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Towards contract based coordination of distributed product development processes with complete substitution

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Abstract

In distributed product development processes system integrators collaborate with suppliers to provide marketable products. We analyze contract structures in their ability to coordinate such processes. A converging supply chain with two suppliers and one system integrator is considered. Each supplier develops one component and faces uncertainty with regard to development results. For the system integrator the specifications of both components are substitutive in terms of the specification of the final product (e.g., recycling quotas). Depending on the resulting specification of the final product, the system integrator generates revenues under a maximum price clause. We apply a game-theoretic framework: the system integrator is the Stackelberg leader and the suppliers are Stackelberg followers. Assuming uniformly distributed development results, we analyze two typical development contracts: a wholesale price contract and a penalty contract. We present numerical illustrations of centralized and decentralized solutions to gain qualitative insights about the optimal decisions and the coordination ability of the two contracts. Accordingly, both contracts coordinate the supply chain only under forced compliance.

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References

  • Andersen FM, Larsen HV, Skovgaard M (2008) Projection of end-of-life vehicles: Development of a projection model and estimates of ELVs for 2005–2030. In: ETC/RWM working paper 2008/2

  • Anupindi R, Bassok Y (1999) Supply contracts with quantity commitments and stochastic demand. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain management. Kluwer Academic Publishers, Norwell, pp 197–232

    Chapter  Google Scholar 

  • Arshinder K, Kanda A, Deshmukh SG (2011) A review on supply chain coordination: coordination mechanisms, managing uncertainty and research directions. In: Choi TM, Cheng TCE (eds) Supply chain coordination under uncertainty. Springer, Heidelberg, pp 39–82

    Chapter  Google Scholar 

  • Habrich-Böcker C (2011) Neue Rolle, neues Rollenverständnis. Automobil Produktion 03(2011):26–27

    Google Scholar 

  • Bertsekas DP, Nedić A, Ozdaglar AE (2003) Convex analysis and optimization. Athena Scientific, Belmont

    Google Scholar 

  • Bolton P, Dewatripont M (2005) Contract theory. The MIT Press, Cambridge

    Google Scholar 

  • Brousseau E, Glachant JM (2002) The economics of contracts: theories and applications. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Cachon GP (2003) Supply chain coordination with contracts. In: de Kok T, Graves S (eds) Handbooks in operations research and management science. Supply chain management. North-Holland, Amsterdam, pp 229–339

    Google Scholar 

  • Fergusson M (2007) End of Life Vehicles (ELV) Directive: an assessment of the current state of implementation by Member States. Study requested by the European Parliament’s Committee on the Environment, Public Health and Food Safety (IP/A/ENVI/FWC/2006-172/Lot 1/C1/SC2)

  • GHK (2006) A study to examine the benefits of the End of Life VehiclesDirective and the costs and benefits of a revision of the 2015 targets for recycling, re-use and recovery under the ELV Directive. Final Report to DG Environment

  • Güler MG, Bilgiç T (2009) On coordinating an assembly system under random yield and random demand. Eur J Oper Res 196:342–350

    Article  Google Scholar 

  • Gurnani H, Gerchak Y (2007) Coordination in decentralized assembly systems with uncertain component yields. Eur J Oper Res 176:1559–1576

    Article  Google Scholar 

  • Handfield RB, Ragatz GL, Petersen KJ, Monczka RM (1999) Involving suppliers in new product development. Calif Manag Rev 42(1):59–82

    Article  Google Scholar 

  • He Y, Zhang J (2008) Random yield risk sharing in a two-level supply chain. Int J Prod Econ 112:769–781

    Article  Google Scholar 

  • He Y, Zhang J (2010) Random yield supply chain with a yield dependent secondary market. Eur J Oper Res 206:221–230

    Article  Google Scholar 

  • Inderfurth K, Clemens J (2014) Supply chain coordination by risk sharing contracts under random production yield and deterministic demand. OR Spectrum 36(2):525–556

    Article  Google Scholar 

  • Kamath RR, Liker JK (1994) A second look at Japanese product development. Harvard Business Rev 72(6):154–170

    Google Scholar 

  • Klein M, Sayama H, Faratin P, Bar-Yam Y (2006) The dynamics of collaborative design: Insights from complex systems and negotiation research. In: Braha D, Minai AA, Bar-Yam Y (eds) Complex engineered systems. Springer, Berlin, pp 158–174

    Chapter  Google Scholar 

  • Kruse J, Volling T, Thomsen C, Ernst R, Spengler TS (2005) Introducing flexible quantity contracts into distributed soc and embedded system design processes. In: Proceedings of the conference on Design, Automation and Test in Europe (DATE’05) (2), pp 938–943

  • Laffont JJ, Martimort D (2002) The theory of incentives: the principal-agent model. Princeton University Press, Princeton

