Multistage stochastic programs, which involve sequences of decisions over time, are usually hard to solve in realistically sized problems. Providing bounds for optimal solution may help in evaluating whether it is worth the additional computations for the stochastic program vs. simplified approaches. In this paper we generalize measures from the two-stage case, based on different levels of available information, to the multistage stochastic programming problems. A set of theorems providing chains of inequalities among the new quantities are proved. Numerical results on a case study related to a simple transportation problem illustrate the described relationships

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Global climate change has encouraged international and regional adoption of pollution taxes and carbon emission reduction policies. Europe has taken the leadership in environmental regulations by introducing the European Union Emissions Trading System (EU-ETS) in 2005 and by promoting a set of policies destined to lower carbon emissions from energy, industrial, and transport sectors. These environmental policies have significantly affected the production choices of these European sectors.\ud \ud Considering this framework, the objective of this paper is to evaluate the effects of the application of environmental policies in a multitiered closed-loop supply chain network where raw material suppliers, manufacturers, consumers, and recovery centers operate. In particular, we assume that manufacturers are subject to the EU-ETS and a carbon tax is imposed on truck transport. In this way, the developed model captures carbon emission regulations, recycling, transportation and technological factors within a unified framework. In particular, it allows for evaluating the impacts of the considered environmental regulations on carbon emissions, product flows, and prices. The proposed model is optimized and solved by using the theory of variational inequalities. \ud \ud \ud Our analysis shows that the combined application of the EU-ETS at the manufacturers' tier and the carbon tax on truck transport implies additional costs for producers that reduce their good provisions. On the other side, this has a positive outcome for {the} environment since CO2 emissions reduce. Moreover, an increase of the efficiency level of the recycling process increments the availability of reusable raw material in the reverse supply chain. Finally, the distance between {a} couple of CLSC tiers plays a very important role. The lower is the distance covered by vehicles, the higher is the production of goods and the lower is the amount of CO2 emitted

“Market Coupling” is currently seen as the most advanced market\ud design in the restructuring of the European electricity market. Market Coupling, by construction, introduces what is generally referred to as an incomplete market: it leaves several constraints out of the market and hence avoids pricing them. This may or may not have important consequences in practice depending on the case on hand. Quasi-Variational Inequality problems and the associated Generalized Nash Equilibrium can be used for representing incomplete markets. Recent papers propose methods for finding a set of solutions of Quasi-Variational Inequality problems. We apply one of these methods to\ud a subproblem of market coupling namely the coordination of counter-trading. This problem is an illustration of a more general question encountered, for instance, in hierarchical planning in production management. We first discuss the economic interpretation of the Quasi-Variational Inequality problem. We then apply the algorithmic approach to a set of stylized case studies in order to illustrate the impact of different organizations of counter-trading. The paper\ud emphasizes the structuring of the problem. A companion paper considers the full problem of Market Coupling and counter-trading and presents a more extensive numerical analysis

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