that is 119901lowastlowast ge 119901lowastProof See the Appendix


Theorem 9 reveals that when demand uncertainty level isrelatively low (ie 119860 le 119908119887) the capital-constrained retailertends to adopt a ldquohigher price but smaller quantityrdquo policycomparing with the well-funded one The rationale can beinferred as follows the loan interest rate causes the increaseof unit purchase cost which leads to the increase of the sellingprice Then a higher price leads to the decrease of marketdemand so a smaller quantity is necessary to control theexpected leftovers

Obviously the same conclusions can be obtained in deter-ministic demandmodels by replacing119908(1+119903) for119908 in (2) and(3) and then compared with (2) and (3) However it shouldbe noted that the optimal order quantity and selling price areall linear with loan interest rate under deterministic demandmodels but they are all nonlinear with loan interest rateunder uncertain demand models More importantly whendemand uncertainty level is relatively high (ie 119860 gt 119908119887)the conclusions in Theorem 9 will not be always establishedwhich will be introduced in Section 7 Results show thatthe capital-constrained retailer is likely to set a lower pricethan the well-funded one in some special situations which isopposite to the conclusion inTheorem 9(b)

6 Numerical Analyses

In this paper models M1 and M2 are both stochastic andthe optimal solutions are all embodied in implicit functionswhich are not intuitional Numerical analyses in this sectionwill contribute to a better understanding of the conclusionswith two main objectives (1) validating the theoretic resultsobtained in Section 5 and (2) comparing the optimal profitsin three models

61 Experimental Parameters To validate the results ob-tained in Section 51 the following parameters are used 119860 =5 119908 = 3 119887 = 2 which satisfies Assumption 2 (ie 119860 le 119908119887)Then in order to validate conclusions obtained in Section 52the market size 119886 should be fixed at a certain value thatsatisfies 119886 gt 119860 + 119908119887(1 + 119903) for any possible value of 119903 isin[0 1] thus 119886 = 50 is chosen to meet this condition Finallyexpected profits in three models are compared by assumingloan interest rate 119903 = 0162 Comparisons of Optimal Solutions Figure 1 shows thechange of stocking factor 119911lowast with regard to market size 119886 inmodel M1 It can be observed that as market size increasesfrom its lower bound 119908119887 + 119860 (ie 11) to infinity the optimalstocking factor will increase from minus119860 to119860 (ie from minus5 to 5)which verifies Proposition 5

Figures 2 and 3 present the relations between the optimalsolutions in models M0 and M1 In Figure 2 when marketsize is smaller than a critical valuewhich is approximately 162(ie the abscissa of intersection of the two lines) the optimalorder quantity inmodelM1will be smaller than that inmodelM0 Otherwise the optimal order quantity in model M1 willbe larger than that in model M0 From Figure 3 it can beobserved that the optimal price in model M1 is always lessthan that in model M0 and the difference between the two

20 40 60 80 100 120 140 160 180 2000a

zlowast

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Figure 1 Stocking factor with market size varying (model M1)

M0M1

15 20 25 30 35 40 45 5010a

0

5

10

15

20

25

Q

Figure 2 Optimal order quantity comparison in models M0 andM1

optimal prices will gradually decrease to zero as market sizeincreases to infinityThe above simulation results support theconclusions of Theorem 7

Figure 4 shows the change of stocking factor 119911lowastlowast withregard to loan interest rate 119903 in model M2 Obviously theoptimal stocking factor is decreasing in loan interest rate inmodel M2 which verifies Proposition 8

Figures 5 and 6 present the relations between the optimalsolutions in models M1 and M2 Based on Theorem 3 theoptimal solutions in model M1 are independent of loaninterest rate so the optimal order quantity and price ofmodelM1 are all horizontal lines as loan interest rate increases fromzero to one (see the dot-dash lines in Figures 5 and 6) Thesimulation results show that the optimal order quantity inmodel M2 is equal to or less than that in model M1 whilethe optimal price in model M2 is equal to or more than that


15 20 25 30 35 40 45 5010a

2

4

6

8

10

12

14

p

M0M1

Figure 3 Optimal price comparison in models M0 and M1

01 02 03 04 05 06 07 08 09 10r

1

12

14

16

18

2

22

24

26

28

3

zlowastlowast

Figure 4 Stocking factor with loan interest rate varying (modelM2)

in model M1 which are consistent with the conclusions inTheorem 9 Moreover it can be observed that the equivalentcases occur only when loan interest rate equals zero and asloan interest rate rises the difference between the optimalsolutions in models M1 and M2 will be enlarged It shouldalso be noted that the optimal solutions in model M2 are allnonlinear with the loan interest rate even though the resultsin our numerical simulations look like linear (see the star-solid lines in Figures 5 and 6)

63 Comparison of Expected Profits In this subsection com-parison of expected profits in three models is performedand the result is shown in Figure 7 It can be found thatexpected profit in riskless model M0 is the highest followedby uncertainty model M1 and expected profit in uncertainty-underfunded model M2 is the lowest Furthermore this

M1M2

01 02 03 04 05 06 07 08 09 10r

20

21

22

23

24

25

26

Q


138

14

142

144

146

148

15

152

154

p

01 02 03 04 05 06 07 08 09 10r

M1M2


quantitative relation is independent of market size So bothdemand uncertainty and capital constraint can be regardedas obstacles for the retailer to overcome which lead to somelosses of efficiency

7 Extension for the Case 119860 gt 119908119887When demand uncertainty level is relatively low that is119860 le 119908119887 some regular results are obtained as shown inTheorems 7 and 9 However when demand uncertainty levelis relatively high that is 119860 gt 119908119887 theoretic results cannot bederived through strict mathematical proofs In this sectionthe case 119860 gt 119908119887 is further studied with the aid of numerical


