چکیده:
In managerial and economic applications, there appear problems in which the goal is to simultaneously optimize several criteria functions (CFs). However, since the CFs are in conflict with each other in such cases, there is not a feasible point available at which all CFs could be optimized simultaneously. Thus, in such cases, a set of points, referred to as 'non-dominate' points (NDPs), will be encountered that are ineffective in relation to each other. In order to find such NDPs, many methods including the scalarization techniques have been proposed, each with their advantages and disadvantages. A comprehensive approach with scalarization perspective is the PS method of Pascoletti and Serafini. The PS method uses the two parameters of as the starting point and as the direction of motion to find the NDPs on the 'non-dominate' frontier (NDF). In bi-objective cases, the point is selected on a special line, and changing point on this line leads to finding all the NDPs. Generalization of this approach is very difficult to three- or more-criteria optimization problems because any closed pointed cone in a three- or more-dimensional space is not like a two-dimensional space of a polygonal cone. Moreover, even for multifaceted cones, the method cannot be generalized, and inevitably weaker constraints must be used in the assumptions of the method. In order to overcome such problems of the PS method, instead of a hyperplane (two-dimensional line), a hypersphere is applied in the current paper, and the parameter is changed over its boundary. The generalization of the new method for more than two criteria problems is simply carried out, and the examples, provided along with their comparisons with methods such as mNBI and NC, ensure the efficiency of the method. A case study in the realm of health care management (HCM) including two conflicting CFs with special constraints is also presented as an exemplar application of the proposed method.
خلاصه ماشینی:
The Quasi-Normal Direction (QND) Method: An Efficient Method for Finding the Pareto Frontier in Multi-Objective Optimization Problems Armin Ghane Kanafi Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran (Received: November 5, 2018; Revised: November 13, 2018; Accepted: April 7, 2019) Abstract In managerial and economic applications, there appear problems in which the goal is to simultaneously optimize several criteria functions (CFs).
In bi-objective cases, the point is selected on a special line, and changing point on this line leads to finding all the NDPs. Generalization of this approach is very difficult to three- or more-criteria optimization problems because any closed pointed cone in a three- or more-dimensional space is not like a two-dimensional space of a polygonal cone.
One problem is the generalization of the method to solve three- or more-objective optimization problems, and the other is that this method does not provide any solution for finding proper NDPs. The current paper examines the first problem and uses a hypersphere instead of a hyperplane to overcome it.
As an iterative method, QND generates a set of points which are considered as approximations of the NDF,YN in which in each iteration the NDP is denoted by YA which presents an estimation of the real NDF, .
The convergence to the NDF and also the distribution of solutions of the QND, mNBI, and NC methods for finding 256, NDP after 103109965, 121624414 and 87449325 TFE for the current problem are illustrated in Figures.