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Selforganizology, 2025, 12(1-2): 1-20
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Using the analytic hierarchy process in an interactive interior point algorithm for mathematical multiple-objective nonlinear programming problems

M. Tlas
Scientific Services Department, Atomic Energy Commission, P. O. Box 6091, Damascus, Syria

Received 30 June 2024;Accepted 26 July 2024;Published online 25 August 2024;Published 1 June 2025
IAEES

Abstract
An interactive interior point method for solving multiple-objective nonlinear programming problems has been proposed. The method uses a single-objective nonlinear variant based on both logarithmic barrier function and Newton's method in order to generate, at each iterate, interior search directions. New feasible points are found along these directions which will be later used for deriving best-approximation to the gradient of the implicitly-known utility function at the current iterate. Using this approximate gradient, a single feasible interior direction for the implicitly-utility function could be found by solving a set of linear equations. It may be easily taken an interior step from the current iterate to the next one along this feasible direction. During the execution of the algorithm, a sequence of interior points will be generated. It has been proved that this sequence converges to an ε-optimal solution, where ε is a predetermined error tolerance known a priori. A numerical multiobjective example is illustrated using this algorithm.

Keywords multiobjective mathematical programming;multi-criteria optimization;interactive methods;interior point methods;barrier function;Newton's method;analytic hierarchy process.

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