A negotiation-based algorithm to coordinate supplier development in decentralized supply chains

ContentslistsavailableatScienceDirect

European

Journal

of

Operational

Research

journalhomepage:www.elsevier.com/locate/ejor

Production,

Manufacturing

and

Logistics

A

negotiation-based

algorithm

to

coordinate

supplier

development

in

decentralized

supply

chains

M.

Proch

a ,∗

,

K.

Worthmann

b

,

J.

Schlüchtermann

a

a_{Faculty of Law, Business Administration and Economics, Universität Bayreuth, 95440 Bayreuth, Germany} b_{Institute for Mathematics, Technische Universität Ilmenau, 98683 Ilmenau, Germany}

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 5 March 2015 Accepted 13 June 2016 Available online 16 June 2016 Keywords:

Supply chain management

Supply chain coordination Supplier development Optimal control Relational view

a

b

s

t

r

a

c

t

In this paper, we study supplier development in a decentralized supply chain with a single manufacturer and a single supplier. Because supplier development usually requires relationship-specific investments, the allocation of investment costs is a critical issue faced by participating firms. Referencing the relational view, we first investigate the effects of relationship-specific investments on the efficiency and effective- ness of supplier development. Next, we formulate and solve a continuous time optimal control model characterizing the decision to invest in supplier development and show that the supplier’s incentive to participate in supplier development critically depends on the manufacturer’s share of investment costs. The findings of our numerical analysis indicate that although the subsidy can lead to significant improve- ment in supply chain performance, subsidizing a constant share of investment costs is not always eco- nomically reasonable from the manufacturer’s point of view. Thus, we provide a negotiation-based algo- rithm that assists the manufacturing firm in gradually increasing the share of investment costs to ensure an efficient level of subsidy, resulting in both perfect supply chain coordination and a win–win situation. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Because manufacturing firms increasingly focus on their core competencies, capable supplier networks play a paramount role ingeneratingcompetitiveadvantage.However,supplierstoooften lackthecapabilitytoperformadequately.Inresponse, manufactur-ersacross a widerangeof industriesdevelop closerrelationships withtheir suppliers and initiate supplier development programs (Wagner, 2010 ). Within the automotive industry, Toyota initially beganprovidingon-site assistancetohelpsupplying firms imple-mentleanmanufacturingconcepts fortechnologicaland organiza-tionalchanges (Marksberry, 2012; Sako, 2004 ). Other automobile manufacturers have followed this collaborative approach to im-provesupplychainperformance,includingChrysler,Daimler,Ford, GeneralMotors,Honda,Nissan,andVolkswagen(Praxmarer-Carus, Sucky, & Durst, 2013; Talluri, Narasimhan, & Chung, 2010 ).Further examplesofsupplierdevelopmentprogramsappliedbycompanies outsidethe automotive industry can be found, among others, at

∗ _{Corresponding author. Tel.: +49 921556195; fax: +49 921556192.}

E-mail addresses: [email protected](M. Proch), karl.worthmann@ tu-ilmenau.de(K. Worthmann),[email protected]

(J. Schlüchtermann).

Boeing,Dell,GeneralElectric,Hewlett-Packard,Motorola,Siemens, andWalmart(Routroy & Pradhan, 2013; Wagner, 2006a ).

Supplier development is broadly defined as “any effort by a buying firm toimprovea supplier’s performance and/or capabili-tiestomeetthemanufacturingfirm’sshort-and/orlong-term sup-ply needs” (Krause, 1999 , p. 206). Following this definition, sup-plier developmentactivities are typically initiated, designed, and administered by the manufacturing firm. Moreover, it is usually assumed that suppliers are eagerly willing to adapt and imple-mentsupplierdevelopmentactivitiesimposedbythemanufacturer (Mortensen & Arlbjørn, 2012 ).However,despitethepotential ben-efits resultingfromsuch participation,supplierdevelopmentmay notalwaysbeapayingpropositionforthesupplier(Kim & Netes- sine, 2013; Krause, Handfield, & Tyler, 2007 ).

Indeed,therearesoundargumentswhysuppliersmightrefrain fromjoininginsupplierdevelopment.Because resources commit-ted to supplier development are mostoften relationship-specific and therefore difficult or even impossible to redeploy outside the particular business relationship, suppliers may see such in-vestmentsasvulnerable to opportunisticexpropriation (Wang, Li, Ross, & Craighead, 2013; Williamson, 1979 ). Therefore, suppliers might be reluctant to modify or improveinternal processes,and insteadpursuetheirownobjectiveswhileparticipatinginsupplier http://dx.doi.org/10.1016/j.ejor.2016.06.029

development(Bai & Sarkis, 2014 ).Becausesupplierdevelopmentis areciprocalapproachthatrequiresmutualrecognition,misaligned objectives andthe hazards ofopportunistic behavior could cause inefficiency in or, even worse, the premature abandonment of the supplier development process (Blonska, Storey, Rozemeijer, Wetzels, & de Ruyter, 2013; Iida, 2012 ).

Giventhisbackground,thepurposeofourresearchisto exam-ine the alignmentof thesupply chainpartners’ objectives to en-hance the supplier developmentprocess. We seekto answer the following questions: How does the risk of partner opportunism affect the supplier’s willingness to participate in manufacturer-initiated supplier development? Arebilateral relationship-specific investmentsaviableincentivetoinducedesirablesupplier behav-ior, whilesimultaneouslyfacilitatingvalue generationwithin sup-plier development? Additionally, how should the mutual invest-ment decisionbe arrangedto improvesupplychaincoordination, while both the supplier andthe manufacturer increase their re-spectiveprofit?

Byansweringthesequestions,ourpapermakesathreefold con-tribution.First,inreferencetotherelationalviewasatheoretical framework,weinvestigatetheeffectofrelationship-specific invest-mentsontheefficiencyandeffectivenessofsupplierdevelopment andshowthatthedeploymentofbilateralrelationship-specific in-vestmentsmightbeanimportantsourceofcompetitiveadvantage. Second,consideringadecentralizedsupplychainconsistingofone manufacturer andone supplier, we formulate a continuous time optimalcontrolmodelcharacterizingthesupplierdevelopment in-vestment decision. Using this model, we find that the supplier’s incentivetoparticipateinsupplierdevelopmentcriticallydepends onthemanufacturer’sshareofinvestmentcosts.Wethencarryout anextensivenumericalanalysisanddemonstratethatalthoughthe manufacturer’s subsidy leads to significant improvement in sup-plychainperformance,subsidizingaconstantshareofinvestment costs isnot alwayseconomicallyreasonablefromthe manufactur-ingfirm’sperspective.Giventhefactthatforanongoing collabora-tivebusinessrelationship,supplychaincoordinationmustresultin enhancingtheprofitability ofboththemanufacturerandthe sup-plier, we third presenta negotiation-based algorithm that assists themanufacturingfirmingraduallyincreasingtheshareof invest-mentcoststoensureanefficientlevelofsubsidy.Theproposed co-ordinationschemecanbe employedasaguideline torealize per-fect supply chain coordination while both the manufacturer and thesupplierincreasetheirrespectiveprofitineachiteration, lead-ingtoawin–winsituation.

The remaining ofthispaperisstructured asfollows.First,the relatedliteratureisbrieflyreviewedinSection 2 beforesome the-oretical background on the performance implications of supplier development is provided in the subsequent Section 3 . Then, the basic optimalcontrol problemis describedin Section 4 before a reference solution is computed in Section 5 assuming a central-ized decision-making process. Next, two cases ofa decentralized decision-makingprocessareconsidered:indirectsupplier develop-ment in Section 6 anddirect supplier development inSection 7 . Subsequently, a negotiation-based coordination algorithm is pro-posed and numerically analyzed in Section 8 . Finally, an exten-siontoascenariowithmultiplesuppliersisbrieflysketchedbefore conclusionsaredrawninSection 9 .

2. Relatedliterature

Thetopicofsupplierdevelopmenthasreceivedconsiderable at-tention from researchers in the past two decades (Talluri et al., 2010 ). Previous research on supplier development demonstrates that manufacturing firms use a variety of activities to develop suppliers’ performance and/or capabilities. With few exceptions (e.g., Hartley & Jones, 1997; Sánchez-Rodríguez, Hemsworth, &

Martínez-Lorente, 2005 ),supplierdevelopmentactivitiesare classi-fiedbythemanufacturer’slevelofcommitmenttoaspecific sup-plier(e.g.,Humphreys, Cadden, Wen-Li, & McHugh, 2011; Krause, 1997; Krause, Scannell, & Calantone, 20 0 0; Monczka, Trent, & Callahan, 1993; Wagner, 2006b ). Accordingly, we distinguish two typesofsupplierdevelopmentactivitiesinthispaper,indirectand directsupplierdevelopment.