    Google Scholar 

  • Lucas R (2001) End-of-life vehicle regulation in Germany and Europe: Problems and perspectives. Wuppertal Papers No. 113

  • Milner JM, Pinker EJ (2001) Contingent labor contracting under demand and supply uncertainty. Manag Sci 47(8):1046–1062

    Article  Google Scholar 

  • Petersen KJ, Handfield RB, Ragatz GL (2003) A model of supplier integration into new product development. J Prod Innov Manag 20(4):284–299

    Article  Google Scholar 

  • Petersen KJ, Handfield RB, Ragatz GL (2005) Supplier integration into new product development coordinating product: process and supply chain design. J Oper Manag 23(3):371–388

    Article  Google Scholar 

  • Ragatz GJ, Handfield RB, Scannell TV (1997) Success factors for integrating suppliers into new product development. J Prod Innov Manag 14(3):190–202

    Article  Google Scholar 

  • Rox J, Schmidt K, Winter A, Spengler TS, Rolf E (2010) Estimating and mitigating design risk in a flexible distributed design process. IEEE Embed Syst Lett 2(2):35–38

    Article  Google Scholar 

  • Schmidt K, Volling T, Spengler TS (2011) Coordination of design-for-recycling activities in decentralized product design processes in the automotive industry. In: Hesselbach J, Herrmann C (eds) Glocalized solutions for sustainability in manufacturing. Springer, Berlin, pp 73–78

    Chapter  Google Scholar 

  • Tsay AA, Nahmias S, Agrawal N (1999) Modeling supply chain contracts: a review. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain management. Kluwer Academic Publishers, Norwell, pp 299–336

    Chapter  Google Scholar 

  • Wagner SM (2003) Intensity and managerial scope of supplier integration. J Supply Chain Manag 39(4):4–15

    Article  Google Scholar 

  • Wasti SN, Liker JK (1997) Risky business or competitive power? Supplier involvement in Japanese product design. J Prod Innov Manag 14(5):337–355

    Article  Google Scholar 

  • Wolters P (2002) Forward sourcing—Entwicklungsbegleitende Lieferantenauswahl. In: Hahn D, Kaufmann L (eds) Handbuch IndustriellesBeschaffungsmanagement. Gabler, Wiesbaden, pp 337–348

    Google Scholar 

  • Xu H (2010) Managing production and procurement through option contracts in supply chains with random yield. Int J Prod Econ 126:306–313

    Article  Google Scholar 

  • Yan X, Zhang M, Liu K (2010) A note on coordination in decentralized assembly systems with uncertaincomponent yields. Eur J Oper Res 205:469–478

    Article  Google Scholar 

Sources of Law

  • AltfahrzeugG: Gesetz über die Entsorgung von Altfahrzeugen. Bundesgesetzblatt Jahrgang 2002 Teil I Nr. 41, ausgegeben zu Bonn am 28. Juni 2002

  • AltfahrzeugV: Verordnung über die Überlassung, Rücknahme und umweltverträgliche Entsorgung von Altfahrzeugen. Altfahrzeug-Verordnung in der Fassung der Bekanntmachung vom 21. Juni 2002 (BGBl. I S. 2214), die zuletzt durch Artikel 17 des Gesetzes vom 31. Juli 2009 (BGBl. I S. 2585) geändert worden ist

  • Directive 2000/53/EC of the European Parliament and of the Council of 18 September 2000 on end-of life vehicles

  • Directive 2005/64/EC of the European Parliament and of the Council of 26 October 2005 on the type-approval of motor vehicles with regard to their reusability, recyclability and recoverability and amending Council Directive 70/156/EEC

Industry Standards

  • DIN ISO 22628 (2002) Straßenfahrzeuge: Recyclingfähigkeit und Verwertbarkeit, Berechnungsmethode. Beuth Verlag GmbH, Berlin

  • FAR (2001) Federal Aquisition Regulations. General Services Administration and Department of Defense and National Aeronautics and Space Administration (1)

  • ISO/TR 14062 (2003) Umweltmanagement: Integration von Umweltaspekten in Produktdesign und -entwicklung. DIN-Fachbericht ISO/TR 14062:2003

  • VDI-Richtlinie 2243 (2002) Recyclingorientierte Produktentwicklung. Verein Deutscher Ingenieure, Beuth Verlag GmbH, Düsseldorf

Download references

Acknowledgments

We would like to thank the anonymous referees for all comments and suggestions that have improved the quality of the paper. This contribution is based on the project “SAKE – Systemanalyse und Koordination flexibler Entwicklungsprozesse für komplexe Eingebettete Systeme” (grant number SP 491/3-1) which was funded by the German Research Foundation (DFG). On behalf of all project partners the authors would like to thank for the support.

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Correspondence to Kerstin Schmidt.

Appendices

Appendix 1

See Table 1.

