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Implementation of optimal schedules in outsourcing with identical suppliers

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Abstract

This paper deals with decentralized decision-making situations in which firms outsource production orders to multiple identical suppliers. Each firm aims to minimize the sum of its completion times. We study whether a central authority can install a mechanism such that strategic interaction leads to a socially optimal schedule. For the case of single demand the shortest-first mechanism implements optimal schedules in Nash equilibrium. We show that for the general case there exists no anonymous mechanism that implements optimal schedules in correlated equilibrium.

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Notes

  1. Jackson (2001) reviews some of the fundamental results in implementation theory.

  2. The price of stability was first studied in Schulzan and Stier Moses (2003). The name was coined by Anshelevich et al. (2008).

  3. We refer to Li and Wang (2007) for a general review of coordination mechanisms for supply chain systems. Implementation theory and more generally game theory have become essential tools in the analysis of supply chains with multiple agents. Cachon and Netessine (2004) survey the applications of game theory to supply chain analysis. Li and Whang (2001) survey game theory models in operations management.

  4. The price of anarchy was first studied in Dubey (1986) and Koutsoupias and Papadimitriou (2009). The name was coined by Papadimitriou (2001).

  5. Minimum mean flow time and minimum sum of completion times are equivalent objectives.

  6. Not all optimal schedules can be obtained by the SFG algorithm. For instance, let \(M=\{m_1,m_2\}\) and \(J=\{1,2,3\}\) with \(p_1<p_2<p_3\). Consider the schedule that processes jobs 1 and 2 (in this order) on \(m_1\) and job 3 on \(m_2\). The schedule is optimal (since it can be obtained by the MFT algorithm) but it cannot be obtained by the SFG algorithm (since jobs 1 and 2 are processed on the same machine).

  7. We omit the formal definition of Nash equilibrium in mixed strategies as it can be easily found in any standard text book on game theory.

References

  • Abed F, Correa JR, Huang CC (2014) Optimal coordination mechanisms for multi-job scheduling games. In: Schulz AS, Wagner D (eds) Algorithms—ESA 2014. Lecture notes in computer science, vol 8737. Springer, Berlin, Heidelberg, pp 13–24

  • Agussurja L, Lau HC (2009) The price of stability in selfish scheduling games. Web Intell Agent Syst 7(4):305–311

    Google Scholar 

  • Angel E, Bampis E, Pascual F, Thibault N (2016) Truthfulness for the sum of weighted completion times. In: Dinh T, Thai M (eds) Computing and combinatorics. COCOON 2016. Lecture notes in computer science, vol 9797. Springer, Cham, pp 15–26

  • Anshelevich E, Dasgupta A, Kleinberg J, Tardos É, Wexler T, Roughgarden T (2008) The price of stability for network design with fair cost allocation. SIAM J Comput 38(4):1602–1623

    Article  MathSciNet  MATH  Google Scholar 

  • Arrow KJ (1950) A difficulty in the concept of social welfare. J Polit Econ 58(4):328–346

    Article  Google Scholar 

  • Aumann Y, Dombb Y (2010) Pareto efficiency and approximate pareto efficiency in routing and load balancing games. In: Proceedings of the 3rd international symposium on algorithmic game theory. pp 66–77

  • Braat J, Hamers H, Klijn F, Slikker M (2016) A selfish allocation heuristic in scheduling: equilibrium and inefficiency bound analysis. Tilburg University, Working Paper

  • Bukchin Y, Hanany E (2007) Decentralization cost in scheduling: a game-theoretic approach. Manuf Serv Oper Manag 9(3):263–275

    Article  Google Scholar 

  • Bukchin Y, Hanany E (2011) Decentralization cost in supply chain jobshop scheduling with minimum flowtime objective. Tel Aviv University, Working Paper

  • Cachon GP, Netessine S (2004) Game theory in supply chain analysis. In: Simchi-Levi D, Wu SD, Shen Z-J (eds) Handbook of quantitative supply chain analysis, vol 34. Springer, New York, pp 13–65

  • Caragiannis I (2013) Efficient coordination mechanisms for unrelated machine scheduling. Algorithmica 66(3):512–540

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen J, Pascual F (2015) Scheduling tasks from selfish multi-tasks agents. In: Träff J, Hunold S, Versaci F (eds) Euro-Par 2015: Parallel Processing. Euro–Par 2015. Lecture Notes in Computer Science, vol 9233. Springer, Berlin, Heidelberg, pp 183–195

  • Cole R, Correa J, Gkatzelis V, Mirrokni V, Olver N (2015) Decentralized utilitarian mechanisms for scheduling games. Games Econ Behav 92:306–326

    Article  MathSciNet  MATH  Google Scholar 

  • Correa JR, Queyranne M (2012) Efficiency of equilibria in restricted uniform machine scheduling with total weighted completion time as social cost. Naval Res Logist 59(5):384–395

