箱约束两两比较方案评价双准则问题的代数解法
Algebraic solution to box-constrained bi-criteria problem of rating alternatives through pairwise comparisons
摘要：
我们考虑一个决策问题，以评估的绝对评级的替代品，根据两个标准进行比较，受框约束的评级。该问题被表述为两个成对比较矩阵的log-Chebyshev逼近由一个共同的一致矩阵（单位秩的对称互反矩阵），以最小化两个矩阵的逼近误差同时。我们重新安排的近似问题作为一个有约束的双目标最优化问题找到一个向量，确定的近似一致矩阵，然后表示的热带代数方面的问题。我们应用热带优化的方法和结果，推导出的约束问题的解析解。该解决方案包括在引入两个新的变量，描述的值的目标函数，并允许减少问题的解决方案的一个系统的参数化不等式构造的未知向量，其中新的变量起参数的作用。我们利用系统的解决方案的存在条件，以获得这些值的参数，属于帕累托前固有的问题。然后，我们解决了未知向量的系统，并采取对应于帕累托前沿的所有解决方案，作为双目标问题的完整解决方案。我们将所得结果应用于所考虑的双准则判定问题，并给出了说明性的例子。
英文摘要：
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.
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