Appendix 2

2.1 2.1 Derivation of (4) and (7)

Let \( \alpha_{i} ,i = 1,2 \) (\( 0 \le \alpha_{i} \le 1 \)) be two independent random variables with pdf g i and cdf G i . We can then write for the pdf \( g_{{s_{i} }} \) and cdf \( G_{{s_{i} }} \) of \( \alpha_{i} \cdot s_{i} \) according to Theorem 1 (see Appendix 3):

$$ \alpha_{i} \cdot s_{i} \sim \quad \quad g_{{s_{i} }} \left( \varepsilon \right) = g_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) \cdot \frac{1}{{s_{i} }}\quad and\quad G_{{s_{i} }} \left( \varepsilon \right) = G_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) $$

By definition of the convolution it follows for the pdf \( h_{{\alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} }} \) and cdf \( H_{{\alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} }} \) of \( \alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} \):

$$ \begin{aligned} h_{{\alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} }} \left( \varepsilon \right) = & \int\limits_{0}^{\varepsilon } {g_{{s_{1} }} \left( \tau \right) \cdot g_{{s_{2} }} \left( {\varepsilon - \tau } \right) \cdot d\tau } = \int\limits_{0}^{\varepsilon } {g_{1} \left( {\frac{\tau }{{s_{1} }}} \right) \cdot \frac{1}{{s_{1} }} \cdot g_{2} \left( {\frac{\varepsilon - \tau }{{s_{2} }}} \right) \cdot \frac{1}{{s_{2} }} \cdot d\tau } \\ H_{{\alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} }} \left( \delta \right) = & \int\limits_{0}^{\delta } {h_{{\alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} }} \left( \varepsilon \right) \cdot d\varepsilon } \\ \end{aligned} $$

Under the assumptions that \( \alpha_{1} \) and \( \alpha_{2} \) are uniformly distributed in \( \left[ {0,\,\,1} \right] \) and \( s_{1} < s_{2} \), we get the following convolution of the pdf’s:

$$ h_{{\alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} }} \left( \varepsilon \right) = \left\{ {\begin{array}{*{20}c} {\frac{\varepsilon }{{s_{1} \cdot s_{2} }}} , & {0 < \varepsilon \le s_{1} } \\ {\frac{1}{{s_{2} }}} , & {s_{1} < \varepsilon \le s_{2} } \\ {\frac{{s_{1} + s_{2} - \varepsilon }}{{s_{1} \cdot s_{2} }}} , & {s_{2} < \varepsilon \le s_{1} + s_{2} } \\ \end{array} } \right. $$

As a result we have to distinguish three cases in order to reformulate the term \( E\lfloor \hbox{min} \{ \alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} ,\hat{S}\} \rfloor\) in (3): \( 0 < \hat{S} \le s_{1} \), \( 0 \le s_{1} < \hat{S} \le s_{2} \), and \( 0 \le s_{2}^{{}} < \hat{S} \le s_{1}^{{}} + s_{2}^{{}} \), depending on the range \( \hat{S} \) lies within.

2.1.1 2.1.1 Case \( 0 < \hat{S} \le s_{1}^{{}} \)

In this case the expected profit function of the supply chain can be reformulated as follows:

$$ \begin{aligned} & \varPi_{{}}^{SC} \left( {s_{1} ,s_{2} } \right) \\ & \quad = e \cdot E\left[ {\hbox{min} \left\{ {\alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} ,\hat{S}} \right\}} \right] - c_{1} \cdot s_{1} - c_{2} \cdot s_{2} \\ & \quad = e \cdot \left( {\int\limits_{0}^{{\hat{S}}} {\varepsilon \cdot \frac{\varepsilon }{{s_{1} \cdot s_{2} }}} \cdot d\varepsilon + \hat{S} \cdot \left( {\int\limits_{{\hat{S}}}^{{s_{1} }} {\frac{\varepsilon }{{s_{1} \cdot s_{2} }} \cdot d\varepsilon + \int\limits_{{s_{1} }}^{{s_{2} }} {\frac{1}{{s_{2} }}} } \cdot d\varepsilon + \int\limits_{{s_{2} }}^{{s_{1} + s_{2} }} {\frac{{s_{1} + s_{2} - \varepsilon }}{{s_{1} \cdot s_{2} }} \cdot d\varepsilon } } \right)} \right) - c_{1} \cdot s_{1} - c_{2} \cdot s_{2} \\ \end{aligned} $$

Interpretation of the term within the bracket: In the first term \( \alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} \le \hat{S} \) holds, while in the second, third and fourth term \( \alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} > \hat{S} \) holds. Due to the minimum formulation, the revenue in the latter term is constrained to \( e \cdot \hat{S} \).

$$ = e \cdot \left( {\hat{S} - \frac{{\hat{S}^{3} }}{{6 \cdot s_{1} \cdot s_{2} }}} \right) - c_{1} \cdot s_{1} - c_{2} \cdot s_{2} $$

Thus, under the condition that \( \alpha_{1} \) and \( \alpha_{2} \) are uniformly distributed in [0, 1] and \( 0 < \hat{S} \le s_{1}^{{}} \), (3) = (4) holds.