    Article  MathSciNet  MATH  Google Scholar 

  • Dubey P (1986) Inefficiency of Nash equilibria. Math Oper Res 11(1):1–8

    Article  MathSciNet  MATH  Google Scholar 

  • Epstein L, Kleiman E (2011) On the quality and complexity of pareto equilibria in the job scheduling game. In: Proceedings of the 10th international conference on autonomous agents and multiagent systems, vol 2. pp 525–532

  • Hoeksma R, Uetz M (2012) The price of anarchy for minsum related machine scheduling. In: Solis–Oba R, Persiano G (eds) 9th Workshop on approximation and online algorithms. lecture notes in computer science studies in economic theory, vol 7164. Springer, Berlin, pp 261–273

  • Horowitz E, Sahni S (1976) Exact and approximate algorithms for scheduling nonidentical processors. J ACM 23(2):317–327

    Article  MathSciNet  MATH  Google Scholar 

  • Ibarra O, Kim C (1977) Heuristic algorithms for scheduling independent tasks on nonidentical processors. J ACM 24(2):280–289

    Article  MathSciNet  MATH  Google Scholar 

  • Immorlica N, Li L, Mirrokni V, Schulz A (2009) Coordination mechanisms for selfish scheduling. Theor Comput Sci 410(17):1589–1598

    Article  MathSciNet  MATH  Google Scholar 

  • Jackson MO (2001) A crash course in implementation theory. Soc Choice Welf 18(4):655–708

    Article  MathSciNet  MATH  Google Scholar 

  • Koutsoupias E, Papadimitriou C (2009) Worst-case equilibria. Comput Sci Rev 3(2):65–69

    Article  Google Scholar 

  • Lee K, Leung JY-T, Pinedo ML (2012) Coordination mechanisms for parallel machine scheduling. Eur J Oper Res 220(2):305–313

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Wang Q (2007) Coordination mechanisms of supply chain systems. Eur J Oper Res 179(1):1–16

    Article  MATH  Google Scholar 

  • Li L, Whang S (2001) Game theory models in operations management and information systems. In: Chatterjee K, Samuelson WF (eds) Game theory and business applications. Kluwer Academic Publishers, New York, pp 95–131

    Google Scholar 

  • Maskin E (1985) The theory of implementation in nash equilibrium: a survey. In: Hurwicz L, Schmeidler D, Sonnenschein H (eds) Social goals and social organization: volume in memory of Elisha Pazner. Cambridge University Press, Cambridge, pp 173–204

  • Myerson RB, Satterthwaite MA (1983) Efficient mechanisms for bilateral trading. J Econ Theory 29(2):265–281

    Article  MathSciNet  MATH  Google Scholar 

  • Papadimitriou C (2001) Algorithms, games, and the internet. In: Proceedings of the 33rd annual acm symposium on the theory of computing, pp 749–753

  • Perakis G, Roels G (2007) The price of anarchy in supply chains: quantifying the efficiency of price-only contracts. Manag Sci 53(8):1249–1268

    Article  MATH  Google Scholar 

  • Schulzan A, Stier Moses N (2003) On the performance of user equilibria in traffic networks. In: Proceedings of the 14th annual ACM-SIAM symposium on discrete algorithms, pp 86–87

  • Smith W (1956) Various optimizers for single-stage production. Naval Res Logist Q 3(1–2):59–66

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CentER and Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands

    Herbert Hamers

  2. Institute for Economic Analysis, CSIC, Bellaterra, Spain

    Flip Klijn

  3. Barcelona GSE Center, Barcelona, Spain

    Flip Klijn

  4. CentER, Tilburg University, Tilburg, The Netherlands

    Flip Klijn

  5. Industrial Engineering and Innovation Sciences, Eindhoven University of Technology, Eindhoven, The Netherlands

    Marco Slikker

Authors
  1. Herbert Hamers
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  2. Flip Klijn
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  3. Marco Slikker
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Correspondence to Flip Klijn.

Additional information

We thank an associate editor and two anonymous reviewers for their comments and suggestions.

The first draft of the paper was written while F. Klijn was visiting CentER and the Department of Econometrics and Operations Research, Tilburg University. He gratefully acknowledges the hospitality of Tilburg University and an extramural fellowship from CentER. Financial support from from AGAUR–Generalitat de Catalunya (2014-SGR-1064 and 2017-SGR-1359), the Spanish Ministry of Economy and Competitiveness through Plan Estatal de Investigación Científica y Técnica y de Innovación 2013–2016 (ECO2014-59302-P and ECO2017-88130-P), and the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563) is also gratefully acknowledged.

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Hamers, H., Klijn, F. & Slikker, M. Implementation of optimal schedules in outsourcing with identical suppliers. Math Meth Oper Res 89, 173–187 (2019). https://doi.org/10.1007/s00186-018-0645-1

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  • DOI: https://doi.org/10.1007/s00186-018-0645-1

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