Inthe caseof indirect supplierdevelopment, the manufactur-ing firm commits no oronly limitedresources to a specific sup-plier.Instead,indirectsupplierdevelopmentmayencompass activ-itiessuch asevaluatingsuppliers’ operations,settingperformance goals,providingperformancefeedback,instillingcompetitive pres-sure, promisingfuture businessbased on goalattainment or rec-ognizingsuppliers’progressbydesignatingthemaspreferred sup-pliers(Krause et al., 20 0 0; Wagner, 2010 ). These activities might encourage suppliers to take additional efforts to better comply withthe manufacturer’s requirements,resulting in unilateral de-ploymentofrelationship-specificinvestments.

Incontrasttoindirectsupplierdevelopment,themanufacturer playsamoreactiveroleinthecaseofdirectsupplierdevelopment. Directsupplierdevelopmentmightincludeactivitiessuchas train-inggiventosuppliers’personnelbymanufacturingfirm represen-tatives, furnishing temporary on-site support to enhance further interaction,providingequipmentandtools,orevendedicating cap-italresourcestosuppliers(Monczka et al., 1993; Wagner & Krause, 2009 ). Thus, direct supplier development presents a more col-laborative approach based on frequentmanufacturer-supplier ex-changes, resultingin bilateral deployment ofrelationship-specific investments.

Empirical research generallysupports the notionthat supplier development plays a critical role in driving performance and/or capabilities improvement on the part of the supplier and con-tributesstrategicallytostrengthenthemanufacturer’s competitive-ness. However, Krause and Ellram (1997) note that manufactur-ers’ success in supplier development varies and that those who aremoresatisfied withtheoutcomeof supplierdevelopment ac-tivities appear to communicate more effectively with and invest moretimeandresourcesinsuppliersthandoless-satisfied compa-nies.As indicatedby Krause, Handfield, and Scannell (1998) , sup-pliersare unlikely to embrace fullya setof changes requiredfor improvementunlessthereis tangibleevidencethat the manufac-turing firm will support the supplier’s efforts with matched re-sources.Thus,successfulsupplierdevelopmentapparentlyrequires bilateral deployment of resources, not only inputs fromthe sup-plying firm (Krause, 1999 ). Similar results are found by Krause et al. (20 0 0) ,Wen-li, Humphreys, Chan, and Kumaraswamy (2003) , Humphreys, Li, and Chan (2004) ,Krause et al. (2007) ,Humphreys et al. (2011) andWagner (2011) ,whoall statethat directsupport by a manufacturing firm is of major significance in determining supplier performance and/or capabilities improvement, thus en-hancingthemanufacturer’scompetitiveness.

Althoughdirectinvolvementby themanufacturing firmseems to be an important antecedent of successful supplier develop-ment,mountinganecdotal evidenceindicatesthat themajorityof manufacturersare generallyvery hesitanttocommitconsiderable resources to external, independent suppliers. As Monczka et al. (1993) determine, manufacturers are reluctant to conduct direct supplier development activities when they fear that competitors maybenefitfromthesupplier’scapabilityimprovements. Further-more,Krause (1997) reportsthat relationship-specificinvestments in suppliers’ operations are rarely used compared with indirect supplierdevelopmentactivities.Inlinewiththis,Krause and Scan- nell (2002) state that manufacturers’ commitment appears to be non-existentwhenaneedfordirectinvestmentsarisesinthe con-textofsupplierdevelopment.SimilarresultsarefoundbyWagner (2006a) ,Carr and Kaynak (2007) ,andWagner and Krause (2009) .

Therealsohavebeensomeanalytical,formalmodeling-oriented studiesof supplier development. Usingrough set theory,Bai and Sarkis (2010) introduceaformalmodeltoinvestigaterelationships betweenorganizational attributes, firms’ involvement in supplier development, and performance outcomes. Using interpretative structural modeling, Govindan, Kannan, and Haq (2010) present a framework to analyze interactions among several critical suc-cess factors of supplier development. Furthermore, Talluri et al. (2010) presenta set ofoptimization models proposed for assist-ingmanufacturersinmakingoptimalresourceallocationdecisions amongdifferentsupplierswhileminimizingthelevelofrisk.Based onaprofit-maximizingframework,Friedl and Wagner (2012) study amanufacturer’sdecisionregardingwhethertodevelopadeficient supplieror switch to an alternative source. Routroy and Pradhan (2013) propose a fuzzyanalytic hierarchyprocess to evaluatethe effectofcriticalsuccessfactorsontheperformanceofsupplier de-velopment.Focusingongreensupplierdevelopment,Dou, Zhu, and Sarkis (2014) introduce a gray analytical network process-based modelto identify supplier development activities that effectively improvesuppliers’ performance. Bai and Sarkis (2014) model co-operativeandnon-cooperativesupplier developmentsituations as a game-theoretic analytical evaluation and explore the effect of returns to scale on investment strategies. Recently, Bai, Dhavale, and Sarkis (2016) introduceanovelintegrationofroughsettheory andfuzzyclusteringmeanstoprovideamethodologyfordecision modelinginthecontextofgreensupplierdevelopment.

Bothempiricalevidenceandanalyticalinvestigationsagreethat manufacturer’sdirectinvolvementiscriticaltothesupplier’s par-ticipation in supplier development. However, development of a theoretical understanding of the effect of bilateral relationship-specificinvestments ontheperformance ofsupplier development andtheapplicationofformaldecision-makingmodelsproposedfor assistingsupplychainpartnersinbalancingsuchinvestmentsinan appropriatemannerhavereceivedlimitedattentioninthesupplier developmentliterature.

In thispaper,we considera manufacturer’sproblemof incen-tivizingsupplierstoparticipateinsupplierdevelopmentprograms. We specifically focus on direct supplier development, i.e., bilat-eral relationship-specific investments, as incentive instrument to achieve supply chain coordination. Thus, our research also con-tributestothestreamofliteratureinoperationsresearch that ex-aminesthecoordinationofsuppliers’cost-reduction effortsin de-centralizedsupplychains.

Severalaspectsofcostreductionhavebeenstudiedinthe con-text of decentralized decisionstructures (Li & Wang, 2007 ). Kim (20 0 0) considersasupplychaininwhichthemanufacturer subsi-dizessupplier’sinnovationthatcaneventually leadtosupplycost reduction and thereby increasing channel profit in a continuous timesetting.Gilbert and Cvsa (2003) studytheeffectofprice com-mitmentto encouragedownstream investments incost-reduction initiatives under the assumption of demand uncertainty. Consid-eringan assemblysystem witha singlemanufacturerand multi-plesuppliers,Bernstein and Kök (2009) examinethe dynamicsof suppliers’investmentsincost reductionthroughprocess improve-ment efforts over the life cycle of a product. Kogan and Tapiero (2009) present an inter-temporal model of co-investment in the supply chain infrastructure and show that supply chain perfor-mancedeterioratesifthefirmsdonotcooperate.Lee, Palekar, and Qualls (2011) studycoordination problemsand corresponding in-centive mechanisms between a retailer and a manufacturer for jointlyinvesting ininformationtechnologiesthat havethe poten-tialtoimprovesupplychainefficiency.Iida (2012) investigatestwo different types of agreements, namely effort sharing agreements andeffortcompensationagreements, toachieve supply chain co-ordinationandadvancecooperativecost-reductionactivities.Using single-periodoligopolyandCournotduopolymodels,Li, Wang, Yin,

Kull, and Choi (2012) examine the impact ofjoint cost-reduction efforts ontheequilibriumoutcome.Kim and Netessine (2013) in-vestigate howincentives to collaborateare impactedby informa-tion asymmetry and contracting strategies, considering a setting whereboththesupplier andthe manufacturerexertcollaborative effortstoreducetheunitcostofacriticalcomponentduring prod-uct development. Recently, Bernstein, Kök, and Meca (2015) in-vestigatethebenefitsandchallengesofknowledgesharing activi-tiesinadecentralizedassemblynetworkinwhichsuppliersinvest inprocessimprovementinitiatives toreduce thefixedproduction costs.