2.1.2 2.1.2 Case \( 0 \le s_{1} < \hat{S} \le s_{2}^{{}} \)

In this case the expected profit function of the supply chain can be reformulated as follows:

$$ \begin{aligned} & \varPi_{{}}^{SC} \left( {s_{1} ,s_{2} } \right) \\ & \quad = e \cdot E\left[ {\hbox{min} \left\{ {\alpha_{1} \cdot s_{1} + \alpha_{2} \cdot s_{2} ,\hat{S}} \right\}} \right] - c_{1} \cdot s_{1} - c_{2} \cdot s_{2} \\ & \quad = e \cdot \left( {\int\limits_{0}^{{s_{1} }} {\varepsilon \cdot \frac{\varepsilon }{{s_{1} \cdot s_{2} }}} \cdot d\varepsilon + \int\limits_{{s_{1} }}^{{\hat{S}}} {\varepsilon \cdot \frac{1}{{s_{2} }} \cdot d\varepsilon + } \hat{S} \cdot \left( {\int\limits_{{\hat{S}}}^{{s_{2} }} {\frac{1}{{s_{2} }}} \cdot d\varepsilon + \int\limits_{{s_{2} }}^{{s_{1} + s_{2} }} {\frac{{s_{1} + s_{2} - \varepsilon }}{{s_{1} \cdot s_{2} }} \cdot d\varepsilon } } \right)} \right) - c_{1} \cdot s_{1} - c_{2} \cdot s_{2} \\ & \quad = e \cdot \left( { - \frac{{3 \cdot \hat{S}^{2} + s_{1}^{2} - 3 \cdot \hat{S} \cdot \left( {s_{1} + 2 \cdot s_{2} } \right)}}{{6 \cdot s_{2} }}} \right) - c_{1} \cdot s_{1} - c_{2} \cdot s_{2} \\ \end{aligned} $$

Under the condition that \( \alpha_{1} \) and \( \alpha_{2} \) are uniformly distributed in [0, 1] and \( 0 \le s_{1} < \hat{S} \le s_{2}^{{}} \), (3) = (7) holds.

2.1.3 2.1.3 Case \( 0 \le s_{2}^{{}} < \hat{S} \le s_{1}^{{}} + s_{2}^{{}} \)

The third case (\( 0 \le s_{2}^{{}} < \hat{S} \le s_{1}^{{}} + s_{2}^{{}} \)) can be excluded from the analysis.

2.2 2.2 Derivation of (6)

In the first case (\( 0 < \hat{S} \le s_{1}^{{}} \)) the optimal intended component specifications \( s_{i,ud}^{SC} \) are given by (5):

$$ \begin{aligned} s_{1,ud}^{SC} = & \frac{{\sqrt e \cdot \hat{S}^{3/2} }}{{\sqrt 6 \cdot \sqrt {c_{1} } \cdot \sqrt {s_{2} } }} \\ s_{2,ud}^{SC} = & \frac{{\sqrt e \cdot \hat{S}^{3/2} }}{{\sqrt 6 \cdot \sqrt {c_{2} } \cdot \sqrt {s_{1} } }} \\ \end{aligned} $$

By inserting \( s_{2,ud}^{SC} \) into \( s_{1,ud}^{SC} \) the optimal intended component specification \( s_{1,ud}^{SC} \) can be reformulated as follows:

$$ \begin{aligned} s_{1,ud}^{SC} = & \frac{{\sqrt e \cdot \hat{S}^{3/2} }}{{\sqrt 6 \cdot \sqrt {c_{1} } \cdot \sqrt {s_{2} } }} \\ = & \frac{{\sqrt e \cdot \hat{S}^{3/2} }}{{\sqrt 6 \cdot \sqrt {c_{1} } \cdot \sqrt {\frac{{\sqrt e \cdot \hat{S}^{3/2} }}{{\sqrt 6 \cdot \sqrt {c_{2} } \cdot \sqrt {s_{1} } }}} }} \\ = & \frac{{c_{2}^{1/3} \cdot e^{1/3} \cdot \hat{S}}}{{6^{1/3} \cdot c_{1}^{2/3} }} \\ \end{aligned} $$

From \( 0 < \hat{S} \le s_{1}^{{}} \) we have:

$$ \begin{aligned} \hat{S} \le & \frac{{c_{2}^{1/3} \cdot e^{1/3} \cdot \hat{S}}}{{6^{1/3} \cdot c_{1}^{2/3} }} \\ \Leftrightarrow & \frac{{6 \cdot c_{1}^{2} }}{{c_{2} }} \le e \\ \end{aligned} $$

2.3 2.3 Derivation of (10)

