Abstract
Hierarchical multilevel multi-leader multi-follower problems are non-cooperative decision problems in which multiple decision-makers of equal status in the upper-level and multiple decision-makers of equal status are involved at each of the lower-levels of the hierarchy. Much of solution methods proposed so far on the topic are either model specific which may work only for a particular sub-class of problems or are based on some strong assumptions and only for two level cases. In this paper, we have considered hierarchical multilevel multi-leader multi-follower problems in which the objective functions contain separable and non-separable terms (but the non-separable terms can be written as a factor of two functions, a function which depends on other level decision variables and a function which is common to all objectives across the same level) and shared constraint. We have proposed a solution algorithm to such problems by equivalent reformulation as a hierarchical multilevel problem involving single decision maker at all levels of the hierarchy. Then, we applied a multi-parametric algorithm to solve the resulting single leader single followers problem.
Highlights
Multilevel multi-leader multi-follower (MLMF) game is a non-cooperative decision system in which there are multiple higher-level decision-makers and many lower-level decision-makers
In the sequential part of the game Stackelberg behavior is assumed, from which the leaders make their decision first by competing in a Nash game constrained by the equilibrium conditions of another Nash game among the followers and the followers react by optimizing their objective functions conditioned on the leaders’ decision
One of the solution approaches in solving multilevel-MLMF games uses a reformulation that replaces the lower-level problems by their optimality conditions which results in a mathematical problem classified as an equilibrium problem with equilibrium constraints (EPECs)
Summary
Multilevel multi-leader multi-follower (MLMF) game is a non-cooperative decision system in which there are multiple higher-level decision-makers (who are referred to as leaders) and many lower-level decision-makers (who are referred to as followers). One of the solution approaches in solving multilevel-MLMF games uses a reformulation that replaces the lower-level problems by their optimality conditions which results in a mathematical problem classified as an equilibrium problem with equilibrium constraints (EPECs). Kassa and Kassa [14] reformulated a class of multilevel single-leader multiple-follower games, that consist of separable terms and nonseparable terms across all the followers parameterized by constant positive weights Motivated by this latest method, the authors have proposed in [25, 26] an equivalent reformulation procedure to solve multiple-leaders multiple-followers problems with any finite level of hierarchy.
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