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When only the moments (mean, variance or t-th moment) of the underline distribution are known, a variety of many max-min optimization models choose actions to maximize the firm’s expected profit against the most unfavorable distribution. We introduce relaxation scalars to reformulate the max-min model as a relaxed model and demonstrate that the closed form solutions (if they exist in the first place) can be quickly identified when we reduce the relaxation scalars to zero. To demonstrate the effectiveness of this new method, we provide closed-form solutions, hitherto unknown, for several distributionally robust inventory models, including the newsvendor problem with mean and t-th moment (for t > 1), the pricing model, the capacity planning model with multiple supply sources, and the two-product inventory system with common component.

Micro, small and medium-sized enterprises (MSMEs) face persistent challenges in raising capitals, and one of the practical reasons could be the high level of ambiguity in this sector. As many not-for-profit organizations or governmental agencies strengthen financial supports to MSMEs, the important issue of stimulating growth while protecting fund providers under ambiguity arises. We propose a robust optimization framework to jointly determine the firm's production planning and financing decisions in a principal-agent model with the presence of distributional ambiguity. We apply the notion of absolute robustness to derive a financing agreement that is both feasibility-robust and performance-robust. We assume that both the firm and the investor base their decisions on two fundamental descriptive statistics: the mean and the variance of the demand. The firm jointly determines the production quantity and financial agreement to maximize the worst-case expected profit, while the investor approves the financial agreement if the worst case expected return can cover the cost of capital. We show that equity financing is one of the robust optimal financing agreements. We also consider loan financing as an alternative. We derive the firm's robust optimal interest rate and production quantity in closed forms. Notably, the robust optimal interest rate depends on the demand variability and the asset recovery ratio, which comprehensively considers the value of collateral, initial capital, and production quantity.

When only the moments (mean, variance or t-th moment) of the underline distribution are known, numerous max-min optimization models can be interpreted as a zero-sum game, in which the decision maker (DM) chooses actions to maximize her expected profit while Adverse Nature chooses a distribution subject to the moments conditions to minimize DM’s expected profit. We propose a new method to efficiently solve this class of zero-sum games under moment conditions. By applying the min-max inequality, our method reformulates the zero-sum game as a robust moral hazard model, in which Adverse Nature chooses both the distribution and actions to minimize DM’s expected profit subject to incentive compatibility (IC) constraints. Under quasi-concavity, these IC constraints are replaced by the first-order conditions, which give rise to extra moment constraints. Interestingly, these extra moment constraints drastically reduce the number of corner points to be considered in the corresponding semi-infinite programming models. We show that in the equilibrium, these moment constraints are binding but have zero Lagrangian multipliers and thus facilitate closed-form solutions in several application examples with different levels of complexity. The high efficiency of the method enables us to solve a large class of zero-sum games and the corresponding max-min robust optimization models.

When only the moments (mean, variance or t-th moment) of the underline distribution are known, numerous max-min optimization models can be interpreted as a zero-sum game, in which the decision maker (DM) chooses actions to maximize her expected profit while Adverse Nature chooses a distribution subject to the moments conditions to minimize DM’s expected profit. We propose a new method to efficiently solve this class of zero-sum games under moment conditions. By applying the min-max inequality, our method reformulates the zero-sum game as a robust moral hazard model, in which Adverse Nature chooses both the distribution and actions to minimize DM’s expected profit subject to incentive compatibility (IC) constraints. Under quasi-concavity, these IC constraints are replaced by the first-order conditions, which give rise to extra moment constraints. Interestingly, these extra moment constraints drastically reduce the number of corner points to be considered in the corresponding semi-infinite programming models. We show that in the equilibrium, these moment constraints are binding but have zero Lagrangian multipliers and thus facilitate closed-form solutions in several application examples with different levels of complexity. The high efficiency of the method enables us to solve a large class of zero-sum games and the corresponding max-min robust optimization models.

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In this study, CO2 seepage of sandstone samples from the Taiyuan-Shanxi Formation coal seam roof in Ordos Basin, China, under temperature-stress coupling was studied with the aid of the TAWD-2000 coal rock mechanics-seepage test system. Furthermore, the evolution law and influencing factors on permeability for CO2 in sandstone samples with temperature and axial pressure were systematically analyzed. The results disclose that the permeability of sandstone decreases with the increase in stress. The lower the stress is, the more sensitive the permeability is to stress variation. High stress results in a decrease in permeability, and when the sample is about to fail, the permeability surges. The permeability of sandstone falls first and then rises with the rise of temperature, which is caused by the coupling among the thermal expansion of sandstone, the desorption of CO2, and the evaporation of residual water in fractures. Finally, a quadratic function mathematical model with a fitting degree of 98.2% was constructed between the temperature-stress coupling effect and the permeability for CO2 in sandstone. The model provides necessary data support for subsequent numerical calculation and practical engineering application. The experimental study on the permeability characteristics for CO2 in sandstone under high temperature and overburden pressure is crucial for evaluating the storage potential and predicting the CO2 migration evolution in underground coal gasification coupling CO2 storage projects.