Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years. Recently Delage et al. [Shortfall Risk Models When Information of Loss Function Is Incomplete, GERAD HEC, Montréal, 2018] consider a situation where a decision maker's true loss function in the definition of SR is unknown but it is possible to elicit a set of plausible utility functions with partial information and consequently propose a robust formulation of SR based on the worst-case utility function. In this paper, we extend this new stream of research to a multi-attribute prospect space since multi-attribute decision-making problems are ubiquitous in practical applications. Specifically, we introduce a preference robust multivariate utility-based shortfall risk measure (PRMSR) and demonstrate that it is law invariant and convex. We then apply the PRMSR to an optimal decision-making problem where the objective is to minimize the PRMSR of a vector-valued cost function and propose some numerical scheme for solving the resulting optimization problem in the case when the underlying exogenous uncertainty is finitely distributed. Finally, we discuss statistical robustness of the PRMSR based optimization model by examining qualitative stability of the estimator of the optimal value obtained with potentially contaminated data. A case study is carried out to examine the performance of the proposed robust model and numerical scheme.

基于效用的短缺风险度量（SR）在过去几年中受到越来越多的关注。最近 Delage 等人。[损失函数信息不完整时的短缺风险模型, GERAD HEC, Montréal, 2018] 考虑这样一种情况，即决策者在 SR 定义中的真实损失函数未知，但可以用部分信息得出一组合理的效用函数，并因此提出基于 SR 的稳健公式最坏情况效用函数。在本文中，我们将这一新的研究流扩展到多属性前景空间，因为多属性决策问题在实际应用中无处不在。具体来说，我们引入了一种偏好稳健的基于效用的多变量短缺风险度量（PRMSR），并证明它是法律不变的和凸的。然后，我们将 PRMSR 应用于一个最优决策问题，其目标是最小化向量值成本函数的 PRMSR，并提出一些数值方案来解决在潜在外生不确定性有限分布的情况下产生的优化问题。最后，我们通过检查使用潜在污染数据获得的最优值的估计量的定性稳定性来讨论基于 PRMSR 的优化模型的统计稳健性。进行案例研究以检查所提出的稳健模型和数值方案的性能。我们通过检查使用潜在污染数据获得的最优值的估计量的定性稳定性来讨论基于 PRMSR 的优化模型的统计稳健性。进行案例研究以检查所提出的稳健模型和数值方案的性能。我们通过检查使用潜在污染数据获得的最优值的估计量的定性稳定性来讨论基于 PRMSR 的优化模型的统计稳健性。进行案例研究以检查所提出的稳健模型和数值方案的性能。