In this context, our papercomplements research that consid-ers incentive instruments to induce desirable supplier behavior, e.g., pricing mechanism, contract design, and subsidies for cost-reduction initiatives. However, unique features of our study, e.g., thespecificsupplier developmentcontextandtheintroductionof anegotiation-basedcoordinationalgorithmthatassiststhe manu-facturingfirmingraduallyincreasingtheshareofinvestmentcosts toensureanefficientlevelofsubsidy,differentiatethispaperfrom theexistingliterature.

3. Theoreticalbackground

Sources of economic rents and competitive advantage have received considerable attention in the strategic management lit-erature. Whereas the industry structure view (e.g., Porter, 1980 ) suggeststhateconomicrentsareprimarilyafunctionofthe struc-tural characteristics of an industry, the resource-based view of a firm (e.g., Barney, 1991; Wernerfelt, 1984 ) argues that economic rents are fundamentally due to firm heterogeneity rather than industrystructure.However,becausecriticalresourcesmayextend beyondfirmboundaries,researchershavealsoadoptedarelational approach to explain how business relationships can be a source of competitive advantage. According to the relational view as proposed by Dyer and Singh (1998) , firms who combine, share and invest in relationship-specific assets, substantial knowledge, complementary resources, and effective governance may realize relational rents that cannot be generated by either firm in iso-lation. This suggests that activities of supplier development, in whichfirmsconvertgeneral-purposeassetssuchasmoney,people skills or managerial knowledge into relationship-specific assets, obviouslyrepresent arent-generating process in accordancewith therelationalview(Krause et al., 2007; Sánchez-Rodríguez, 2009 ). However,inspiteorevenbecauseoftheirrelevancefor compa-nies, relationship-specific assets entail considerablerisk and thus are two-sided. As proposed by transaction cost economics (e.g., Williamson, 1979 ),investmentsinspecializationaredifficultto re-deploy outside the focal relationship because specializing a re-source lowersits value foralternative uses (Crosno & Dahlstrom, 2008; Wang et al., 2013 ).Assuch,relationship-specificinvestments lockintheinvestorandenablethereceivertoopportunistically ex-ploitor expropriatethe investments’value by usingexpost bar-gainingorthreatsoftermination(Lui, Wong, & Liu, 2009; Rokkan, Heide, & Wathne, 2003 ). Therefore, the investing firm sees high levels ofunilateralrelationship-specific investmentsasvulnerable toopportunisticexpropriation,particularina dynamicand uncer-tainbusinessenvironment(Hawkins, Wittmann, & Beyerlein, 2008; Sambasivan, Siew-Phaik, Mohamed, & Leong, 2013 ).

Thus,inthecaseofunilateralrelationship-specificinvestments, e.g., indirect supplier development, supplierstend to assign con-siderable resources to eradicate orat leastminimize the hazards ofopportunistic behaviorby themanufacturer.Thisin turn influ-encesthe supplier’s transactioncosts, e.g., exantecosts of draft-ing,negotiating andsafeguardingan agreementandexpostcosts ofadjustingcontractstorespondtounexpectedcontingencies, re-sulting in decreased efficiency of supplier developmentactivities

(Vázquez, Iglesias, & Bosque, 2007; Xie, Suh, & Kwon, 2010 ). Additionally,concernsaboutpartneropportunismmightalso tem-perasupplier’sincentivetocontributeresourcesto manufacturer-initiated supplier development activities in the first place, a de-cision that might lead to underinvestment and thus potentially undermine the manufacturer’s effort to improve supplier perfor-mance(Artz, 1999; Rokkan et al., 2003 ).

According to the relational view,the employment ofeffective governancemayinfluencetransactioncostsandthewillingnessof firms to engage in supplier development initiatives, a condition thatcould beanimportantsourceofcompetitiveadvantage(Dyer & Singh, 1998; Li, Humphreys, Yeung, & Cheng, 2007 ).Inthefirst case, firms achieve an advantage by incurring lower transaction costs to realizea given level of supplier developmentspecificity. In thesecond case, appropriate safeguard mechanismsencourage companiestomakehigherinvestmentsinrelationship-specific as-sets(Dyer, 1996b; Vázquez et al., 2007 ).Followingthislineof rea-soning, firms’ability toalign aconsiderablelevel of relationship-specific investments with an appropriate safeguard mechanism couldenhanceefficiencyandeffectivenessofsupplierdevelopment activities andthereby shouldbecriticaltothesuccessofsupplier development.

Although firms can select a variety of safeguard mechanisms, legalcontractsaretypicallyconsidered theprimaryformal means for safeguarding transactions. Contracts are formalized, legally binding agreements that explicitlyspecifytheobligations ofeach firm(Artz, 1999 ).Ifonefirmviolatesthetermsofthecontract,the other hastherighttogotoathirdpartytoimposecorrective ac-tion. Thus, contracts can prevent opportunistic behavior through legal force (Liu, Luo, & Liu, 2009 ). The drawback to contractual mechanismsisthat asthe transactionbecomesmorecomplex,so too must the contract protecting the exchange – and the costs of writing, monitoring andenforcing the contract increase (Dyer, 1997 ).

Another approach to managing opportunistic behavior in manufacturer-supplier relationships is to design incentive struc-tures that deter opportunistic behavior. In Telser ’s (1980) termi-nology,astrong disincentiveforpartner opportunismcanbe cre-ated bydesigning self-enforcing agreementsthat make long-term gains from the ongoing relationship exceed potential short-term payoffsfromactingopportunistically,makingtheuseoflegal con-tractsredundant.Scholarsusuallyarguethat self-enforcing agree-ments area lesscostlyandmoreeffectivemeansofsafeguarding relationship-specificinvestments(Artz, 1999; Dyer, 1996a ).Within self-enforcing agreements, contracting costs are avoided because firmsbehave inamore trustworthyfashion.Therefore,specifying everydetailoftheagreementinacontractisnotnecessary.In ad-dition,monitoringcostsare lowerbecauseself-enforcementrelies onself-monitoringratherthanexternalorthirdpartymonitoring. Finally,self-enforcing agreements lower thecosts associated with complex adaptation because firms are able to adjust the agree-mentinastraightforwardmannertorespondtounforeseenmarket changes(Dyer & Singh, 1998 ).

Several researchers suggest that firms can accomplish self-enforcing agreementsby making bilateral relationship-specific in-vestments (e.g., Anderson & Weitz, 1992; Gundlach, Achrol, & Mentzer, 1995; Jap & Anderson, 2003; Palmatier, Dant, & Grewal, 2007 ).Itisarguedthatinvestmentsmadebybothsidesofan ex-changeserveasmutualhostagesorascrediblecommitmentsthat motivatefirmstomaketherelationshipwork.Ontheonehand, bi-lateral relationship-specificinvestments strengthenthe bonds be-tween companies andcontribute to a stablerelationship because reciprocal actions are considered indications of each firm’s com-mitment totherelationship.Onthe otherhand,bilateral credible commitmentstendtodiminishthepotential threatofpartner op-portunismbecauseopportunisticbehaviorbyonepartycanbemet

byretaliationfromtheother,asituationthatcouldleadtothe for-feiture ofboth the buyer’s andthe supplier’s investments’actual value(Xie et al., 2010 ).

Therefore,ifbothmanufacturer andsupplier investinsupplier development, a self-enforcing agreement will exist that should makethe installationof an additionalgovernance mechanism re-dundant.Withfewer opportunismconcerns andlower safeguard-ing costs, supplier development becomes more efficient, more prone to joint action and includes greater expectations of conti-nuity, all ofwhich contribute to enhanced performance. In other words, direct supplier development provides an incentive to the supplier to behave in a more trustworthy fashion to maintain andcontinuetherelationshipuntilthevalueofits investmentsis recouped.

