Various multicriteria sorting methods have been proposed in the literature to assign the feasible alternatives into predefined categories. We consider here problems involving a set of totally ordered categories representing different achievement levels in the satisfaction of criteria. As in many existing methods, the assignment rule of an alternative to a category is based on the comparison of its performance vector to reference profiles defining lower bounds of the categories. Within this standard setting we address a new problem that consists in finding how to modify a given solution, within a combinatorial set of alternatives, to upgrade it in the upper category (or higher) at minimum cost. We also consider the problem of identifying the sequence of solutions that minimize the total cost while satisfying some budget constraint at every step, and the problem of determining how to modify the current solution to save money while staying in the same category. We first propose a general approach based on mixed integer (linear or quadratic) programming to solve these problems. Then, we implement this approach on various multiobjective combinatorial problems, such as multi-agent assignment problems and multiobjective knapsack problems. Numerical tests are provided to establish the feasibility of the approach on instances of different sizes.

中文翻译：

---

组合域上多准则排序方法的最小成本改进和最大增益稳定性

文献中提出了各种多标准排序方法，以将可行的替代方案分配到预定义的类别中。我们在这里考虑涉及一组完全有序的类别的问题，这些类别代表满足标准的不同成就水平。与许多现有方法一样，类别替代项的分配规则基于其性能向量与定义类别下限的参考配置文件的比较。在这个标准设置中，我们解决了一个新问题，即寻找如何在一组备选方案组合中修改给定解决方案，以最低成本将其升级到更高级别（或更高级别）。我们还考虑了在每一步都满足一些预算约束的同时，确定最小化总成本的解决方案序列的问题，以及确定如何修改当前解决方案以在保持同一类别的同时节省资金的问题。我们首先提出了一种基于混合整数（线性或二次）规划的通用方法来解决这些问题。然后，我们在各种多目标组合问题上实施这种方法，例如多智能体分配问题和多目标背包问题。提供了数值测试来确定该方法在不同大小的实例上的可行性。例如多智能体分配问题和多目标背包问题。提供了数值测试来确定该方法在不同大小的实例上的可行性。例如多智能体分配问题和多目标背包问题。提供了数值测试来确定该方法在不同大小的实例上的可行性。