In the second case \( 0 \le s_{1} < \hat{S} \le s_{2}^{{}} \) the optimal intended component specifications \( s_{i,ud}^{SC} \) are given by (8) and (9):

$$ \begin{aligned} s_{1,ud}^{SC} = & \frac{{3 \cdot \left( {e \cdot \hat{S} - 2 \cdot c_{1} \cdot s_{2} } \right)}}{2 \cdot e} \\ s_{2,ud}^{SC} = & \frac{{\sqrt e \cdot \sqrt {3 \cdot \hat{S}^{2} - 3 \cdot \hat{S} \cdot s_{1} + s_{1}^{2} } }}{{\sqrt 6 \cdot \sqrt {c_{2} } }} \\ \end{aligned} $$

2.3.1 2.3.1 Upper bound \( e < \frac{{6 \cdot c_{1}^{2} }}{{c_{2} }} \)

By inserting \( s_{2,ud}^{SC} \) into \( s_{1,ud}^{SC} \) the optimal intended component specification \( s_{1,ud}^{SC} \) can be reformulated as follows:

$$ \begin{aligned} s_{1,ud}^{SC} = & \frac{{3 \cdot \left( {e \cdot \hat{S} - 2 \cdot c_{1} \cdot s_{2} } \right)}}{2 \cdot e} \\ = & \frac{{3 \cdot \left( {e \cdot \hat{S} - 2 \cdot c_{1} \cdot \left( {\frac{{\sqrt e \cdot \sqrt {3 \cdot \hat{S}^{2} - 3 \cdot \hat{S} \cdot s_{1} + s_{1}^{2} } }}{{\sqrt 6 \cdot \sqrt {c_{2} } }}} \right)} \right)}}{2 \cdot e} \\ = & \frac{{3 \cdot \left( {3 \cdot c_{1}^{2} \cdot \hat{S} - 2 \cdot c_{2} \cdot e \cdot \hat{S} + \sqrt { - 3 \cdot c_{1}^{4} \cdot \hat{S}^{2} + 2 \cdot c_{1}^{2} \cdot c_{2} \cdot e \cdot \hat{S}^{2} } } \right)}}{{2 \cdot \left( {3 \cdot c_{1}^{2} - 2 \cdot c_{2} \cdot e} \right)}} \\ \end{aligned} $$

From \( 0 \le s_{1}^{{}} < \hat{S} \) we have:

$$ \begin{aligned} \hat{S} > & \frac{{3 \cdot \left( {3 \cdot c_{1}^{2} \cdot \hat{S} - 2 \cdot c_{2} \cdot e \cdot \hat{S} + \sqrt { - 3 \cdot c_{1}^{4} \cdot \hat{S}^{2} + 2 \cdot c_{1}^{2} \cdot c_{2} \cdot e \cdot \hat{S}^{2} } } \right)}}{{2 \cdot \left( {3 \cdot c_{1}^{2} - 2 \cdot c_{2} \cdot e} \right)}} \\ \Leftrightarrow & e < \frac{{6 \cdot c_{1}^{2} }}{{c_{2} }} \\ \end{aligned} $$

2.3.2 2.3.2 Lower bound 2 · c 2 ≤ e

In the determination of the lower bound (2 · c 2 ≤ e) s SC1,ud  = 0 is assumed.

Proof by contradiction

  1. 1.

    Case: s SC1,ud  < 0

Contradiction to the initial definition of s SC1,ud (\( 0 \le s_{1} < \hat{S} \le s_{2}^{{}} \)).

  1. 2.

    Case: s SC1,ud  > 0

For s SC1,ud  > 0 it follows:

$$ \begin{aligned} s_{1,ud}^{SC} = & \frac{{3 \cdot \left( {3 \cdot c_{1}^{2} \cdot \hat{S} - 2 \cdot c_{2} \cdot e \cdot \hat{S} + \sqrt { - 3 \cdot c_{1}^{4} \cdot \hat{S}^{2} + 2 \cdot c_{1}^{2} \cdot c_{2} \cdot e \cdot \hat{S}^{2} } } \right)}}{{2 \cdot \left( {3 \cdot c_{1}^{2} - 2 \cdot c_{2} \cdot e} \right)}} > 0 \\ \Rightarrow & e > \frac{{2 \cdot c_{1}^{2} }}{{c_{2} }} \\ \end{aligned} $$