Table 1 Applicability of conclusions inTheorems 7 and 9 under the case of 119860 gt 119908119887119860 Theorem 7(a) Theorem 7(b) Theorem 9(a) Theorem 9(b)600sim800 radic radic radic radic800sim1292 radic radic radic times1292sim3200 times radic radic timesge3200 times radic radic radic

15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t

Figure 7 Optimal expected profit comparison in models M0 M1and M2

experiments Consistent with Section 6 119908 = 3 119887 = 2 areused

Under the high demand uncertainty case of 119860 gt 119908119887Table 1 shows the applicability of conclusions in Theorems7 and 9 Results are more intricate and irregular which aresummarized in Observation 10

Observation 10 In the high demand uncertainty case that is119860 gt 119908119887 the following happen(a) Conclusions in Theorems 7(b) and 9(a) are all appli-

cable(b) Conclusions in Theorem 7(a) are applicable only

when demand uncertainty level is less than a criticalvalue (ie approximately 1292 in our example)

(c) Conclusions in Theorem 9(b) are inapplicable onlywhen demand uncertainty level falls within a specificinterval (ie approximately 800sim3200 in our exam-ple)

The inapplicable cases in Table 1 are further explained asfollows For Theorem 7(a) when demand uncertainty levelis high enough (ie higher than 1292) the optimal orderquantity in model M1 will be always larger than that inmodel M0 which means two lines in Figure 2 will no longerintersect An example with 119860 = 15 is shown in Figure 8

25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1

Figure 8 Optimal order quantity comparison inmodelsM0 andM1(119860 = 15)

For Theorem 9(b) when demand uncertainty level ishigher than 800 but lower than 3200 a small interval ofmarket size 119886 (see Table 2) incurs the result 119901lowastlowast lt 119901lowast whenassigned with specific values of loan interest rate 119903 whichis opposite to the result in Theorem 9(b) with 119860 le 119908119887 Anexample is given in Figure 9 with choosing 119860 = 20 119886 = 33 Itcan be observed that the case 119901lowastlowast lt 119901lowast will occur for somehigher values of 119903 that is approximately 083sim100

In summary the capital-constrained retailer is likely toset a lower price than the well-funded one when demanduncertainty level falls within a subinterval of 119860 gt 119908119887 whichis different from the result obtained in the case119860 le 119908119887wherethe capital-constrained retailer always sets a higher price thanthe well-funded one Therefore it can be further concludedthat the impact of capital constraint on retailerrsquos pricingdecision can be influenced by different levels of demanduncertainty

8 Conclusions and Future Research

This paper combines pricing decision into the ldquocapital-constrained newsvendorrdquo problem and investigates retailerrsquosintegrated ordering and pricing policies in the presence ofcapital constraint Results show that when market size isextremely small the retailer will not borrow from the externalbank to order any quantity Otherwise it will borrow to


Table 2 The interval of market size incurs 119901lowastlowast lt 119901lowast under 119860 gt 119908119887119860 600ndash800 810 1000 2000 3000 3190 ge3200119886 mdash (2010 2015) (2200 2283) (3200 3395) (4200 4247) (4390 4392) mdash

01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2

Figure 9 Optimal price comparison inmodelsM1 andM2 (119860 = 20119886 = 33)

order and its optimal order quantity and selling price can beuniquely determined

Specifically this paper develops threemodels (ie risklessmodel uncertainty model and uncertainty-underfundedmodel) to investigate the impacts of demand uncertainty andcapital constraint on retailerrsquos integrated ordering and pricingpolicies under a low and high demand uncertainty scenariorespectively Theoretical results show that when demanduncertainty level is relatively low (ie 119860 le 119908119887) the retailerfaced with demand uncertainty always sets a lower price thanthe riskless one while its order quantity may be smaller orlarger than the riskless retailerrsquos which depends on the level ofmarket sizeThe retailer with capital constraint always adoptsa ldquohigher price but smaller quantityrdquo policy comparing withthe well-funded one However when demand uncertaintylevel is relatively high (ie 119860 gt 119908119887) numerical results showthat the retailer faced with demand uncertainty will alwaysorder a larger quantity than the riskless one when demanduncertainty level is high enough (above a critical value) andthe retailer with capital constraint is likely to set a lower pricethan the well-funded one when demand uncertainty levelfalls within a specific interval The differences of conclusionsin these two scenarios also indicate the impact of capitalconstraint on retailerrsquos pricing decision can be influencedby different levels of demand uncertainty What is more bycomparing the optimal profits in threemodels it is concludedthat both demand uncertainty and capital constraint canbe regarded as obstacles for the retailer to tackle whichinevitably lead to some losses of efficiency

This paper ends with a discussion of the limitations andpossible directions First the case when demand uncertaintylevel is relatively high is studied only with numerical experi-ments due to obstacles of mathematical proofs Second somesimplification has beenmade in this paper such as the supplychain structure salvage value and shortage cost and eventhe retailerrsquos internal capital endowment Various directionscan be explored in future research More complex scenarioscan be considered such as multichannel for ordering orselling and multiperiod problem Investigating the order-ing and pricing policies of the capital-constrained retailerwith considering different internal capital endowments isalso meaningful for guiding diverse industrial practices Inaddition other demand formulations (eg the multiplicativecase) or other financing schemes (eg the trade creditfinancing) can be introduced to do some comparison studies

Appendix

Proof of Theorem 3 Based on (7) for any given 119911 120587M1 isconcave in 119901 so the unique optimal 119901lowast(119911) can be obtainedby solving 120597120587M1120597119901 = 0 which is

119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)

Then substituting 119901lowast(119911) into (6) and taking the first-orderderivative of 120587M1 with respect to 119911 based on the chain ruleyields

119889120587M1119889119911 = 120597120587M1120597119911 + 120597120587M1120597119901lowast (119911)119889119901lowast (119911)119889119911

= 12119887 ((119886 + 119911 minus Δ (119911) + 119908119887) (1 minus 119865 (119911)) minus 2119908119887) (A2)

