2. Methodological Foundations

In this chapter, we set the groundwork for the understanding and application of the methodological tools that are utilized for the supply chain network models with quality competition in this book. We first overview the basics of variational inequality theory and the connections with optimization. We provide conditions for existence and uniqueness of solutions, along with the definitions of the essential properties. We relate the variational inequality problem to game theory since game theory models are developed throughout this book in order to formulate competition among supply chain network decision-makers. In addition, we recall the fundamentals of projected dynamical systems theory and the relationships with variational inequality theory in order to enable the description of dynamic interactions among decision-makers in supply chains. For completeness, we also provide results on stability analysis. We discuss some fundamentals of multicriteria decision-making since supply chain decision-makers may be faced with multiple criteria, even conflicting ones, that they wish to optimize. Finally, we present algorithms that are used for solving the supply chain network models with quality competition.