Inserting s SC1,ud into s SC2,ud yields:

$$ \begin{aligned} s_{2,ud}^{SC} = & \frac{{\sqrt e \cdot \sqrt {3 \cdot \hat{S}^{2} - 3 \cdot \hat{S} \cdot s_{1} + s_{1}^{2} } }}{{\sqrt 6 \cdot \sqrt {c_{2} } }} \\ = & \frac{{\sqrt e \cdot \sqrt {3 \cdot \hat{S}^{2} - 3 \cdot \hat{S} \cdot \left( {\frac{{3 \cdot \left( {e \cdot \hat{S} - 2 \cdot c_{1} \cdot s_{2} } \right)}}{2 \cdot e}} \right) + \left( {\frac{{3 \cdot \left( {e \cdot \hat{S} - 2 \cdot c_{1} \cdot s_{2} } \right)}}{2 \cdot e}} \right)^{2} } }}{{\sqrt 6 \cdot \sqrt {c_{2} } }} \\ = & \frac{{e \cdot \hat{S}}}{{2 \cdot \sqrt { - 3 \cdot c_{1}^{2} + 2 \cdot c_{2} \cdot e} }} \\ \end{aligned} $$

Inserting \( e > \frac{{2 \cdot c_{1}^{2} }}{{c_{2} }} \) in s SC2,ud we have:

$$ \begin{aligned} & \frac{{\left( {\frac{{2 \cdot c_{1}^{2} }}{{c_{2} }}} \right) \cdot \hat{S}}}{{2 \cdot \sqrt { - 3 \cdot c_{1}^{2} + 2 \cdot c_{2} \cdot \left( {\frac{{2 \cdot c_{1}^{2} }}{{c_{2} }}} \right)} }} < s_{2,ud}^{SC} \\ & \quad \Rightarrow \frac{{c_{1} }}{{c_{2} }} \cdot \hat{S} < s_{2,ud}^{SC} \\ \end{aligned} $$

From the assumption s 1 < s 2 follows:

$$ c_{1} > c_{2} \quad \quad \Rightarrow \frac{{c_{1} }}{{c_{2} }} > 1\quad \quad \Rightarrow \hat{S} < s_{2,ud}^{SC} $$

Contradiction to \( \hat{S} \le s_{2}^{{}} \).

Assuming s 1 = 0 we determine the lower bound as follows:

By inserting s SC1,ud  = 0 into s SC2,ud the optimal intended component specification s SC2,ud can be reformulated as follows:

$$ \begin{aligned} s_{2,ud}^{SC} = & \frac{{\sqrt e \cdot \sqrt {3 \cdot \hat{S}^{2} - 3 \cdot \hat{S} \cdot s_{1} + s_{1}^{2} } }}{{\sqrt 6 \cdot \sqrt {c_{2} } }} \\ s_{1,ud}^{SC} = &\,0\quad \quad \Rightarrow s_{2,ud}^{SC} = \frac{{\sqrt e \cdot \hat{S}}}{{\sqrt 2 \cdot \sqrt {c_{2} } }} \\ \end{aligned} $$

From \( 0 \le s_{1} < \hat{S} \le s_{2} \) we have:

$$ \begin{aligned} \hat{S} \le & \frac{{\sqrt e \cdot \hat{S}}}{{\sqrt 2 \cdot \sqrt {c_{2} } }} \\ \Leftrightarrow & 2 \cdot c_{2} \le e \\ \end{aligned} $$

2.4 2.4 Derivation of (13)

Let α i (0 ≤ α i  ≤ 1) be an independent random variable with pdf g i and cdf G i . We can then write for the pdf \( g_{{s_{i} }} \) and cdf \( G_{{s_{i} }} \) of α i  · s i according to Theorem 1 (see Appendix 3):

$$ \alpha_{i} \cdot s_{i} \quad \sim \quad g_{{s_{i} }} \left( \varepsilon \right) = g_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) \cdot \frac{1}{{s_{i} }}\quad and\quad G_{{s_{i} }} \left( \varepsilon \right) = G_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) $$

The expected profit π WSP i of supplier S i in (12) is reformulated as follows:

$$ \begin{aligned} & \pi_{i}^{WSP} \left( {s_{i} } \right) \\ & \quad = e_{i} \cdot E\left[ {\hbox{min} \left\{ {\alpha_{i} \cdot s_{i} ,\hat{s}_{i} } \right\}} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = e_{i} \cdot \left[ {\int\limits_{0}^{{\hat{s}_{i} }} {\varepsilon \cdot g_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) \cdot \frac{1}{{s_{i} }} \cdot d\varepsilon } + \hat{s}_{i} \cdot \left( {1 - G_{i} \left( {\frac{{\hat{s}_{i} }}{{s_{i} }}} \right)} \right)} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = e_{i} \cdot \left[ {\int\limits_{0}^{{\hat{s}_{i} }} {\varepsilon \cdot g_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) \cdot \frac{1}{{s_{i} }} \cdot d\varepsilon } + \hat{s}_{i} - \hat{s}_{i} \cdot G_{i} \left( {\frac{{\hat{s}_{i} }}{{s_{i} }}} \right)} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ \end{aligned} $$

