Electricity cost has become a critical concern of data center operations with the rapid increasing of information processing demand. Data center microgrid (DCMG) is a promising way to reduce electric energy consumption from traditional fossil fuel generators and the billing cost, by effectively utilizing local renewable energy, e.g., wind power. However, uncertainties of wind power generation and real-time workload of data center would have significant impacts on the operational efficiency of DCMG, especially when it is in the island mode. For this reason, a novel affinely adjustable policy based robust multi-objective optimization model under flexible uncertainty set is proposed in this paper, which simultaneously optimizes wind power curtailment, the operation cost, and the over-plus level of computation resource, while considering uncertainties of both the wind power and real-time workload. Through numerical simulation studies, the validity of robust multi-objective optimization model for the island operation of DCMG is verified. Besides, the effectiveness of the proposed methods, i.e., the novel affinely adjustable policy and the flexible uncertainty set, in handling uncertainties are evaluated. Compared to the conventional robust multi-objective optimization model, the proposed approach reduces the operating costs of about 10% in average while maintaining similar reliability in numerical simulations. Moreover, the complex quantitative relationship among these multiple objectives is further investigated. Simulation results indicate the minimization of wind power curtailment and over-plus level of computation resource increases about 25% of the operation cost. These quantitative relationships can well support the decision making of DCMG operation management.