Letting 119866(119911) = (119886 + 119911 minus Δ(119911) + 119908119887)(1 minus 119865(119911)) minus 2119908119887 thesecond-order derivative of 119866(119911) is

1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)

Thus 119866(119911) is concave and unimodal in 119911 Further since119866(119860) = minus2119908119887 lt 0 and 119866(minus119860) = 119886 minus 119908119887 minus 119860 gt 0(Assumption 1) there always exists a unique 119911lowast isin [minus119860119860]that satisfies 119866(119911lowast) = 0 Obviously when 119911 lt 119911lowast 119889120587M1119889119911 gt0 and when 119911 gt 119911lowast 119889120587M1119889119911 lt 0 By letting (A2) equal zeroa unique 119911lowast that maximizes 120587M1 can be obtained as shown in(8) where 119867(119911lowast) = 119886 + 119911lowast minus Δ(119911lowast) Then the optimal price119901lowast(119911lowast) is determined by substituting 119911lowast into (A1) and theoptimal order quantity is thus 119876lowast = 119886 minus 119887119901lowast(119911lowast) + 119911lowast basedon the definition of stocking factor 119911


Proof of Theorem 4 For any given 119911 1205972120587M21205971199012 = minus2119887 lt 0so 120587M2 is also concave in 119901The unique optimal119901lowastlowast(119911) can beobtained by solving 120597120587M2120597119901 = 0 Then substituting 119901lowastlowast(119911)into (13) and taking the first-order derivative of 120587M2 withrespect to 119911 based on the chain rule yields

119889120587M2119889119911 = 12119887 ((119886 + 119911 minus Δ (119911) + 119908119887 (1 + 119903)) (1 minus 119865 (119911))minus 2119908119887 (1 + 119903))

(A4)

Letting119866(119911) = (119886+119911minusΔ(119911)+119908119887(1+119903))(1minus119865(119911))minus2119908119887(1+119903)the second-order derivative of 119866(119911) is the same as shown in(A3) so119866(119911) is concave and unimodal in 119911 Further it can beeasily calculated that 119866(119860) = minus2119908119887(1 + 119903) lt 0 and 119866(minus119860) =119886 minus 119860 minus 119908119887(1 + 119903)

Case (a) When 119886 gt 119860 + 119908119887(1 + 119903) 119866(minus119860) gt 0 the proof issimilar to the proof of Theorem 3

Case (b)When 119860 + 119908119887 lt 119886 le 119860 + 119908119887(1 + 119903) 119866(minus119860) le 0 inconjunction with 1198661015840(minus119860) = 1 minus 119891(minus119860)(119886 minus 119860 + 119908119887(1 + 119903)) lt1 minus 119908119887119860 minus 1199081198871199032119860 lt 0 (Assumptions 1 and 2) it can beconcluded that 119866(119911) le 0 over [minus119860 119860] thus 120587M2 is decreasingin 119911 over [minus119860 119860] and the optimal stocking factor is 119911lowastlowast =minus119860 Then the optimal selling price and order quantity canbe determined as 119901lowastlowast(minus119860) = (119886 minus 119860 + 119908119887(1 + 119903))2119887 and119876lowastlowast = (119886 minus 119860 minus 119908119887(1 + 119903))2 respectively Since 119886 le 119860 +119908119887(1 + 119903) we have 119876lowastlowast le 0 Because the order quantitymust be nonnegative that is 119876lowastlowast ge 0 and based on 119876lowastlowast =(119886 + 119911lowastlowast + Δ(119911lowastlowast) minus 119908119887(1 + 119903))2 larger 119876lowastlowast requires larger119911lowastlowast causing a decrease in 120587M2 Thus 119876lowastlowast = 0 is the optimaldecision for Case (b)

Proof of Proposition 5 For part (a) taking the first-orderderivative of (8) with respect to 119886 yields

119891 (119911lowast) 120597119911lowast120597119886= (1 + 120597119911lowast120597119886 minus 119865 (119911lowast) (120597119911lowast120597119886)) (119886 + 119911lowast minus Δ (119911lowast) + 119908119887) minus (119886 + 119911lowast minus Δ (119911lowast) minus 119908119887) (1 + 120597119911lowast120597119886 minus 119865 (119911lowast) (120597119911lowast120597119886))(119886 + 119911lowast minus Δ (119911lowast) + 119908119887)2

(A5)

Equation (A5) can be simplified as

(119891 (119911lowast) (119886 + 119911lowast minus Δ (119911lowast) + 119908119887)22119908119887 + 119865 (119911lowast)

minus 1) 120597119911lowast120597119886 = 1(A6)

Letting 119871(119911lowast) = 119891(119911lowast)(119886+119911lowast minusΔ(119911lowast)+119908119887)22119908119887+119865(119911lowast)minus1taking the first-order derivative of 119871(119911lowast) concerning 119911lowast yields120597119871 (119911lowast)120597119911lowast= 2119891 (119911lowast) (119886 + 119911lowast minus Δ (119911lowast) + 119908119887) (1 minus 119865 (119911lowast))2119908119887+ 119891 (119911lowast)

(A7)

Since 119886+119911lowastminusΔ(119911lowast)+119908119887 is increasing in 119911lowast and 119911lowast isin [minus119860119860]the minimum value of 119886+119911lowastminusΔ(119911lowast)+119908119887 is 119886minus119860+119908119887whichis strictly positive based on Assumption 1 Therefore (A7) isstrictly positive whichmeans 119871(119911lowast) is increasing in 119911lowastWhen119911lowast = minus119860 based on Assumptions 1 and 2 the minimum valueof 119871(119911lowast) is obtained as

119871 (minus119860) = (119886 minus 119860 + 119908119887)24119860119908119887 minus 1 gt (2119908119887)24119860119908119887 minus 1= 119908119887 minus 119860119860 ge 0

(A8)

Until now it can be concluded that 119871(119911lowast) is strictly positivefor 119911lowast isin [minus119860119860]Then based on (A6) 120597119911lowast120597119886 gt 0 is proved

