We consider a decision-making problem to evaluate absolute ratings of alternatives that are compared in pairs according to two criteria, subject to box constraints on the ratings. The problem is formulated as the log-Chebyshev approximation of two pairwise comparison matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank), to minimize the approximation errors for both matrices simultaneously. We rearrange the approximation problem as a constrained bi-objective optimization problem of finding a vector that determines the approximating consistent matrix, and then represent the problem in terms of tropical algebra. We apply methods and results of tropical optimization to derive an analytical solution of the constrained problem. The solution consists in introducing two new variables that describe the values of the objective functions and allow reducing the problem to the solution of a system of parameterized inequalities constructed for the unknown vector, where the new variables play the role of parameters. We exploit the existence condition for solutions of the system to derive those values of the parameters that belong to the Pareto front inherent to the problem. Then, we solve the system for the unknown vector and take all solutions that correspond to the Pareto front, as a complete solution of the bi-objective problem. We apply the result obtained to the bi-criteria decision problem under consideration and present illustrative examples.

中文翻译：

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通过成对比较对评级备选方案的框约束双标准问题的代数解

我们考虑一个决策问题来评估备选方案的绝对评级，这些备选方案根据两个标准成对比较，并受到评级的框约束。该问题被公式化为两个成对比较矩阵的对数切比雪夫逼近通过一个公共一致矩阵（单位秩的对称倒数矩阵），以同时最小化两个矩阵的逼近误差。我们将近似问题重新排列为一个约束双目标优化问题，即找到一个确定近似一致矩阵的向量，然后用热带代数表示该问题。我们应用热带优化的方法和结果来推导出约束问题的解析解。该解决方案包括引入两个描述目标函数值的新变量，并允许将问题简化为为未知向量构造的参数化不等式系统的解，其中新变量扮演参数的角色。我们利用系统解的存在条件来导出属于问题固有的帕累托前沿的那些参数值。然后，我们求解未知向量的系统，并将对应于帕累托前沿的所有解作为双目标问题的完整解。我们将获得的结果应用于正在考虑的双标准决策问题，并提供说明性示例。