4. Basicmodel

Weconsideraparticulartwo-stagesupplychainsituationwith asupplierSandamanufacturerM,inwhichMassembles compo-nentsfromSandsellsthefinalproducttothemarket.Lettheprice distribution function p:R→R, which establishes a connection betweenthe productionquantity dand its sale pricep,be given by p

(

d

)

= a−bd where coefficientsa > 0 andb > 0denote the prohibitive price, e.g., the maximum willingnessto pay, and the priceelasticityofthecommodity,respectively.Thissituationmight becomparablewithanoligopolisticormonopolisticmarket condi-tion,inwhichafirmcanincreasemarketdemandbyloweringthe sale price. Notethat we do not distinguish marketdemand from the productionquantity of the manufacturer because themarket priceisendogenoustothequantity sold.Ignoring fixedcosts, the manufacturer’sprofitis

d·

(

p

(

d

)

−cM−cSC

)

. (1) Here, cM denotes the manufacturer’s unit production costs, whereas cSC represents the supply costs per unit charged by S. Because the manufacturer’s goal is profit maximization, the pro-duction quantity d chosen by M is determined by differentiating (1) w.r.t.dandsettingthe resultingexpressionequalto zero,i.e.,

p

(

d

)

−cM−cSC−bd

!

= 0 , (2)

whichyields the optimum productionquantity d ₌ a−cM−cSC

2b and

the optimal sale price p

(

d

)

=a+cM+cSC

2 . Since (1) is a quadratic

andconcave function, (2) is a necessary and sufficient condition forprofitmaximization.

Typically,MiscontractuallyobligedtoSforacertainperiod,or Ttimeunits.Assumingthatsupplycosts cSCareconstantoverthe contractperiod[0,T],themanufacturer’soverallprofitis

JM

0 :=T·

(

d ·

(

p

(

d

)

−cM−cSC

)

)

=

(

a−cM−cSC

)

2 4 b ·T.

Furthermore,letussupposethatthesupplycostsconsistofthe supplier’sfixed profitmargin randthesupplier’sunit production costscS,i.e., cSC =r+cS.Thus,Mcommitstopayaconstant mar-ginabovetheexpectedunitproductioncostsofS,nomatterwhat levelofdisrealized.Similarapproachestospecifythesupplycosts havebeenproposed byBernstein and Kök (2009) ,Li et al. (2012) , andKim and Netessine (2013) .Wedonotconsiderthedetailed ne-gotiationsofaparticularprofitmarginandsimplyconsiderras ex-ogenouslygiven. Moreover,thesupplierproducesthecomponents tosatisfy d;thus,Sdoesnotmakeaproductionquantitydecision. Hence,thesupplier’sprofitis

JS

0:=d ·r·T=

a−cM−

(

r+cS

)

2 b ·r·T,

andtheoverallprofitofthesupplychainis

JSC

0 :=J0M+J0S=

(

a−cM−cS

)

2−r2 4 b ·T.

WefurtherassumethatMwantstodecreasecS byestablishing supplier development projects on the supplier’s side to increase themarketshare,whichmightlead toan increasedoverall profit ofthesupplychain.Tothisend, thesustainableeffectofsupplier developmentonthesupplier’sunitproductioncostscS ismodeled bycS

(

x

)

=c0xm,wherec0>0denotesthesupplier’sunit

produc-tioncostatthebeginningofthecontractperiod,mthesupplier’s learningrate,andxmeasurestheundertakeneffort,e.g.,the cumu-lativenumberofrealizedsupplierdevelopmentprojects.Theeffort ismodeledasatime-dependentfunctionx:[0_,T]_→R≥0governed

bydynamics ˙

x

(

t

)

:= d

dtx

(

t

)

=u

(

t

)

, x

(

0

)

=x0= 1 , (3)

withu:[0,T) → [0,

ω

]toreflectthat xincreasesduringthe con-tract period. Indeed, the ordinary differential Eq. (3) is easy to solve,i.e., x

(

t

)

=x0+0tu

(

s

)

dsassuming uto be sufficiently

reg-ular such that the solution of the differential equation uniquely existsandis(atleast)absolutelycontinuous.Here,u(t)represents theeffort attime t,with a capacitylimit of

ω

> 0, e.g., the re-sourceavailabilityintermsoftime,manpowerorbudget.Because anaccurate determinationof

ω

is notcriticaltoourdiscussion,a presumptionismadethat

ω

isexogenouslyassessedtobefeasible totheproblem.Thelearningratem₌ln_ln(1−_χθ )_,

θ

_∈[0,1)and

χ

_> 1,can be interpreted asfollows.If, e.g.,parameters

θ

₌0_.05and

χ

= 2 are used,theeffort xmustbe doubled to reduce the sup-plier’sproductioncostscS

(

x

)

=c0xmperunitby5percent.Similar

modelsofcostreductionthroughlearninghavebeenproposedby Yelle (1979) , Fine and Porteus (1989) ,Kim (20 0 0) , Bernstein and Kök (2009) ,andLi et al. (2012) .

In summary, cS

(

x

(

t

))

=c0x

(

t

)

m is time varying, continuously

decreasing,strictlypositive, andconvex.It isimportantto realize that, consequently, not only the optimal quantity offered d(cSC) butalso the respectiveoptimal sale price andthe profitbecome timedependent. Moreover, addingcSDu(t), cSD ≥ 0,to theoverall profitof thesupplychainallows forintegratingthecosts of sup-plierdevelopmentintotheproposedmodel.Hence,a centraltask istounderstandhowsupplierdevelopmentcontributestothetotal profit,i.e.,whetherimprovingthecoststructurecStogenerate fur-therrevenuesoutweighs theadditionalcosts ofsupplier develop-ment.Answering thisquestionnaturallyleadstoseekingthebest solutionundertheguidingprincipleofprofitmaximization.Inthe subsequent section, an optimal control problem is formulated to rigorouslydeducethesolution.

5. Supplierdevelopmentinacentralizedsupplychain

In this section, we assume the existence of a central entity managingthe supply chainas an integrated systemin which all parameters,e.g., theoptimalamountofeffortinvested insupplier development,aresimultaneouslychosen.Wecalltheresulting so-lutionoftheproblemthecentralizedsolutionbecauseitisbasedon acentralizeddecision-makingprocess.

Employing thevariables andparameters ofthe preceding sec-tion,theprofitfunctionJSC_:L1

(

_[0,T

)

_,R

)

_{→ Rdefinedas}

JSC

(

_u

)

_:= T

0

(

a−cM−c0x

(

t

)

m

)

2−r2

4 b −cSDu

(

t

)

d t (4)

mustbe maximizedsubjectto thecontrol constraints0 ≤u(t) ≤

ω

,t_∈[0,T),andthesystemdynamics(3) ._L1

(

_[0,T

)

_,R

)

_istheset

ofmeasurablefunctions forwhich thecondition ₀T

|

u

(

t

)

|

dt<∞ holds.Mathematically speaking, the central entity must solve an optimal control problem to determine the optimal control func-tion u, i.e., the centralized optimal collaboration strategy, such thattheaccumulatedprofitJSC_{(·)ismaximized.}

Because investments cSDu(t) pay off over time due to an im-proved cost structure, the optimal control function u is struc-turallyoftheshape

u

(

t

)

:=

ω

if t<t

0 if t≥t , (5)

with t ∈ [0, T] not yet determined. Here, t = 0 corresponds to thecaseinwhichsupplierdevelopmentdoesnotincreasethe ac-cumulatedprofit,i.e., theconsideredtime horizonT (contract pe-riod)istooshortsuchthattheachievablecostreductiondoesnot outweigh the requiredcapital effort₀TcSCu

(

t

)

dt. Hence, the op-timalcontrolproblemtobesolved correspondstofindingt such that the system-wide optimum is attained. The claims based on heuristic argumentscan bededuced ina rigorousmanner by us-ingPontryagin’smaximumprinciple(see,e.g.,Kim, 20 0 0 )and not-ingthat JSC _{(·) iscontinuouswhereas parametert islimitedtoa} compact(closedandbounded)interval.

Notethat the productionquantity d(t) attime t issolely cho-sen by Mwithout considering any collaborationeffects. This as-sumptiononthedecision-makingprocessjustifiestheoptimalsale priceusedinthe abovecalculations. Albeitthephenomenon ofa so-called double marginalization maylead to a suboptimal solu-tion(Li, Li, & Cai, 2013 ),thisassumptionismadetoassessthe ef-ficiencyoftheproposedcoordinationmechanismseparately. Remark 5.1. Although JSC _{(·) is optimized w.r.t. u∈L}1

(

_[0,T

)

_,

[0_,

ω

]

)

_, i.e., itis a setof measurableand boundedfunctions, the optimalcontrolfunctionu (·)ispiecewiseconstantandbang-bang withone jumpatt.Hence,theresultingsolutioniseasily imple-mentable and corresponds to (full) cooperation interms of sup-plierdevelopmentuntiltimet.Then,theimprovedcoststructure isexploitedwithoutmakingfurtherinvestments.