For part (b) based on the proof of Theorem 3 it is clearthat 119866(119911) is concave and unimodal in 119911 and 119866(119860) lt 0119866(minus119860) = 119886 minus 119908119887 minus 119860 gt 0 When 119886 rarr 119908119887 + 119860 we have119866(minus119860) rarr 0+ Two cases should be considered

Case (1) minus119860 infinitely approaches the smaller root of 119866(119911)when the unique optimal stocking factor 119911lowast over [minus119860 119860]willbe the larger root of 119866(119911) which is more than minus119860Case (2) minus119860 infinitely approaches the larger root of 119866(119911)when 119911lowast over [minus119860 119860] will be minus119860

The sign of the first-order derivative can be used todistinguish the above two cases because1198661015840(minus119860) gt 0 holds inCase (1) while 1198661015840(minus119860) lt 0 holds in Case (2) The first-orderderivative of 119866(119911) is as follows1198661015840 (119911) = 120597119866 (119911)120597119911

= (1 minus 119865 (119911))2 minus 119891 (119911) (119886 + 119911 minus Δ (119911) + 119908119887) (A9)

Based on Assumptions 1 and 2 it can be further obtained that

1198661015840 (minus119860) = 1 minus 12119860 (119886 minus 119860 + 119908119887) lt 1 minus 119908119887119860 le 0 (A10)

This supports Case (2) Thus lim119886rarr119908119887+119860119911lowast = minus119860Further when 119886 rarr +infin using LrsquoHospitol principle the

right side of (8) equals one Solving 119865(119911lowast) = 1 yields 119911lowast =119860


Proof of Proposition 6 Based on (9) and (10) the optimalorder quantity in model M1 is

119876lowast = 119886 + 119911lowast + Δ (119911lowast) minus 1199081198872 (A11)

Taking the first-order derivative of119876lowast with respect to 119886 basedon Proposition 5 yields

120597119876lowast120597119886 = 12 (1 + (1 + 119865 (119911lowast)) 120597119911lowast

120597119886 ) gt 0lim119886rarr119908119887+119860

119876lowast = 119908119887 + 119860 minus 119860 + 0 minus 1199081198872 = 0(A12)

This completes the proof of Proposition 6

Proof of Theorem 7 For part (a) define Δ119876 = 119876lowast minus 1198760 =(119911lowast + Δ(119911lowast))2 Obviously Δ119876 is strictly increasing in 119911lowastBy Proposition 5 119911lowast is strictly increasing in 119886 thus Δ119876 isstrictly increasing in 119886 Then lim119886rarr119908119887+119860Δ119876 = minus1198602 lt 0 andlim119886rarr+infinΔ119876 = 119860 gt 0 can be obtained based on Propositions5 and 6 So there always exists a critical value more than119860 + 119908119887 when 119886 lt yields Δ119876 lt 0 and when 119886 gt yields Δ119876 gt 0 Further under the critical value unique119911lowast isin [minus119860119860] is determined by (8) based onTheorem 3 whichcan be explicitly described as

119911lowast3 minus 3119860119911lowast2 + (31198602 minus 4119860 minus 4119860119908119887) 119911lowast+ 41198602 ( minus 3119908119887) minus 1198603 = 0 (A13)

and 119911lowast must simultaneously solve Δ119876 = (119911lowast + Δ(119911lowast))2 = 0which is equivalent to

119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)

It is easy to verify that (A14) has only one root over [minus119860 119860]then based on unique 119911lowast (A13) can uniquely determinethe value of Therefore the solution of the equation setcomprised of (A13) and (A14) can determine unique underthe constraint 119911lowast isin [minus119860119860] which is the conclusion of part(a)

For part (b) based on (2) and (9)

119901lowast = 1199010 + 119911lowast minus Δ (119911lowast)2119887 (A15)

It is clear that 120597(119911lowast minus Δ(119911lowast))120597119911lowast = 1 minus 119865(119911lowast) ge 0 so themaximum value of 119911lowastminusΔ(119911lowast) can be obtained by letting 119911lowast =119860 which is zero Thus 119901lowast le 1199010 holds By Proposition 5 thecase 119911lowast = 119860 occurs only when 119886 approaches infinity

Proof of Proposition 8 Equation (14) can be rewritten as thefollowing formulation

119908119887 (1 + 119903) (1 + 119865 (119911lowastlowast)) = 119867 (119911lowastlowast) (1 minus 119865 (119911lowastlowast)) (A16)

Taking the first-order derivative of (A16) with respect to 119903yields

119908119887 (1 + 119865 (119911lowastlowast)) + 119908119887 (1 + 119903) 119891 (119911lowastlowast) 120597119911lowastlowast120597119903= (1 minus 119865 (119911lowastlowast))2 120597119911lowastlowast120597119903 minus 119891 (119911lowastlowast)119867 (119911lowastlowast) 120597119911lowastlowast120597119903

(A17)

Further

120597119911lowastlowast120597119903= 119908119887 (1 + 119865 (119911lowastlowast))(1 minus 119865 (119911lowastlowast))2 minus 119891 (119911lowastlowast)119867 (119911lowastlowast) minus 119908119887 (1 + 119903) 119891 (119911lowastlowast)

(A18)

Obviously the numerator is positive Letting the denomina-tor be 119879(119911lowastlowast) taking the first-order derivative of 119879(119911lowastlowast) withrespect to 119911lowastlowast yields 120597119879(119911lowastlowast)120597119911lowastlowast = minus3119891(119911lowastlowast)(1 minus 119865(119911lowastlowast))which is nonpositive Since 119886 gt 119860 +119908119887(1 + 119903) the maximumvalue of 119879(119911lowastlowast) is119879 (minus119860) = 3119860 minus 119886 minus 119908119887 (1 + 119903)2119860 lt 119860 minus 119908119887 (1 + 119903)119860

le 0(A19)

Hence 119879(119911lowastlowast) is always negative over [minus119860 119860]Thus (A18) isnegative and the proof is completed

