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Theo phương pháp này, chúng tôi sẵn sàng chấp nhận một giải pháp không tối ưu cho các giá trị danh nghĩa của dữ liệu nhằm đảm bảo rằng giải pháp vẫn khả thi và gần tối ưu khi dữ liệu thay đổi. Một mối quan tâm với phương pháp như vậy là nó có thể quá bảo thủ. Trong bài báo này, chúng tôi đề xuất một phương pháp cố gắng làm cho sự trao đổi này trở nên hấp dẫn hơn; nghĩa là, chúng tôi nghiên cứu các cách để giảm cái chúng tôi gọi là giá của tính kiên cường. Cụ thể, chúng tôi điều chỉnh một cách linh hoạt mức độ bảo thủ của các giải pháp kiên cường theo các giới hạn xác suất của những vi phạm ràng buộc. Một khía cạnh hấp dẫn của phương pháp của chúng tôi là phương pháp định hình kiên cường mới cũng là một vấn đề tối ưu hóa tuyến tính. Do đó, chúng tôi tự nhiên mở rộng các phương pháp của mình đến các vấn đề tối ưu hóa rời rạc một cách dễ xử lý. Chúng tôi báo cáo các kết quả số cho một bài toán tối ưu hóa danh mục đầu tư, một vấn đề ba lô, và một vấn đề từ thư viện Net Lib. \u003C\u002Fjats:p>","\u003Cjats:p> A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. 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We introduce and study the price of fairness, which is the relative system efficiency loss under a “fair” allocation assuming that a fully efficient allocation is one that maximizes the sum of player utilities. We focus on two well-accepted, axiomatically justified notions of fairness, viz., proportional fairness and max-min fairness. 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It is an intuitive framework in which to formulate problems as perceived by decision makers and to incorporate the knowledge of experts. At the same time, it is a precise description of information that can be stored and manipulated by a computer. We develop an algorithm that can evaluate any well-formed influence diagram and determine the optimal policy for its decisions. Since the diagram can be analyzed directly, there is no need to construct other representations such as a decision tree. As a result, the analysis can be performed using the decision maker's perspective on the problem. Questions of sensitivity and the value of information are natural and easily posed. 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Customers waiting to be served may abandon the queue, and there is a cost penalty associated with such abandonments. Service rates, abandonment rates, and abandonment penalties are generally different for the different classes. The problem studied is that of dynamically scheduling the various classes. We consider the Halfin-Whitt heavy traffic regime, where the total arrival rate and the number of servers both become large in such a way that the system's traffic intensity parameter approaches one. An approximating diffusion control problem is described and justified as a purely formal (that is, nonrigorous) heavy traffic limit. The Hamilton-Jacobi-Bellman equation associated with the limiting diffusion control problem is shown to have a smooth (classical) solution, and optimal controls are shown to have an extremal or “bang-bang” character. Several useful qualitative insights are derived from the mathematical analysis, including a “square-root rule” for sizing large systems and a sharp contrast between system behavior in the Halfin-Whitt regime versus that observed in the “conventional” heavy traffic regime. 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