Pontryagin’s maximum principle is used to solve the optimal control problem(4) .To thisend, the so-calledHamiltonian_H

(

_·

)

_, whichisdefinedas

H

(

x,u,

λ

)

:=

(

a−cM−c0xm

)

2−r2

4 b −cSDu+

λ

u,

is needed to formulate the necessary optimality conditions. This yieldsthesystemdynamics

˙

x

(

t

)

=H_λ

(

x

(

t

)

,u

(

t

)

,

λ

(

t

))

=u

(

t

)

,

theso-calledadjoint

λ

:[0,T]→R,whichischaracterizedby ˙

λ

(

t

)

=−Hx

(

x

(

t

)

,u

(

t

)

,

λ

(

t

))

=mc0x

(

t

)

m−1

(

_a−c M−c0x

(

t

)

m

)

2 b (6) andthetransversalitycondition

λ

(

T

)

=0 . (7)

Solvingtheoptimalcontrolproblemyieldsthestructural prop-erty (5) ofthe optimalcontrol function u.Then, the (absolutely continuous)statetrajectory

x

(

t

)

=

1 +

ω

t t∈ [0 ,t

)

1 +

ω

t t∈ [ t ,T] (8)

can be computed using the system dynamics (3) andthe initial conditionx

(

0

)

₌x0 =1.Inparticular,x(t)≥1holdsforallt∈[0,

T]. Hence, Eq. (6) impliesthat theadjoint

λ

(·) exhibitsa strictly negativederivative,i.e.,

λ

˙

(

t

)

<0sincethe inequalitiesm< 0and a_>cM+c0implya>cM +c0x

(

t

)

mforallt∈[0,T].Moreover,

us-ing (8) ,thesystemdynamicsoftheadjoint (6) ,andthe transver-salitycondition(7) allowustocalculatetheadjoint

λ

(

t

)

=mc0

(

1 +

ω

t

)

m−1

(

a−cM−c0

(

1 +

ω

t

)

m

)

Fig.1. Illustration of the switching condition (9).

for all t _∈[t ,T].Then, based on the fact that m _< 0 holds, the adjointEq. (6) implies

λ

(t),t ∈ [0,t],

λ

(

t

)

=

λ

(

t

)

− t t ˙

λ

(

s

)

d s =

λ

(

t

)

−mc0

(

a−cM

)

2 b t t x

(

s

)

m−1_{d s+}mc20 2 b t t x

(

s

)

2m−1_{d s} =

λ

(

t

)

−c0

(

a−cM

)((

1 +

ω

t

)

m−

(

1 +

ω

t

)

m

)

2

ω

b +c20

((

1 +

ω

t

)

2m−

(

1 +

ω

t

)

2m

)

4

ω

b .

Hence, the switching time t can be computed by solving the equation

Hu

(

x

(

t

)

,u

(

t

)

,

λ

(

t

))

=−cSD+

λ

(

t

)

= 0 andisthusgivenbythesolutionoftheequation

mc0

(

1 +

ω

t

)

m−1

(

a−cM−c0

(

1 +

ω

t

)

m

)

2 b ·

(

t −T

)

=cSD (9)

with respect to t. Consequently,the optimal control function u andtheresultingprofitJSC_{(u)aredetermined.}

Economically, the adjoint variable

λ

can be interpreted as a shadowpricethatrepresentstherateofinfinitesimalchangeofthe performance measure,i.e., the marginal revenue, withrespect to aninfinitesimalchangeofthestatevariablex(·).Hence,bymeans ofthe adjointvariable

λ

andthesupplier developmentcosts cSD, the economic efficiency (profitability) of further investments in supplierdevelopmentcanbeassessed.Asexpressedbythe switch-ing condition (9) , attime t the adjoint variable

λ

(t) equals the supplier developmentcosts cSD. Hence, cost-reduction efforts af-ter time t are not economically reasonable, see Fig. 1 for an illustration.

Assumption 5.1. Throughoutthispaperit istacitly assumedthat supplier development can increase the supply chain profit. Fur-thermore, it is supposed that full cooperation, i.e., u¯

(

t

)

₌

ω

for all t ∈ [0, T), is not the optimalsolution. Mathematically speak-ing, this implies the existence of a collaboration strategy u∈ L1

(

_[0,T

)

_,R

)

_{satisfying 0 ≤ u(t) ≤}

ω

_{, t ∈ [0, T), such that the}

inequality

JSC

(

_u

)

_{> max}

{

_JSC

0 ,JSC

(

u¯

)

}

holdsandthusensuresthat the(global)maximumisattainedfor aswitchingtimet locatedintheopeninterval(0,T).Here,JSC

0 and

JSC

(

_u¯

)

_{representt = 0(no collaboration)andt = T (full} collabo-ration),respectively.

Because an optimalsolution exists and Assumption 5.1 holds, everyoptimalsolutionmustsatisfythenecessaryoptimality

condi-tionsresultingfromPontryagin’smaximumprinciple.Furthermore, note that the solution of (9) is unique, because the adjoint and thustheleft handsideofthisequation arestrictly monotonically decreasing.Hence, takingthestructuralproperty(5) intoaccount, the switching time t ∈ (0, T) and the corresponding optimal controlfunctionu (·)areuniquelydetermined.

In conclusion, considering the profit in dependence of the switchingtimet,thededucedqualitativebehaviordirectlyimplies thatthesupplychainprofitstrictlyincreasesforswitchingtimest ∈[0,t ),whicharesmallerthantheoptimalswitchingtimet,and thenstrictlydecreasesforswitchingtimest_∈(t ,T],seeFig. 2 for anillustration.

Inshort,an elementaryproofshowingthatthenecessary con-ditionresultingfromPontryagin’smaximumprinciple isalso suf-ficientintheconsideredsettingwaspresented.Analternativeline of argumentation analogous to Chiang (1992) , which structurally fits the optimalcontrol problemto be solved and thus could be applied,wouldleadtothesameconclusion.

Thedecision-makingprocess inacentralized supplychain en-suressystemefficiencyandoptsfortheoptimumlevelofsupplier development,i.e.,maximizesthetotal(expected)profitofthe sup-plychain.Thus,thecentralizedsolution servesasabenchmarkfor thefollowinganalysis.

6. Indirectsupplierdevelopmentinadecentralizedsupply chain

Next,weconsiderthe decision-makingprocessin a decentral-izedsupplychain,whichdiffersfromthecentrallyplannedonein two fundamentalaspects. First, thereis no informationexchange duringtheplanning phase,resulting inasymmetrical distribution of information, e.g., information about the supplier’s cost struc-turemaybeunknowntothemanufacturer.Second,everydecision maker ina supply chain typically hasdifferent objectives, which mayleadtoconflictingstrategicorientations.Thepresenceofboth issuescouldcauseinefficiency inthesupplychain(Corbett, 2001; Iida, 2012 ).Consequently,thesolutionofthedecision-making pro-cessin a decentralizedsupply chaincould deviate fromthe cen-tralizedsolution.

Wefirstanalyzethesupplier’soptimaldecision-makingprocess undertheassumption ofindirectsupplier development,in which thesuppliermustbear theinvestedeffortalone.Intuitively,Swill determine the optimal supplier development level to reduce the unit production costs of the components, considering the manu-facturer’soptimalreactionintermsofprocurementquantity.Then, thesupplier’s cost-reductionefforts arerealized.Next,Mchooses the optimal production quantity based on the resulting supply price.Oursolutionapproachformalizestheabovereasoning.

Fig.2. The supply chain profit J SC_{is depicted in dependence of the switching time t .}

6.1.Indirectsupplierdevelopment

In the case of indirect supplier development the supplier’s profitJS_:L1

(

_[0,T

)

_,R

)

_{→R,whichisdefinedas}

JS

(

_u S

)

:= T 0 a−cM−

(

r+c0x

(

t

)

m

)

2 b ·r−cSDuS

(

t

)

d t, (10)

ismaximizedsubjectto thesystemdynamics(3) andthecontrol constraints0 ≤ uS(t) ≤

ω

,t ∈ [0, T),where theindexSindicates that uS represents the collaboration strategy from the supplier’s point of view.Note that also the cumulative number of realized supplierdevelopmentprojectsx(·)attime tdependsofthe cho-sen control function uS in view of the differential equation (3) . However,we droppedtheindexSinordertostreamlinethe pre-sentation.Insummary,Ssolvesanoptimalcontrolproblemto de-terminethe supplier’s optimal control function u_S such that the supplier’sprofitJS

(

_u

S

)

ismaximized.