Proof of Theorem 9 For part (a) based on (8) and (14) when119903 = 0119876lowastlowast = 119876lowast holds Since 119903 isin [0 1] and119876lowast is independentof 119903 as long as 120597119876lowastlowast120597119903 lt 0 is proved 119876lowastlowast le 119876lowast will beproved From (15) and (16)

120597119876lowastlowast120597119903 = 12 ((1 + 119865 (119911lowastlowast)) 120597119911lowastlowast

120597119903 minus 119908119887) (A20)

120597119911lowastlowast120597119903 lt 0 has been proved in Proposition 8 thus yielding120597119876lowastlowast120597119903 lt 0For part (b) similar to the proof for part (a) when 119903 = 0119901lowastlowast = 119901lowast holds Hence as long as 120597119901lowastlowast120597119903 ge 0 is proved119901lowastlowast ge 119901lowast will be automatically proved From (15) taking the

first-order derivative of 119901lowastlowast with respect to 119903 yields120597119901lowastlowast120597119903 = 12119887 ((1 minus 119865 (119911lowastlowast)) 120597119911

lowastlowast

120597119903 + 119908119887) (A21)

Substituting (A18) into (A21) yields

120597119901lowastlowast120597119903 = 12119887 (119908119887 (1 minus 1198652 (119911lowastlowast))

(1 minus 119865 (119911lowastlowast))2 minus 119891 (119911lowastlowast)119867 (119911lowastlowast) minus 119908119887 (1 + 119903) 119891 (119911lowastlowast) + 119908119887) (A22)


Let119877(119911lowastlowast) = 119908119887(1minus1198652(119911lowastlowast))((1minus119865(119911lowastlowast))2minus119891(119911lowastlowast)119867(119911lowastlowast)minus119908119887(1 + 119903)119891(119911lowastlowast)) +119908119887 To prove 120597119901lowastlowast120597119903 ge 0 is equivalent toproving 119877(119911lowastlowast) ge 0 which can be simplified as follows

(1 minus 1198652 (119911lowastlowast)) le 119891 (119911lowastlowast)119867 (119911lowastlowast)+ 119908119887 (1 + 119903) 119891 (119911lowastlowast)minus (1 minus 119865 (119911lowastlowast))2

(A23)

Further (A23) can be reduced to

119911lowastlowast2 minus 10119860119911lowastlowast + 91198602 minus 4119860 (119886 + 119908119887 (1 + 119903)) le 0 (A24)

Next (A24) will be proved to be always true for 119911lowastlowast isin[minus119860119860] Let119880(119911lowastlowast) = 119911lowastlowast2minus10119860119911lowastlowast+91198602minus4119860(119886+119908119887(1+119903))It is clear that when 119911lowastlowast = 119860119880(119860) = minus4119860(119886+119908119887(1+119903)) lt 0Based on Proposition 5 it can be inferred that 119911lowastlowast = minus119860occurs only when the market size infinitely approaches itslower bound that is 119886 rarr 119860+119908119887(1+119903) Further 119886 gt 119860+2119908119887must hold in order to ensure that 119886 gt 119860 + 119908119887(1 + 119903) forany 119903 isin [0 1] Thus 119886 rarr 119860 + 119908119887(1 + 119903) occurs only when119886 rarr 119860 + 2119908119887 in conjunction with 119903 rarr 1 Therefore

119880 (minus119860) = 201198602 minus 4119860 (119860 + 2119908119887 + 119908119887 (1 + 1))= 16119860 (119860 minus 119908119887) (A25)

Based onAssumption 2 (A25) is nonpositive Since119880(119911lowastlowast) isa quadratic function with convexity it can be concluded that119880(119911lowastlowast) le 0 over [minus119860 119860] Until now (A24) has been provedto be always true ensuring that 120597119901lowastlowast120597119903 ge 0 holds in (A22)Theorem 9 is thus proved

Notations

119886 Demand intercept which representsmarket size119887 Price coefficient which represents slope ofthe linear demand curve120576 Random term in demand functionminus119860119860 Lower and upper bound of 120576 respectivelywhose absolute values measure the level ofdemand uncertainty119891(120576) 119865(120576) PDF and CDF of 120576 respectively119903 Loan interest rate119863 Market demand119894 Index taking a value among 0 1 and 2120587M119894 Retailerrsquos profit function in model M119894119908 Unit wholesale price announced by themanufacturer119901 Unit selling price set by the retailer119876 Order quantity of the retailer

1198760 1199010 Retailerrsquos optimal order quantity and sellingprice in model M0 respectively119876lowast 119901lowast Retailerrsquos optimal order quantity and sellingprice in model M1 respectively119876lowastlowast 119901lowastlowast Retailerrsquos optimal order quantity and sellingprice in model M2 respectively119911 Stocking factor whose definition will beexplained in the ensuing chapters119911lowast 119911lowastlowast Optimal stocking factor in models M1 andM2 respectively

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant nos 71473193 71403031) andFundamental Research Funds for the Central Universities ofChina (Grant no skz2014010)

References

[1] F Hu C-C Lim and Z Lu ldquoOptimal production and procure-ment decisions in a supply chain with an option contract andpartial backordering under uncertaintiesrdquoAppliedMathematicsand Computation vol 232 no 1 pp 1225ndash1234 2014

[2] Y Li C Wei and X Cai ldquoOptimal pricing and order policieswith B2B product returns for fashion productsrdquo InternationalJournal of Production Economics vol 135 no 2 pp 637ndash6462012

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

[4] H Fu B Dan and X Sun ldquoJoint optimal pricing and order-ing decisions for seasonal products with weather-sensitivedemandrdquo Discrete Dynamics in Nature and Society vol 2014Article ID 105098 8 pages 2014

[5] H Sun ldquoThe research on financing problems of PRCrsquos SMEsrdquoInternational Business and Management vol 10 no 2 pp 69ndash74 2015