Using the following substitution

$$ \begin{aligned} & \phi \left( \varepsilon \right) = \frac{\varepsilon }{{s_{i} }} = \alpha_{i} \\ & \quad \Rightarrow \frac{{d\alpha_{i} }}{d\varepsilon } = \frac{1}{{s_{i} }}\quad \Leftrightarrow \quad d\alpha_{i} = \frac{1}{{s_{i} }}d\varepsilon \quad \Leftrightarrow \quad d\varepsilon = s_{i} \cdot d\alpha_{i} \\ \end{aligned} $$

results in these new bounds of integration:

$$ \Rightarrow \varphi \left( {\hat{s}_{i} } \right) = \frac{{\hat{s}_{i} }}{{s_{i} }}\quad \Rightarrow \varphi \left( 0 \right) = 0 $$

It follows:

$$ \begin{aligned} & \pi_{i}^{WSP} \left( {s_{i} } \right) \\ & \quad = e_{i} \cdot \left[ {\int\limits_{0}^{{\frac{{\hat{s}_{i} }}{{s_{i} }}}} {\alpha_{i} \cdot s_{i} \cdot g_{i} \left( {\alpha_{i} } \right) \cdot \frac{1}{{s_{i} }} \cdot s_{i} \cdot d\alpha_{i} } + \hat{s}_{i} - \hat{s}_{i} \cdot G_{i} \left( {\frac{{\hat{s}_{i} }}{{s_{i} }}} \right)} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = e_{i} \cdot \left[ {s_{i} \cdot \int\limits_{0}^{{\frac{{\hat{s}_{i} }}{{s_{i} }}}} {\alpha_{i} \cdot g_{i} \left( {\alpha_{i} } \right) \cdot d\alpha_{i} } + \hat{s}_{i} - \hat{s}_{i} \cdot G_{i} \left( {\frac{{\hat{s}_{i} }}{{s_{i} }}} \right)} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ \end{aligned} $$

Under the assumption that α i is uniformly distributed in [0, 1] with

$$ \begin{aligned} g_{i} \left( {\alpha_{i} } \right) = & \frac{1}{1 - 0} = 1 \\ G_{i} \left( {\frac{{\hat{s}_{i} }}{{s_{i} }}} \right) = & \frac{{\frac{{\hat{s}_{i} }}{{s_{i} }} - 0}}{1 - 0} = \frac{{\hat{s}_{i} }}{{s_{i} }} \\ \end{aligned} $$

it follows:

$$ \begin{aligned} & \pi_{i}^{WSP} \left( {s_{i} } \right) \\ & \quad = e_{i} \cdot \left[ {s_{i} \cdot \int\limits_{0}^{{\frac{{\hat{s}_{i} }}{{s_{i} }}}} {\alpha_{i} \cdot 1 \cdot d\alpha_{i} } + \hat{s}_{i} - \hat{s}_{i} \cdot \frac{{\hat{s}_{i} }}{{s_{i} }}} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = e_{i} \cdot \left[ {\frac{{\hat{s}_{i}^{2} }}{{2 \cdot s_{i} }} + \hat{s}_{i} - \hat{s}_{i} \cdot \frac{{\hat{s}_{i} }}{{s_{i} }}} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = e_{i} \cdot \left[ {\hat{s}_{i} - \frac{{\hat{s}_{i}^{2} }}{{2 \cdot s_{i} }}} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = e_{i} \cdot \hat{s}_{i} - \frac{{e_{i} \cdot \hat{s}_{i}^{2} }}{{2 \cdot s_{i} }} - \left( {c_{i} \cdot s_{i} } \right) \\ \end{aligned} $$

Thus, under the condition that α i is uniformly distributed in [0, 1], (12) = (13) holds.

2.5 2.5 Derivation of (26)

The expected profit \( \uppi_{i}^{P} \) of supplier S i in (25) is given as follows:

$$ \pi_{i}^{P} \left( {s_{i} } \right) = e_{i} \cdot E\left[ {\hbox{min} \left\{ {\alpha_{i} \cdot s_{i} ,\hat{s}_{i} } \right\}} \right] - p_{i} \cdot E\left[ {\hbox{max} \left\{ {\hat{s}_{i} - \alpha_{i} \cdot s_{i} ,0} \right\}} \right] - \left( {c_{i} \cdot s_{i} } \right) $$

The second term of the expected profit can be reformulated as follows:

$$ \begin{aligned} & p_{i} \cdot E\left[ {\hbox{max} \left\{ {\hat{s}_{i} - \alpha_{i} \cdot s_{i} ,0} \right\}} \right] \\ & \quad = p_{i} \cdot E\left[ {\hat{s}_{i} - \hbox{min} \left\{ {\alpha_{i} \cdot s_{i} ,\hat{s}_{i} } \right\}} \right] \\ & \quad = p_{i} \cdot \left( {\hat{s}_{i} - E\left[ {\hbox{min} \left\{ {\alpha_{i} \cdot s_{i} ,\hat{s}_{i} } \right\}} \right]} \right) \\ & \quad = p_{i} \cdot \hat{s}_{i} - p_{i} \cdot E\left[ {\hbox{min} \left\{ {\alpha_{i} \cdot s_{i} ,\hat{s}_{i} } \right\}} \right] \\ \end{aligned} $$