Proceeding analogously to the centralized approach, the sup-plier’sadjointequationischaracterizedby

˙

λ

S

(

t

)

=

rmc0x

(

t

)

m−1

2 b

andthecorrespondingadjointisgivenby

λ

S

(

t

)

=

_rmc 0(1+ωtS)m−1 2b ·

(

t−T

)

t∈ [ tS,T]

λ

S

(

tS

)

− rc0 2ωb

((

1 +

ω

tS

)

m−

(

1 +

ω

t

)

m

)

t∈ [0 ,tS

)

. Here,t_S mustsatisfy

λ

S

(

tS

)

=cSD,i.e.,

rmc0

(

1 +

ω

t_S

)

m−1

2 b ·

(

tS−T

)

=cSD. (11)

Hence,thesupplier’soptimalcontrolfunctionu_S andthesupplier’s profitJS

(

_u

S

)

aredetermined.

In the indirect supplier development case, it is implicitly as-sumed that the investment decision is solely up to the supplier. Thisassumptionisjustified,becauseScoversallsupplier develop-mentcosts₀TcSDuS

(

t

)

dt,whereas Mbenefits fromthesupplier’s cost-reductioneffortsbyreducedsupplycostswithoutcommitting anyresources to supplier development. Indeed,M would opt for fullcooperation,i.e.,u¯

(

t

)

₌

ω

forallt_∈[0,T).Fromthesupplier’s pointofview,however,cost-reductioneffortsaftertimet_S donot amortizeduringthecontractperiodandthusarenoteconomically reasonable,seeFig. 3 .Hence,thecollaborationstopsaftert_S time units.

Consequently, thecorresponding manufacturer’sprofitis given by JM

(

_u S

)

:= T 0

(

a−cM−r−c0x

(

t

)

m

)

2 4 b d t. SummingupJS

(

_u S

)

+JM

(

uS

)

yieldsJSC

(

uS

)

.

6.2. Comparisonwiththecentralizedsolution

Wenextcomparethecentralizedsolution,cf.Section 5 ,withthe outcomeofthedecentralizeddecision-makingprocess inthecase of indirect supplier development. Here, it can be observed that thestructural property(5) is maintained.However, theswitching timet_S maychange.

Proposition 6.1. Let Assumption 5.1 hold. Then, the supply chain profitobtained in thecentralizeddecision-makingprocess character-izedbyt issuperiorincomparisontoitscounterpartobtainedinthe decentralizedsettingcharacterizedbyt_S.

Proof. SinceithasbeenshowninSection 5 thattheoptimal solu-tionisunique,showingthatt =t_S holdsissufficienttoprovethe assertion.UsingEqs. (9) and(11) leadsto

r·f

(

t_S

)

=

(

a−cM−c0

(

1 +

ω

t

)

m

)

·f

(

t

)

with strictly decreasing function f:[0,T]→R≥0 defined as f:=

t →mc₀(1+ωt)m−1_·(t−T)

2b .Hence, ifr<

(

a−cM−c0

(

1+

ω

t

)

m

)

holds,

the switching time for the centralized solution t will be strictly largerthanitscounterpartt_S forthedecentralizedone.

Becausethe prohibitivepricea isstrictlylarger thanthecosts per unit cM+cSC =cM +r+c0 withoutany supplier development

– anecessaryconditionforthemanufacturer– theinequalitya_> cM + cSC

(

t

)

= cM + r+ c0

(

1+

ω

t

)

m holdsforallt ∈ [0,T] andthus

inparticularfort ∈(0, T],i.e.,inthecasethat supplier develop-ment cancontribute to thesupplychain profit. Hence,the asser-tionfollows.

Inotherwords,ifthesupplierdefraysallsupplierdevelopment costs, i.e., indirect supplier development, S will choose a smaller switchingtimet_S,t_S<t ,andthuswillstopthecollaboration“too early”.

7. Directsupplierdevelopmentinadecentralizedsupplychain Next,weinvestigatethedecentralizeddecision-makingprocess undertheassumption ofdirectsupplier development.Hereto, we suppose that the manufacturer covers a certain share

α

cSD,

α

∈ (0, 1], of the supplier development costs cSD to align potentially contradictoryobjectivesandthusalleviatetheinefficiencies occur-ringintheindirectsupplierdevelopmentcase.Thisassumptionis notcompletelynew:withintheautomobileindustry,forinstance, manufacturers offer assistance by providing training, equipment andtoolsto their suppliers, orsharingthe monetary costs of in-vestments (Bernstein et al., 2015; Sako, 2004 ). Moreover, similar approachestospecifya subsidyforcost-reduction initiativeshave alsobeen proposed by Iida (2012) , Li et al. (2012) , andBernstein and Kök (2009) .

Fig.3. The supply chain profit J SC_{(solid line), the supplier’s profit J} S_{(dotted line), and the manufacturer’s profit J} M_{(dashed line) are depicted in dependence of the switching}

time t .

7.1. Directsupplierdevelopment

Incorporating thecost allocation factor

α

,the supplier’s profit function(10) ischangedto JS

(

_u S

)

:= T 0 a−cM−

(

r+c0x

(

t

)

m

)

2 b ·r−

(

1 −

α

)

cSDuS

(

t

)

d t,

whilethemanufacturer’sprofitfunctionisgivenby

JM

(

_u M

)

:= T 0

(

a−cM−r−c0x

(

t

)

m

)

2 4 b −

α

cSDuM

(

t

)

d t,

wheretheindexMindicates thatuM (·) representsthe collabora-tionstrategyfromthemanufacturer’spointofview.

In the case of direct supplier development the manufacturer supports the supplier’s cost-reduction efforts with, e.g., matched resources, and thus actively participates in the decentralized decision-makingprocess.Hence, boththesupplierandthe manu-facturer solve an optimalcontrol problemto determinetheir op-timal control functions u_S and u_M maximizing their individual profitsJS

(

_u

S

)

andJM

(

uM

)

,respectively.

Dueto theadaptationof thecostfunctionalJS_{,the righthand} side of the supplier’s switching condition(11) is multiplied with thefactor

(

1−

α

)

,i.e.,

rmc0

(

1 +

ω

tS

)

m−1

2 b ·

(

tS−T

)

=

(

1 −

α

)

cSD, (12) which yields t_S and, consequently, the supplier’s optimal control functionu_S.

Applying thesamereasoningasinthecentralizedapproach,it canbe observedthatthestructuralproperty(5) alsoholdsforM. Thus,themanufacturer’sadjointequationischaracterizedby

˙

λ

M

(

t

)

=mc0x

(

t

)

m−1·

(

a−cM−r−c0x

(

t

)

m

)

/

(

2 b

)

(13)

andthemanufacturer’sadjointisgivenby

λ

M

(

t

)

=

⎧

⎪

⎨

⎪

⎩

mc0(1+ωtM)m−1·(a−cM−r−c0(1+ωtM)m) 2b ·

(

t−T

)

for t∈[ tM,T]

λ

M

(

tM

)

− c0(a−cM−r)[(1+ωtM) m_−(1+ωt)m_] 2ωb +c2 0[(1+ωtM)2m−(1+ωt)2m] 4ωb for t∈[0 ,tM

)

. (14) Themanufacturer’soptimalswitchingtimet_Misdeterminedby theswitchingcondition

λ

M

(

tM

)

=

α

cSD,i.e.,

mc0

(

1 +

ω

t_M

)

m−1·

(

a−cM−r−c0

(

1 +

ω

t_M

)

m

)

2 b ·

(

tM−T

)

=

α

cSD,

(15)

which characterizes the manufacturer’s optimal control function u_M.