[6] P Kouvelis and W Zhao ldquoFinancing the newsvendor suppliervs bank and the structure of optimal trade credit contractsrdquoOperations Research vol 60 no 3 pp 566ndash580 2012

[7] M Dada and Q Hu ldquoFinancing newsvendor inventoryrdquo Oper-ations Research Letters vol 36 no 5 pp 569ndash573 2008

[8] Y Feng Y Mu B Hu and A Kumar ldquoCommodity optionspurchasing and credit financing under capital constraintrdquoInternational Journal of Production Economics vol 153 no 1 pp230ndash237 2014

[9] X Yan and Y Wang ldquoA newsvendor model with capitalconstraint and demand forecast updaterdquo International Journalof Production Research vol 52 no 17 pp 5021ndash5040 2014

[10] B Jing X Chen and G G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[11] B Jing and A Seidmann ldquoFinance sourcing in a supply chainrdquoDecision Support Systems vol 58 no 1 pp 15ndash20 2014


[12] X Chen ldquoAmodel of trade credit in a capital-constrained distri-bution channelrdquo International Journal of Production Economicsvol 159 no 1 pp 347ndash357 2015

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

[14] L Zhao and A Huchzermeier ldquoOperationsndashfinance interfacemodels a literature review and frameworkrdquo European Journalof Operational Research vol 244 no 3 pp 905ndash917 2015

[15] X Xu and J R Birge ldquoJoint production and financing decisionsmodelling and analysisrdquo Working Paper Graduate School ofBusiness University of Chicago 2004

[16] J A Buzacott and R Q Zhang ldquoInventory management withasset-based financingrdquo Management Science vol 50 no 9 pp1274ndash1292 2004

[17] P Kouvelis and W Zhao ldquoThe newsvendor problem and price-only contract when bankruptcy costs existrdquo Production andOperations Management vol 20 no 6 pp 921ndash936 2011

[18] Z Molamohamadi N Ismail Z Leman and N ZulkiflildquoReviewing the literature of inventory models under tradecredit contactrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 975425 19 pages 2014

[19] G G Cai X Chen and Z Xiao ldquoThe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Productionand Operations Management vol 23 no 4 pp 583ndash598 2014

[20] N Yan B Sun H Zhang and C Liu ldquoA partial credit guar-antee contract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journal ofProduction Economics vol 173 pp 122ndash133 2016

[21] B Zhang D Wu L Liang and D L Olson ldquoSupply chain lossaverse newsboy model with capital constraintrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 46 no 5pp 646ndash658 2016

[22] M N Katehakis B Melamed and J J Shi ldquoCash-flow baseddynamic inventory managementrdquo Production and OperationsManagement vol 25 no 9 pp 1558ndash1575 2016

[23] BWang D-CHuang H Li and J-Y Ding ldquoOptimal decisionsand financing strategies selection of supply chain with capitalconstraintrdquo Mathematical Problems in Engineering vol 2016Article ID 6597259 14 pages 2016

[24] Y Li T Chen and B Xin ldquoOptimal financing decisions of twocash-constrained supply chains with complementary productsrdquoSustainability vol 8 no 5 pp 429ndash445 2016

[25] T M Whitin ldquoInventory control and price theoryrdquo Manage-ment Science vol 2 no 1 pp 61ndash68 1955

[26] E SMills ldquoUncertainty andprice theoryrdquoTheQuarterly Journalof Economics vol 73 no 1 pp 116ndash130 1959

[27] J A VanMieghem andMDada ldquoPrice versus production post-ponement capacity and competitionrdquoManagement Science vol45 no 12 pp 1631ndash1649 1999

[28] J Chen and P C Bell ldquoCoordinating a decentralized supplychain with customer returns and price-dependent stochasticdemand using a buyback policyrdquo European Journal of Opera-tional Research vol 212 no 2 pp 293ndash300 2011

[29] M Xu and Y Lu ldquoThe effect of supply uncertainty in price-setting newsvendor modelsrdquo European Journal of OperationalResearch vol 227 no 3 pp 423ndash433 2013

[30] W Jammernegg and P Kischka ldquoThe price-setting newsvendorwith service and loss constraintsrdquoOmega vol 41 no 2 pp 326ndash335 2013

[31] P Abad ldquoDetermining optimal price and order size for a pricesetting newsvendor under cycle service levelrdquo InternationalJournal of Production Economics vol 158 pp 106ndash113 2014

[32] W Chung S Talluri and R Narasimhan ldquoOptimal pricing andinventory strategies with multiple price markdowns over timerdquoEuropean Journal of Operational Research vol 243 no 1 pp130ndash141 2015

[33] J Shi G Zhang and K K Lai ldquoOptimal ordering and pricingpolicy with supplier quantity discounts and price-dependentstochastic demandrdquo Optimization vol 61 no 2 pp 151ndash1622012

[34] C Wang and X Chen ldquoOptimal ordering policy for a price-setting newsvendorwith option contracts under demanduncer-taintyrdquo International Journal of Production Research vol 53 no20 pp 6279ndash6293 2015

[35] W Xing S Wang and L Liu ldquoOptimal ordering and pricingstrategies in the presence of a B2B Spot marketrdquo EuropeanJournal of Operational Research vol 221 no 1 pp 87ndash98 2012

[36] D A Serel ldquoProduction and pricing policies in dual sourcingsupply chainsrdquo Transportation Research Part E Logistics andTransportation Review vol 76 pp 1ndash12 2015

[37] K Pan K K Lai L Liang and S C H Leung ldquoTwo-periodpricing and ordering policy for the dominant retailer in a two-echelon supply chain with demand uncertaintyrdquoOmega vol 37no 4 pp 919ndash929 2009

[38] H Li and A Thorstenson ldquoA multi-phase algorithm for ajoint lot-sizing and pricing problem with stochastic demandsrdquoInternational Journal of Production Research vol 52 no 8 pp2345ndash2362 2014

