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
Jackson (2001) reviews some of the fundamental results in implementation theory.
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.
Minimum mean flow time and minimum sum of completion times are equivalent objectives.
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).
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.
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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
Keywords
- Game theory
- Outsourcing
- Scheduling
- Efficiency
- Implementation
- Nash equilibrium
- Correlated equilibrium
- Price of stability