Assigning orders to suppliers with linear piecewise concave costs

Purpose: Once a set of suppliers has been determined, according to criteria of quality, price and reliability, among others, there remains the problem of assigning orders to the selected suppliers, in order to cover the needs at the lowest cost. We consider the case in which the needs of a component for a set of plants should be covered by suppliers with linear piecewise concave cost functions, a lower bound on the order size for the non-zero deliveries and a capacity constraint. The purpose is to design procedures for solving this problem.Design/methodology/approach: With the aim of providing practical tools to solve the problem of assigning orders to suppliers with linear piecewise concave costs, two mixed integer linear programs are proposed.Findings: The two MILP models are compared through an extensive computational experiment. This shows that both models, with a slight advantage for one of them, can be solved within a very short time, even when the dimensions of the instance largely exceed those that can occur in real cases.Originality/value: The paper proposes novel models that can be used to solve the problem to optimality in reasonable times and with standard optimization software.

The Price of Mediation

Discrete Algorithms International audience We study the relationship between correlated equilibria and Nash equilibria. In contrast to previous work focusing on the possible benefits of a benevolent mediator, we define and bound the Price of Mediation (PoM): the ratio of the social cost (or utility) of the worst correlated equilibrium to the social cost (or utility) of the worst Nash. We observe that in practice, the heuristics used for mediation are frequently non-optimal, and from an economic perspective mediators may be inept or self-interested. Recent results on computation of equilibria also motivate our work. We consider the Price of Mediation for general games with small numbers of players and pure strategies. For two player, two strategy games we give tight bounds in the non-negative cost model and the non-negative utility model. For larger games (either more players, or more pure strategies per player, or both) we show that the PoM can be arbitrary. We also have many results on symmetric congestion games (also known as load balancing games). We show that for general convex cost functions, the PoM can grow exponentially in the number of players. We prove that the PoM is one for linear costs and at most a small constant (but can be larger than one) for concave costs. For polynomial cost functions, we prove bounds on the PoM which are exponential in the degree.