[39] X Wu and J Zhang ldquoJoint ordering and pricing decisions fornew repeat-purchase productsrdquo Discrete Dynamics in Natureand Society vol 2015 Article ID 461959 8 pages 2015

[40] Y Xia ldquoResponding to supplier temporary price discountsin a supply chain through ordering and pricing decisionsrdquoInternational Journal of Production Research vol 54 no 7 pp1938ndash1950 2016

[41] B Dan Q He K Zheng and R Liu ldquoOrdering and pricingmodel of retailersrsquo preventive transshipment dominated bymanufacturer with conditional returnrdquo Computers amp IndustrialEngineering vol 100 no 1 pp 24ndash33 2016

[42] B-B Cao Z-P Fan H Li and T-H You ldquoJoint inventorypricing and advertising decisions with surplus and stockoutloss aversionsrdquo Discrete Dynamics in Nature and Society vol2016 Article ID 1907680 14 pages 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Research ArticleOptimal Ordering and Pricing Policies for Seasonal ProductsImpacts of Demand Uncertainty and Capital Constraint

Jinzhao Shi12 Richard Y K Fung2 and Jursquoe Guo1

1School of Management Xirsquoan Jiaotong University No 28 Xianning West Road Xirsquoan Shaanxi 710049 China2Department of Systems Engineering amp Engineering Management City University of Hong Kong Tat Chee AvenueKowloon Hong Kong

Correspondence should be addressed to Jursquoe Guo guojuemailxjtueducn

Received 31 May 2016 Accepted 25 September 2016

Academic Editor Jean J Loiseau

Copyright copy 2016 Jinzhao Shi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With a stochastic price-dependent market demand this paper investigates how demand uncertainty and capital constraint affectretailerrsquos integrated ordering and pricing policies towards seasonal products The retailer with capital constraint is normalized tobe with zero capital endowment while it can be financed by an external bank The problems are studied under a low and highdemand uncertainty scenario respectively Results show that when demand uncertainty level is relatively low the retailer facedwith demand uncertainty always sets a lower price than the riskless one while its order quantity may be smaller or larger thanthe riskless retailerrsquos which depends on the level of market size When adding a capital constraint the retailer will strictly prefera higher price but smaller quantity policy However in a high demand uncertainty scenario the impacts are more intricate Theretailer faced with demand uncertainty will always order a larger quantity than the riskless one if demand uncertainty level is highenough (above a critical value) while the capital-constrained retailer is likely to set a lower price than the well-funded one whendemand uncertainty level falls within a specific interval Therefore it can be further concluded that the impact of capital constrainton the retailerrsquos pricing decision can be influenced by different demand uncertainty levels

1 Introduction

With rapid development of science and technology moreand more products become fashionable or seasonal goodswith short sales cycle low salvage value and high demanduncertainty [1] For instance cut flowers clothing and evensmartphones are all common products with fashionableor seasonal attributes and inventory management of theseproducts is crucial aswell as complicated [2ndash4]What ismorethe vast of enterprises in the industry are small- andmedium-sized enterprises (SMEs) often with shortage of liquid funds[5] Especially under the influence of global financial crisismassive enterprises all over the world suffer from capitalshortage even bankruptcy throughout manufacturers dis-tributors and retailers Therefore in the latest decade inter-faces of operations and financial decisions receive substantialinterest and the so called ldquocapital-constrained newsvendorrdquoproblem is widely concerned [6]

Towards ldquocapital-constrained newsvendorrdquo problemmany scholars study the retailerrsquos integrated decisionon operations and financing in the presence of capitalconstraint including studies on optimal order quantity [7]purchase timing [8 9] and financing mode selection [10ndash12]These researches are all based on classical newsvendormodels where market prices and demand distributions areexogenously given However in most practical cases marketdemands are price-dependent thus characterizing priceeffect into the newsvendor problem is necessary [2 13] Forinstance demand uncertainty and demand-price elasticityof smartphones are both relatively high so it is crucial forsmartphone companies (eg Samsung Huawei and Xiaomi)to make an integrated decision on capacity and pricingbefore releasing a new smartphone to the market With arandom price-dependent market demand the current papercombines pricing decision into the ldquocapital-constrainednewsvendorrdquo problem and investigates retailerrsquos integrated

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 1801658 13 pageshttpdxdoiorg10115520161801658






2 Literature Review






















max119901

120587M0 = (119901 minus 119908) (119886 minus 119887119901) (1)


1199010 = 119886 + 1199081198872119887 (2)



1198760 = 119886 minus 1199081198872 (3)


max119876119901

120587M1 = 119901119864min (119876 and 119863) minus 119908119876 (4)


max119876119901



max119911119901




1205972120587M11205971199012 = minus2119887 (7)










max119876119901

120587M2 = 119901119864min (119876 and 119863) minus 119908 (1 + 119903)119876 (11)


max119876119901



max119911119901


























(17)

















20 40 60 80 100 120 140 160 180 2000a

zlowast

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5


M0M1

15 20 25 30 35 40 45 5010a

0

5

10

15

20

25

Q






15 20 25 30 35 40 45 5010a

2

4

6

8

10

12

14

p

M0M1


01 02 03 04 05 06 07 08 09 10r

1

12

14

16

18

2

22

24

26

28

3

zlowastlowast




M1M2

01 02 03 04 05 06 07 08 09 10r

20

21

22

23

24

25

26

Q


138

14

142

144

146

148

15

152

154

p

01 02 03 04 05 06 07 08 09 10r

M1M2






15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t









25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1








01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














2 Literature Review






















max119901

120587M0 = (119901 minus 119908) (119886 minus 119887119901) (1)


1199010 = 119886 + 1199081198872119887 (2)



1198760 = 119886 minus 1199081198872 (3)


max119876119901

120587M1 = 119901119864min (119876 and 119863) minus 119908119876 (4)


max119876119901



max119911119901




1205972120587M11205971199012 = minus2119887 (7)










max119876119901

120587M2 = 119901119864min (119876 and 119863) minus 119908 (1 + 119903)119876 (11)


max119876119901



max119911119901


























(17)

















