Лексикографічна згортка багатьох критеріїв як надкритерій їх паретівської згортки

Abstract

В статтi розглядається лексикографiчна згортка багатьох критерiїв в один векторний критерiй. Ця згортка одержана на основi умови попарної рiзної важливостi критерiїв. Також розглянута вiдповiдна лексикографiчнiй згортцi критерiїв задача вiдшукання альтернативи, оптимальної в нiй, — задача лексикографiчної оптимiзацiї. В статтi доведено, що лексикографiчна згортка багатьох критерiїв є надкритерiєм паретiвської згортки критерiїв. На основi цього доведення показано, що розв’язання задачi багатокритерiального вибору за паретiвською згорткою зводиться до розв’язання задач лексикографiчної оптимiзацiї. Розглянуто також лексикографiчне лiнiйне програмування i побудована двоїста задача, як задача лiнiйного програмування з векторними змiнними i доведенi теореми двоїстостi. Описано варiант симплексного алгоритму стосовно задачi лексикографiчного лiнiйного програмування.

This article focuses on research, constructing new models, and developing methods to solve problems that involve analyzing these models. They tackle essential concerns regarding multiple criteria selection, considering both theoretical and practical aspects. Introducing a concept known as the super criterion has improved the selection process. This concept evaluates alternatives based on a shared set of options. Studies have demonstrated that any substitute that satisfies the super criterion is also the best choice based on the initial criteria for the identical range of alternatives. The choice is a significant aspect of purposeful activity. Almost every complex practical problem of choice is multi-criteria. Typically, it’s challenging to find an alternative that meets all the criteria. Combining various criteria into a single one with agreed-upon conditions by all parties involved is a practical approach to simplify the process. Various conditions lead to different criteria convolutions, resulting in distinct challenges for multiple criteria optimization. A commonly used method for evaluating alternatives is the Paretian convolution, which involves pairwise balancing all the criteria. One way to combine multiple criteria into a single vector is by assigning pairwise different levels of importance to each of them. The type of convolution used in this context is called a lexicographic convolution. It involves solving a lexicographic optimization problem to determine the best possible alternative. Note that the criterion order given by this convolution is a complete order on the set of options. The article considers the lexicographic convolution of multiple criteria into one vector criterion. Studies have demonstrated that multiple criteria combination results in a superior criterion compared to relying solely on the Paretian convolution. This proof indicates that solving the problem of multi-criteria selection through Paretian convolution can be simplified by solving lexicographic optimization issues instead. In optimization, one area of study is lexicographic linear programming, which involves creating a dual problem that uses vector variables and is demonstrated as a linear programming problem. Proofs related to duality have been presented within this framework. Furthermore, this article explains a version of the simplex algorithm used for lexicographic linear programming.