Payoff allocation methods for several operational research games

Abstract

From the start of this century, enforcing and maintaining collaborations became one of the trends in business and management, ranging from production and inventory to transportation and network communication. Because of the economies of scale, many enterprises might collaborate to generate more value and save costs instead of working on their own. The rapid change in science and information technologies enables users to communicate and share data more efficiently through different digital platforms. These emerging technologies open several opportunities for companies and institutions to allow potential coordination and collaboration. This development has created the sharing economy which enables individuals, households, businesses, and organisations to engage in collaborative production, distribution, and consumption of goods and services. The interactions among players willing to cooperate are the underlying theme of Cooperative Game Theory. In this field, payoff (yield/cost) sharing problems aim at producing solutions with two main criteria: fairness and stability. These principles lead to some popular solution concepts such as the Shapley value and the core, which concern distributing the payoff to members of a coalition. The main contributions of this thesis are the developments of some computationally efficient methods to calculate these solution concepts in the generalised minimum spanning tree, linear production, and multidimensional integer knapsack games. These are the subclasses of operational research games where the characteristic functions of these cooperative games are described by linear or integer linear programming formulations. The minimum-cost spanning tree game is a particular class of cooperative games defined on a graph, where each player owns a vertex. Solutions of the game represent ways to distribute the total cost of a minimum-cost spanning tree among all the players. When we partition the graph into clusters, the generalised minimum spanning tree problem is to determine a minimum-cost tree including exactly one vertex from each cluster.
The first chapter introduces and studies the generalised minimum spanning tree game and some of its properties. We propose a constraint generation algorithm to calculate a stable payoff distribution and present some computational results obtained using the proposed algorithm. Another class of operational research games, called linear production game, is concerned with allocating the total payoff of an enterprise among the owners of the resources in a fair way. With cooperative game theory provides a mathematical framework for sharing the benefit of the cooperation, the Shapley value is one of the widely used solution concepts as a fair measurement in this area. Finding the exact Shapley values for linear production games is, however, challenging when the number of players exceeds 30. This chapter describes the deploy of linear programming sensitivity analysis for more efficient computation. We also propose the stratified sampling technique to estimate the Shapley value for large-scale linear production games. Computational results show the effectiveness of these compared to other existing methods. The integer optimisation game, where the characteristic function is generated by solving an integer optimisation problem, forms an important class in cooperative game theory. In this chapter, we introduce a new subclass of integer optimisation games namely the multidimensional integer knapsack game. Finding the Shapley value is challenging for most integer optimisation games with a large number of players, particularly for the multidimensional integer knapsack game due to the structure of its characteristic function. We describe an algebraic approach using Grobner bases to be able to compute the Shapley value efficiently. Some computational experiments are presented to show the potential of the proposed method compared to CPLEX, the state-of-the-art commercial solver on some randomly generated instances.

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ORCID for Tri-Dung Nguyen: orcid.org/0000-0002-4158-9099

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