20 40 60 80 100 120 140 160 180 2000a

zlowast

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5


M0M1

15 20 25 30 35 40 45 5010a

0

5

10

15

20

25

Q






15 20 25 30 35 40 45 5010a

2

4

6

8

10

12

14

p

M0M1


01 02 03 04 05 06 07 08 09 10r

1

12

14

16

18

2

22

24

26

28

3

zlowastlowast




M1M2

01 02 03 04 05 06 07 08 09 10r

20

21

22

23

24

25

26

Q


138

14

142

144

146

148

15

152

154

p

01 02 03 04 05 06 07 08 09 10r

M1M2






15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t









25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1








01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















max119901

120587M0 = (119901 minus 119908) (119886 minus 119887119901) (1)


1199010 = 119886 + 1199081198872119887 (2)



1198760 = 119886 minus 1199081198872 (3)


max119876119901

120587M1 = 119901119864min (119876 and 119863) minus 119908119876 (4)


max119876119901



max119911119901




1205972120587M11205971199012 = minus2119887 (7)










max119876119901

120587M2 = 119901119864min (119876 and 119863) minus 119908 (1 + 119903)119876 (11)


max119876119901



max119911119901


























(17)

















20 40 60 80 100 120 140 160 180 2000a

zlowast

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5


M0M1

15 20 25 30 35 40 45 5010a

0

5

10

15

20

25

Q






15 20 25 30 35 40 45 5010a

2

4

6

8

10

12

14

p

M0M1


01 02 03 04 05 06 07 08 09 10r

1

12

14

16

18

2

22

24

26

28

3

zlowastlowast




M1M2

01 02 03 04 05 06 07 08 09 10r

20

21

22

23

24

25

26

Q


138

14

142

144

146

148

15

152

154

p

01 02 03 04 05 06 07 08 09 10r

M1M2






15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t









25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1








01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198760 = 119886 minus 1199081198872 (3)


max119876119901

120587M1 = 119901119864min (119876 and 119863) minus 119908119876 (4)


max119876119901



max119911119901




1205972120587M11205971199012 = minus2119887 (7)










max119876119901

120587M2 = 119901119864min (119876 and 119863) minus 119908 (1 + 119903)119876 (11)


max119876119901



max119911119901


























(17)

















20 40 60 80 100 120 140 160 180 2000a

zlowast

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5


M0M1

15 20 25 30 35 40 45 5010a

0

5

10

15

20

25

Q






15 20 25 30 35 40 45 5010a

2

4

6

8

10

12

14

p

M0M1


01 02 03 04 05 06 07 08 09 10r

1

12

14

16

18

2

22

24

26

28

3

zlowastlowast




M1M2

01 02 03 04 05 06 07 08 09 10r

20

21

22

23

24

25

26

Q


138

14

142

144

146

148

15

152

154

p

01 02 03 04 05 06 07 08 09 10r

M1M2






15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t









25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1








01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























(17)

















20 40 60 80 100 120 140 160 180 2000a

zlowast

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5


M0M1

15 20 25 30 35 40 45 5010a

0

5

10

15

20

25

Q






15 20 25 30 35 40 45 5010a

2

4

6

8

10

12

14

p

M0M1


01 02 03 04 05 06 07 08 09 10r

1

12

14

16

18

2

22

24

26

28

3

zlowastlowast




M1M2

01 02 03 04 05 06 07 08 09 10r

20

21

22

23

24

25

26

Q


138

14

142

144

146

148

15

152

154

p

01 02 03 04 05 06 07 08 09 10r

M1M2






15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t









25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1








01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















20 40 60 80 100 120 140 160 180 2000a

zlowast

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5


M0M1

15 20 25 30 35 40 45 5010a

0

5

10

15

20

25

Q






15 20 25 30 35 40 45 5010a

2

4

6

8

10

12

14

p

M0M1


01 02 03 04 05 06 07 08 09 10r

1

12

14

16

18

2

22

24

26

28

3

zlowastlowast




M1M2

01 02 03 04 05 06 07 08 09 10r

20

21

22

23

24

25

26

Q


138

14

142

144

146

148

15

152

154

p

01 02 03 04 05 06 07 08 09 10r

M1M2






15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t









25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1








01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










15 20 25 30 35 40 45 5010a

2

4

6

8

10

12

14

p

M0M1


01 02 03 04 05 06 07 08 09 10r

1

12

14

16

18

2

22

24

26

28

3

zlowastlowast




M1M2

01 02 03 04 05 06 07 08 09 10r

20

21

22

23

24

25

26

Q


138

14

142

144

146

148

15

152

154

p

01 02 03 04 05 06 07 08 09 10r

M1M2






15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t









25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1








01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











15 20 25 30 35 40 45 5010a

M2

M0M1

0

50

100

150

200

250

Profi

t









25 30 35 40 45 5020a

5

10

15

20

25

30

35

Q

M0M1








01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











01 02 03 04 05 06 07 08 09 10r

905

91

915

92

925

93

935

94

p

M1M2





Appendix


119901lowast (119911) = 119886 + 119911 minus Δ (119911) + 1199081198872119887 (A1)





1205972119866 (119911)1205971199112 = minus3119891 (119911) (1 minus 119865 (119911)) le 0 (A3)





(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(A4)






(A5)



minus 1) 120597119911lowast120597119886 = 1(A6)


(A7)



(A8)















120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120597119886 ) gt 0lim119886rarr119908119887+119860






119911lowast2 + 1198602 + 6119860119911lowast = 0 (A14)









(A17)

Further


(A18)


le 0(A19)




120597119903 minus 119908119887) (A20)



lowastlowast

120597119903 + 119908119887) (A21)







(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(A23)






Notations



Competing Interests


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









optimal ordering and pricing policies for seasonal

Documents