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1

A new geometric approach to multiobjective linear programming problems

Kaci Mustapha, Radjef Sonia

Mathematica Applicanda

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2023

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Vol. 51, no. 1

3--12

EN

This paper is a follow-up to a previous work where we developed a new geometric approach to sensitivity analysis. In this paper, we present a simple method to determine whether a given multiobjective linear programming problem (MOLPP) has an ideal solution (i.e. all of the objective functions are optimized simultaneously) without having to calculate the optimal value of each objective function. First, we divide the space of linear forms into a finite number of sets based on a fixed convex polygonal subset of R2 using an equivalency relationship. All the elements from a given equivalency class have the same optimal solution. Next, we characterize the equivalence classes of the quotient set using a geometric approach to sensitivity analysis. Finally, a numerical example is given to illustrate the method.

PL

W tym artykule przedstawiamy nową metodę rozwiązywania problemów programowania liniowego z wieloma celami (MOLPP), która eliminuje potrzebę obliczania optymalnej wartości każdej funkcji celu. Metoda ta jest kontynuacją naszych wcześniejszych prac dotyczących analizy wrażliwości, gdzie opracowaliśmy nowe podejście geometryczne. Pierwszym krokiem naszego podejścia jest podział przestrzeni form liniowych na skończoną liczbę zbiorów opartych na stałym wypukłym podzbiorze wielokąta R2. Dokonujemy tego za pomocą relacji równoważności, która zapewnia, że wszystkie elementy z danej klasy równoważności mają takie same rozwiązanie optymalne. Następnie charakteryzujemy klasy równoważności zbioru ilorazowego za pomocą podejścia geometrycznego do analizy wrażliwości. Ten krok jest kluczowy w identyfikacji rozwiązania idealnego dla MOLPP. Korzystając z tego podejścia, możemy określić, czy dana MOLPP ma rozwiązanie idealne, bez konieczności obliczania optymalnej wartości każdej funkcji celu. Jest to znacząca poprawa w stosunku do istniejących metod, ponieważ znacznie zmniejsza złożoność obliczeniową i czas wymagany do rozwiązania MOLPP. Aby zilustrować naszą metodę, przedstawiamy numeryczny przykład, który dowodzi jej skuteczności. Nasza metoda jest prosta, ale potężna i może być łatwo zastosowana do szerokiego zakresu MOLPP. Niniejsza praca przyczynia się do dziedziny optymalizacji poprzez przedstawienie nowego podejścia do rozwiązywania MOLPP, które jest wydajne, skuteczne i łatwe do zaimplementowania.

2

Decision maker's preferences modeling for multiple objective stochastic linear programming problems

Bellahcene Fatima

Operations Research and Decisions

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2019

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Vol. 29, No. 3

5--16

EN

A method has been suggested which solves a multiobjective stochastic linear programming problem with normal multivariate distributions in accordance with the minimum-risk criterion. The approach to the problem uses the concept of satisfaction functions for the explicit integration of the preferences of the decision-maker for different achievement level of each objective. Thereafter, a nonlinear deterministic equivalent problem is formulated and solved by the bisection method. Numerical examples with two and three objectives are given for illustration. The solutions obtained by this method are compared with the solutions given by other approaches.

3

Nondifferentiable (Phi,rho)-type I and generalized (Phi,rho)-type I functions in nonsmooth vector optimization

Antczak T.

Journal of Applied Analysis

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2013

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Vol. 19, nr 2

247--270

EN

In this paper, new classes of nondifferentiable generalized invex functions are introduced. Further, nonsmooth vector optimization problems with functions belonging to the introduced classes of (generalized) (Phi,rho)-type I functions are considered. Sufficient optimality conditions and duality results for such classes of nonsmooth vector optimization problems are established. It turns out that the presented results are proved also for such nonconvex vector optimization problems in which not all functions constituting them possess the fundamental property of invexity.

4

Second order duality in multiobjective programming

Ahmad I., Husain Z.

Journal of Applied Analysis

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2008

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Vol. 14, nr 1

131-148

EN

A nonlinear multiobjective programming problem is considered. Weak, strong and strict converse duality theorems are established under generalized second order (F, α, ρ, d)-convexity for second order Mangasarian type and general Mond-Weir type vector duals.

5

The notion of V-r-invexity in diferentiable multiobjective programing

Antczak T.

Journal of Applied Analysis

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2005

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Vol. 11, nr 1

63-79

EN

In this paper, a generalization of convexity, V-r-invexity, is considared in the case of nonlinear multiobective programming problems whwer the functions involved are differentiale. The assumptions on Pareto solutions are relaxed by means of V-r-invex functions. Also some duality results are obtained for such optimization problems.

6

A logarithmic barrier function method for solving nonlinear multiobjective programming problems

Tlas M., Abdul Ghani B.

Control and Cybernetics

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2005

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Vol. 34, no 2

487-504

EN

An interior point method for solving nonlinear multiobjective programming problems, over a convex set contained in the real space R^n, has been developed in this paper. In this method a new strictly concave logarithmic barrier function has been suggested in order to transform the orginal problem into a sequence of unconstrained subproblems. These subproblems can be solved using Newton method for determining Newton's directions along which line searches are performed. It also has been proved that the number of iterations required by the suggested algorithm to converge to an [epsilon]-optimal solution is 0(m|ln[epsilon]|), depending on predetermined error tolerance [epsilon] and the number of constraints m.

7

Robust goal programming

Kuchta D.

Control and Cybernetics

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2004

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Vol. 33, no 3

501-510

EN

In the paper a new approach to goal programming is presented: the robust approach, applied so far to a single-objective linear programming. It is a "pessimistic" approach, meant to find a solution which will be reasonably good even in a bad case, but it is based on the assumption that almost never everything goes bad - the decision maker can control and simulate the pessimistic aspect of the decision situation. The pessimism refers here to uncertain coefficients in the goal functions. It is assumed that in each case only a certain number of them can take on unfavourable values - but we do not know which ones. A robust solution, i.e. the one which will be good even in the most pessimistic case among those considered to be possible - is determined, using only the linear programming methods.

8

Second-order necessary conditions of the Kuhn-Tucker type in multiobjective programming problems

Aghezzaf B.

Control and Cybernetics

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1999

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Vol. 28, no 2

213-224

EN

In this paper, we are concerned with a multiobjective programming problem with inequality constraints. We develop second-order necessary condition of the Kuhn-Tucker type for effciency and prove that the condition holds under a. constraint qualification. Moreover, we give some conditions which ensure that the constraint qua,lifica.tion holds.