Ingeneral, t_M andu_M do notcoincide witht_S andu_S, respec-tively. However, direct supplier development is a reciprocal ap-proachthat requiresa mutuallyagreed collaborationstrategy be-tweentheparticipatingfirms.Consequently,Mcannotpursue the manufacturer’soptimalcollaborationstrategywithoutconsidering thesupplier’soptimalcollaborationstrategy, andviceversa.Here, weheavilyexploitthestructuralproperty(5) :bothSandM mono-tonically increase their respectiveprofit untilt = min

{

t_S,t_M

}

, be-cause their respective investments in supplier development pay off duringthe consideredtime interval[0,T]duetoan improved cost structure. Hence, both firms willingly collaborate until that time. However, for t>min

{

t_S,t_M

}

further cost-reduction efforts do not amortize during the contract period from the perspec-tive of at least one firm, cf. the switching conditions (12) and (15) . Because prolonging supplier development is not economi-cally reasonablefor at least one firm, the collaboration stops at t₌min

{

t_S,t_M

}

. Mathematically speaking, the (mutually agreed) collaboration strategy on supply chain level is limited by min

{

t_S,t_M

}

. In conclusion, the (mutually agreed)optimal control functionu_SC isstructurallyofshape

uSC

(

t

)

:=

ω

if t< min

{

t_S,t_M

}

0 if t≥min

{

t_S,t_M

}

,

asillustratedinFig. 4 .

Accordingly,theindividualfirms’profitsarenowdeterminedby u_SC.Summingup JS

(

_u

SC

)

+JM

(

uSC

)

yields theoverall supplychain profitJSC

(

_u

SC

)

.

Proposition7.1. LetAssumption5.1holdandlet

α

_∈[0,1]begiven. Then, if the individual firms’ switching times t_S∈ [0,T] and t_M∈ [0,T]do not coincide,the centralizeddecision-makingprocess char-acterizedbyt leadsto a (strictly)highersupplychainprofitJSC_(u) than itscounterpart JSC

(

_u

SC

)

with switching timetSC:= min

{

tS,tM

}

obtainedinthecaseof(in)directsupplierdevelopment.

Proof. For

α

=0,i.e.,indirectsupplierdevelopment,theassertion followsdirectlyfromProposition 6.1 .Hence,let

α

becontainedin theinterval (0,1]. Since ithasbeenshown inSection 5 that the optimalsolutionisunique,theequalityt =min

{

t_S,t_M

}

musthold. Withoutlossofgenerality,letusassumethatt =t_S holds;a sim-ilarargumentationprovestheclaimincaseoft = t_M.Then,using theswitchingconditions(9) and(12) yields

Fig.4. The supplier’s profit J S _{(dotted line), the manufacturer’s profit J} M_{(dashed line), and the (mutually agreed) collaboration strategy (solid line) – represented by the}

optimal control function u

SC– for α= 0 .5 using the parameter set introduced in Table1.

2 b·

α

cSD=mc0

(

1+

ω

t

)

m−1

(

t −T

)

·

(

a−cM−r−c0

(

1+

ω

t

)

m

)

.

(16) Becauset_S=t_M andt ₌min

{

t_S,t_M

}

hold,the manufacturer’s op-timal switching time will be strictly larger than its counterpart forthe centralized solution, i.e., t_M>t . Thus, the manufacturer’s switchingcondition(15) implies

2 b·

α

cSD<mc0

(

1 +

ω

t

)

m−1

(

t −T

)

·

(

a−cM−r−c0

(

1 +

ω

t

)

m

)

,

acontradictionto(16) .Hence,t =min

{

t_S,t_M

}

follows,which com-pletestheproof.

Bymeansofthefirms’switchingconditions(12) and(15) ,both thesupplier andthe manufacturer can assess the economic effi-ciency offurther investments in supplier development. Based on theseinformationand themutually agreed switchingtime t_SC:₌ min

{

t_S,t_M

}

,threecasescanbedistinguished:

(1)t_S <t_M,i.e.,

λ

S

(

tSC

)

=

(

1−

α

)

cSDand

λ

M

(

tSC

)

>

α

cSD, (2)t_S >t_M,i.e.,

λ

S

(

tSC

)

>

(

1−

α

)

cSDand

λ

M

(

tSC

)

=

α

cSD,and (3)t_S =t_M,i.e.,

λ

S

(

tSC

)

=

(

1−

α

)

cSDand

λ

M

(

tSC

)

=

α

cSD.

Since

λ

M

(

tSC

)

+

λ

S

(

tSC

)

>cSD holds inthe first two cases, fur-thercostreductioneffortspayoff duringthecontractperiodfrom thesupplychainperspective,i.e.,eitherthemanufacturerM(Case 1)orthesupplier (Case2) isinterestedinextendingthe collabo-rationby adaptingthe

α

-value.InCase3neitherfirmhasa prof-itableunilateraldeviationfromthe(mutuallyagreed)collaboration strategy.Following thislineofreasoning,we caneven provethat thesupplychainprofitobtainedinthecentralizeddecision-making process characterized by t,i.e., the centralizedsolution,coincides withits counterpartobtainedin thedecentralizedsetting charac-terizedbyt_SC foranappropriatelychosencostallocationfactor. Theorem7.1. LetAssumption5.1hold.Then, thereuniquelyexistsa costallocationfactor

α

∈(0,1)suchthatt_S andt_M,determined ac-cordinglytotheswitchingconditions(12)and(15),respectively, coin-cide,i.e.,t_S=t_M.Moreover,theresulting(mutuallyagreed)switching

timet_SC:=min

{

t_S,t_M

}

coincideswith theoptimalswitching timet ofthecentralizedsolution.

Proof. For

α

₌0_, the manufacturer’s optimal switching time t_M equalsthe finaltime Tofthecontractual periodandt_S <T holds in view of Assumption 5.1 . Moreover, t_M<T holds for all

α

∈ (0, 1] according to the manufacturer’s switching condition (15) . Inaddition, becausethe left handside of(15) is strictly decreas-ingwithrespecttot_M,themanufacturer’s optimalswitchingtime t_M=t_M

(

α

)

strictly decreasesontheintervalofadmissiblecost al-locationfactors

α

_∈[0,1].Similarly,inviewofAssumption 5.1 and accordingto the supplier’s switching condition(12) ,t_S= T holds if and only if

α

=1. Furthermore,because the left hand side of (12) is strictly decreasing in t_S, the supplier’s optimal switching timet_S= t_S

(

α

)

increasesontheintervalofadmissiblecost alloca-tionfactors

α

∈[0,1].

Let us define the (allocation) function aS: [0, 1] → [0, T] by aS

(

α

)

= tS where tS satisfies the supplier’s switching condition (12) .Analogously,aM:[0,1]→[0,T]isdefinedasaM

(

α

)

=tMwith t_Mchoseninaccordancewith(15) .Theabovereasoningshowsthat aS

(

1

)

= T andaM

(

0

)

= T holdandthataS isstrictlymonotonically increasing onits domain[0, 1]whileaM isstrictly monotonically decreasingonitsdomain[0,1].

Hence,thecontinuousfunction f:[0_,1]_→R, f

(

α

)

_→aM

(

α

)

− aS

(

α

)

,isstrictlymonotonicallydecreasingwithf(0)>0andf(1)< 0.Usingthemean valuetheoremshowstheexistence of

α

such that f

(

α

)

₌0holdsor,equivalently,aM

(

α

)

=aS

(

α

)

,i.e.,the in-dividualfirms’switchingtimescoincideforthecostallocation fac-tor

α

.

Then, using thiscost allocation factor andadding the switch-ing conditions (12) and (15) shows that the resulting switching timet_SC=t_S=t_M satisfiesthe switchingcondition(9) .Because t isuniquely determinedby(9) ,t_SC=t holds. Hence,theresulting controlfunctionsandtherespectivesupplychainprofitsalso coin-cide,whichshowstheassertion.

Table1

List of parameter values (basic scenario).

T a b c M c 0 r c SD ω m

60 200 0.01 70 100 15 10 0,0 0 0 1 −0.1

As a conclusion that can be drawn from Proposition 7.1 and Theorem 7.1 , it is desirable that the desired switching times t_S andt_Mcoincideinordertoachievethesystem-wideoptimum,i.e., t_S=t_M=t .Todemonstratetheimpactofdirectsupplier develop-mentontheindividualfirms’switchingtimes,anumericalanalysis isconductedinthesubsequentsection.

7.2. NumericalanalysisandmanagerialinsightspartI

The above reasoningshowsthat Mcan prolongthe collabora-tionandthusincreasethesupplychainprofitofthedecentralized supplychainbysubsidizingacertainshareofthesupplier develop-mentcosts.We examinenumericalexamplesusingtheparameter valuesinTable 1 toobtainmoremanagerialinsights.