Thus, we can reformulate the expected profit as follows:

$$ \begin{aligned} \pi_{i}^{P} \left( {s_{i} } \right)& = e_{i} \cdot E\left[ {\hbox{min} \left\{ {\alpha_{i} \cdot s_{i} ,\hat{s}_{i} } \right\}} \right] - p_{i} \cdot \hat{s}_{i} + p_{i} \cdot E\left[ {\hbox{min} \left\{ {\alpha_{i} \cdot s_{i} ,\hat{s}_{i} } \right\}} \right] - \left( {c_{i} \cdot s_{i} } \right) \\ & = \left( {e_{i} + p_{i} } \right) \cdot E\left[ {\hbox{min} \left\{ {\alpha_{i} \cdot s_{i} ,\hat{s}_{i} } \right\}} \right] - p_{i} \cdot \hat{s}_{i} - \left( {c_{i} \cdot s_{i} } \right) \\ \end{aligned} $$

Due to the derivation of (13) it follows:

$$ \begin{aligned} & \pi_{i}^{P} \left( {s_{i} } \right) \\ & \quad = \left( {e_{i} + p_{i} } \right) \cdot \left[ {s_{i} \cdot \int\limits_{0}^{{\frac{{\hat{s}_{i} }}{{s_{i} }}}} {\alpha_{i} \cdot 1 \cdot d\alpha_{i} } + \hat{s}_{i} - \hat{s}_{i} \cdot \frac{{\hat{s}_{i} }}{{s_{i} }}} \right] - \left( {p_{i} \cdot \hat{s}_{i} } \right) - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = \left( {e_{i} + p_{i} } \right) \cdot \left[ {\frac{{\hat{s}_{i}^{2} }}{{2 \cdot s_{i} }} + \hat{s}_{i} - \hat{s}_{i} \cdot \frac{{\hat{s}_{i} }}{{s_{i} }}} \right] - \left( {p_{i} \cdot \hat{s}_{i} } \right) - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = \left( {e_{i} + p_{i} } \right) \cdot \left[ {\hat{s}_{i} - \frac{{\hat{s}_{i}^{2} }}{{2 \cdot s_{i} }}} \right] - \left( {p_{i} \cdot \hat{s}_{i} } \right) - \left( {c_{i} \cdot s_{i} } \right) \\ & \quad = e_{i} \cdot \hat{s}_{i} - \frac{{\left( {e_{i} + p_{i} } \right) \cdot \hat{s}_{i}^{2} }}{{2 \cdot s_{i} }} - \left( {c_{i} \cdot s_{i} } \right) \\ \end{aligned} $$

Thus, under the condition that α i is uniformly distributed in [0, 1], (25) = (26) holds.

Appendix 3

Theorem 1

Let α i i = 1, 2 (0 ≤ α i  ≤ 1) be two independent random variables with pdf g i and cdf G i . Then pdf \( g_{{s_{i} }} \) and cdf \( G_{{s_{i} }} \) of α i  · s i are given by:

$$ \begin{aligned} pdf\sim & g_{{s_{i} }} \left( \varepsilon \right) = g_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) \cdot \frac{1}{{s_{i} }} \\ cdf\sim & G_{{s_{i} }} \left( \varepsilon \right) = G_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) \\ \end{aligned} $$

Proof of Theorem 1

$$ \begin{aligned} & \alpha_{i} \sim g_{i} \,,\,\,G_{i} \\ & \alpha_{i} \cdot s_{i} \sim g_{{s_{i} }} \,,\,\,G_{{s_{i} }} \\ \end{aligned} $$

Determination of cdf:

$$ \begin{aligned} G_{{s_{i} }} \left( \varepsilon \right) & = P\left( {\alpha_{i} \cdot s_{i} \le \varepsilon } \right) \\ & = P\left( {\alpha_{i} \le \frac{\varepsilon }{{s_{i} }}} \right) \\ & = G_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) \\ \end{aligned} $$

Determination of pdf:

$$ \begin{aligned} \frac{\partial }{\partial \varepsilon }G_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) & = G_{i} '\left( {\frac{\varepsilon }{{s_{i} }}} \right) \cdot \frac{1}{{s_{i} }} \\& = g_{i} \left( {\frac{\varepsilon }{{s_{i} }}} \right) \cdot \frac{1}{{s_{i} }} \\ & = g_{{s_{i} }} \left( \varepsilon \right) \\ \end{aligned} $$

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Schmidt, K., Volling, T. & Spengler, T.S. Towards contract based coordination of distributed product development processes with complete substitution. J Bus Econ 84, 665–714 (2014). https://doi.org/10.1007/s11573-014-0727-3

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