In this paper, we consider a bi-attribute decision making problem where the decision maker’s (DM’s) objective is to maximize the expected utility of outcomes with two attributes but where the true utility function which captures the DM’s risk preference is ambiguous. To tackle this ambiguity, we propose a maximin bi-attribute utility preference robust optimization (BUPRO) model where the optimal decision is based on the worst-case utility function in an ambiguity set of plausible utility functions constructed using partially available information such as the DM’s specific preference for certain lotteries. Specifically, we consider a BUPRO model with two attributes, where the DM’s risk attitude is bivariate risk-averse and the ambiguity set is defined by a linear system of inequalities represented by the Lebesgue–Stieltjes integrals of the DM’s utility functions. To solve the inner infinite-dimensional minimization problem, we propose a continuous piecewise linear approximation approach to approximate the DM’s unknown true utility. Unlike the univariate case, we partition the domain of the utility function into a set of small non-overlapping rectangles and then divide each rectangle into two triangles by either the main diagonal (Type-1) or the counter diagonal (Type-2). The inner minimization problem based on the piecewise linear utility function can be reformulated as a mixed-integer linear program and the outer maximization problem can be solved efficiently by the derivative-free method. In the case that all the small triangles are partitioned either in Type-1 or in Type-2, the inner minimization can be formulated as a finite dimensional linear program and the overall maximin as a single mixed-integer program. To quantify the approximation errors, we derive, under some mild conditions, the error bound for the difference between the BUPRO model and the approximate BUPRO model in terms of the ambiguity set, the optimal value and the optimal solutions. Finally, we carry out some numerical tests to examine the performance of the proposed models and computational schemes. The results demonstrate the efficiency of the computational schemes and highlight the stability of the BUPRO model against data perturbations.

中文翻译：

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双属性效用首选项鲁棒优化：一种连续分段线性近似方法


在本文中，我们考虑了一个双属性决策问题，其中决策者 （DM） 的目标是最大化具有两个属性的结果的预期效用，但捕获 DM 风险偏好的真正效用函数是模棱两可的。为了解决这种歧义性，我们提出了一个 maximin 双属性效用偏好稳健优化 （BUPRO） 模型，其中最佳决策基于一组模棱两可的合理效用函数中的最坏情况效用函数，该模棱两可的效用函数使用部分可用的信息（例如 DM 对某些彩票的特定偏好）构建。具体来说，我们考虑一个具有两个属性的 BUPRO 模型，其中 DM 的风险态度是双变量风险厌恶的，而歧义集由 DM 效用函数的 Lebesgue-Stieltjes 积分表示的线性不等式系统定义。为了解决内部无限维最小化问题，我们提出了一种连续的分段线性近似方法来近似 DM 的未知真实效用。与单变量情况不同，我们将效用函数的域划分为一组不重叠的小矩形，然后按主对角线 （Type-1） 或计数器对角线 （Type-2） 将每个矩形划分为两个三角形。基于分段线性效用函数的内部最小化问题可以重新表述为混合整数线性规划，外部最大化问题可以通过无导数方法高效求解。在所有小三角形都以 Type-1 或 Type-2 划分的情况下，内部最小化可以表述为有限维线性规划，将总体 maximin 表述为单个混合整数规划。 为了量化近似误差，我们推导出了在一些温和条件下，BUPRO 模型和近似 BUPRO 模型之间在模糊集、最优值和最优解方面的差值的边界误差。最后，我们进行了一些数值测试，以检验所提出的模型和计算方案的性能。结果证明了计算方案的效率，并突出了 BUPRO 模型对数据扰动的稳定性。