Becausewe areinterestedinthedependenceoftheindividual firms’ switching times on the cost allocation factor

α

, we begin our numerical analysis by evaluating the switching conditions given by Eqs. (12) and (15) for all admissible

α

values, i.e.,

α

∈ [0, 1]. Fig. 5 shows that both t_S∗ and t_M∗ change continuously andmonotonically withrespectto

α

.If

α

= 0holds, Scoversthe supplier developmentcosts completely. Hence, Mcan exploit the achievedcostreductionsforfree,whichimpliest_M=T.Conversely, for

α

= 1,Sbenefitsfromtheincreasing quantity suppliedatthe market,whereas Mbearsall supplierdevelopmentcostsresulting int_S =T.WealsointegratedthesupplychainprofitJSC

(

_u

SC

)

.Here, weobservethatthesystem-wideoptimumisattainedfort_S=t_M. Given0<t_S<t_M= T for

α

= 0,0<t_M<t_S= T for

α

= 1andthe fact that t_S and t_M change continuously andmonotonically with respectto

α

,thesupplychainpartners’objectivescan bealigned, i.e., t_S = t_M, as shown in Theorem 7.1 . Thus, the manufacturing firm can induce the centralized solution by choosing the cost allocationfactor

α

_∈(0,1)appropriately.

To analyze the effect of direct supplier development on sup-ply chain performance moredeeply, we vary theparameters b ∈ {0.007, 0.008,0.009, 0.01,0.011, 0.012, 0.013}, r∈{12, 13,14, 15, 16,17,18},cSD ∈ {70000,80000,90000,100000,110000,120000, 130000},andm∈{−0.13,−0.12,−0.11,−0.1,−0.09,−0.08,−0.07}, resulting in a total number of 74_{=2401 instances. For each}

pa-rameter combination, we compute both the supply chain profit JSC

(

_u

SC

)

resulting fromdirect supplier development(

α

=

α

) and the corresponding profit JSC

(

_u

S

)

resulting from indirect supplier development(

α

₌0),andthencomparetherespectiveprofits.The

depicted histogram in Fig. 6 shows the absolute frequency with whicha percentage ofprofitincrease is observed within our pa-rameterset.Forallcomputedinstances,theratiosarepositive.The arithmeticmeanis19_.76percent,withastandarddeviationof5_.14 percentandamedianof19.12percent,whichshowsthatadirect involvement of the manufacturer leadsto a significant improve-mentofsupplychainperformance.

According to Kim (20 0 0) , a system-wide optimum might not necessarilybe optimalto each individual firm ina supply chain. Giventhisbackground,wealsocomputetheindividualfirms’ prof-its forthe considered parameter set andcompare the profits re-sultingfromdirectsupplierdevelopment,i.e.,JS

(

_u

SC

)

andJM

(

uSC

)

, with the corresponding profits resulting from indirect supplier development, i.e., JS

(

_u

S

)

and JM

(

uS

)

, respectively. At 6.38 per-cent,witha standard deviation of 8.18 percentand a medianof 5.96 percent, the manufacturer’s average-profit increase ratio is substantially lower than the corresponding ratio forthe supplier (34.95 percent, with a standard deviation of 5.96 percent and a medianof34.85percent).Moreover,althoughthesupplier’sratios arestrictly positive,themanufacturersuffers lossesin595out of 2401instances,cf.Fig. 7 .

Inconclusion,theresultsofournumericalanalysisaretwofold. On the one hand, supply chain coordination can be achievedby adoptingdirectsupplierdevelopment,resulting ina possibly pro-longed collaboration phase and the optimal system-wide profit corresponding to our benchmark – the centralized solution. This observation is clearly not a coincidence but is rather rigorously proved inTheorem 7.1 . Ontheother hand, theresultsshow that covering₀T

α

cSDuSC

(

t

)

dt ofthesupplierdevelopmentcostsisnot always economicallyreasonablefrom themanufacturer’s point of view. In other words, even though direct supplier development withaconstantcostallocationfactor

α

leadstoasignificant im-provement ofthe overall supply chainprofit in comparisonwith indirect supplier development (

α

= 0), i.e., the strict inequality JSC

(

_u

SC

)

>JSC

(

uS

)

holds, the manufacturer’s profit resulting from indirectsupplierdevelopment,i.e.,JM

(

_u

S

)

,mightbesuperior com-pared to its counterpart resulting from direct supplier develop-ment,i.e., JM

(

_u

SC

)

, see the depictedhistograms in Figs. 6 and7 , respectively.

Giventhefact,that themutuallyagreed collaborationstrategy mustincreasetheprofitability ofboth firms,a possibleremedyis basedonthefollowing idea:Fort∈[0,t_S

)

thesupplier investsin supplier developmentanyway (

λ

S(t) > cSD,

α

= 0) andthus does not requirefurther incentives(subsidies). This(simple) finding is thebasicideaofthenegotiation-basedcoordinationalgorithm de-velopedinthesubsequentsection. Here,themanufacturer gradu-allyincreasesthecostallocationfactor

α

inordertoensurean ef-ficientlevelofdirectsupplierdevelopment(subsidy)soasnofirm

Fig.5. Optimal switching times t

Fig.6. Comparison of indirect and direct supplier development – supply chain profit.

Fig.7. Comparison of indirect and direct supplier development – manufacturer’s profit.

hasa profitableunilateral deviationfrom theset of system-wide optimalactions.

8. Coordinatingsupplierdevelopment

In this section, we propose a negotiation-based algorithm that can be employed as a guideline to realize (perfect) sup-ply chain coordination while both the manufacturer and the supplier increase their respective profit in each iteration step, leading to a win–win situation. First, the sequence of events is described.Thereafter,thestepsperformedbythesupplierandthe manufacturerineachiterationarepresentedinmoredetail.

Starting from

α

= 0,both SandM solvetheir optimalcontrol problemtodeterminet_S andt_M,respectively.Notethat for

α

=0,

0≤t_S<t_M=T isensuredinviewofAssumption 5.1 .Hence,both firmscollaborate until time t_S=min

{

t_S,t_M

}

_, i.e., they realize the outcomeofthe decentralizedoptimizationcorrespondingto indi-rectsupplierdevelopment,seeSection 6 .Next,based onthe sen-sitivity information available from the adjoint and its derivative, Mdeterminesan adapted(increased)

α

-value.Thisnewlysetcost allocationfactor holdsfor allsupplier developmentcosts fromt_S onwards,i.e., M offers to increase the amount of effortinvested insupplierdevelopmentfromt_S onwardstoextendthe collabora-tion.Then,thenegotiationprocessstartsagain,resultingina pro-longedcollaborationphaseandthusincreasingoverallprofit.With thisoverviewinmind,wenowpresentthecoordinationprocessin detail.

8.1. Anegotiation-basedalgorithm

The proposed algorithm consists of the following six steps, which are repeated until a stopping criterion is satisfied. Each iterationstepcausessome additionalexpenses(negotiationcosts), denoted by

Ξ

>0. Moreover, the following (technical)condition is neededin orderto ensure proper functioningof the proposed algorithm.

Assumption 8.1. Let t¯_∈[0_,t

)

be the optimal switching time resulting from indirect supplier development, see Section 6 . Moreover, letthe optimalswitching timeof thecentralized solu-tionandthe correspondingmanufacturer’s adjointbe denoted by t and

λ

_M,respectively.Then,supposethat

λ

_Misstrictlyconvexon [t¯,t

)

.

Remark 8.1. The manufacturer’s adjoint

λ

_M is twice (infinitely many times) continuously differentiable on [0,t_M

)

. Hence, Assumption 8.1 is equivalent to strict positivityof

λ

_M on [t¯,t

)

,

i.e., ¨

λ

M

(

t

)

=− mc0

ω

2 b

((

1 −m

)(

a−cM−r

)(

1 +

ω

t

)

−m_−c 0

(

1 −2 m

))

×

(

1 +

ω

t

)

2m−2> 0 .

This conditionis satisfied if, and only if,

(

1₋m

)(

a₋cM −r

)(

1+

ω

t¯

)

−m_>c

0

(

1−2m

)

holds. To this end, checking the inequality

(

1−m

)(

a−cM −r

)

>c0

(

1−2m

)

suffices.

The algorithm is constructed such that, ignoring negotiation costs, i.e.,

Ξ

=0, the manufacturer’s and the supplier’s profit