Another approach to multiobjective programming problems withF-convex functions

In this paper, optimality conditions for multiobjective programming problems havingF-convex objective and constraint functions are considered. An equivalent multiobjective programming problem is constructed by a modification of the objective function. Furthermore, anF—Lagrange function is introduced for a constructed multiobjective programming problem, and a new type of saddle point is introduced. Some results for the new type of a saddle point are given.     

A new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems

Most real-life decision-making activities require more than one objective to be considered. Therefore, several studies have been presented in the literature that use multiple objectives in decision models. In a mathematical programming context, the majority of these studies deal with two objective functions known as bicriteria optimization, while few of them consider more than two objective functions. In this study, a new algorithm is proposed to generate all nondominated solutions for multiobjective discrete optimization problems with any number of objective functions. In this algorithm, the search is managed over (p − 1)-dimensional rectangles where p represents the number of objectives in the problem and for each rectangle two-stage optimization problems are solved. The algorithm is motivated by the well-known ε-constraint scalarization and its contribution lies in the way rectangles are defined and tracked. The algorithm is compared with former studies on multiobjective knapsack and multiobjective assignment problem instances. The method is highly competitive in terms of solution time and the number of optimization models solved.     

完全分层多目标规划的基线算法

本文采用基线算法求解完全分层多目标规划问题。给出了简单完全分层多目标规划基线算法的求解步骤，并对其进行了修正，从而得到完全分层多目标规划的宽容基线算法。并给出了两个计算实例。     

Characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method

In this paper, a new approximation method is introduced to characterize a so-called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with (F,ρ)-convex functions of order 2. In this method, an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point. In order to prove the equivalence between the original multiobjective programming problem and its associated F-approximated vector optimization problem, the suitable (F,ρ)-convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem.     

Multiobjective fractional programming problems involving (p,r)?ρ?(η,θ)-invex function

In this paper we move forward in the study of multiobjective fractional programming problem and established sufficient optimality conditions under the assumption of (p,r)????(??,??)-invexity. Weak, strong and strict converse duality theorems are also derived for three type of dual models related to multiobjective fractional programming problem involving aforesaid invex function.     

Multiobjective linear programming with context-dependent preferences

Multiobjective linear programming algorithms are typically based on value maximization. However, there is a growing body of experimental evidence showing that decision maker behavior is inconsistent with value maximization. Tversky and Simonson provide an alternative model for problems with a discrete set of choices. Their model, called the componential context model, has been shown to capture observed decision maker behavior. In this paper, an interactive multiobjective linear programming algorithm is developed which follows the rationale of Tversky and Simonson. The algorithm is illustrated with an example solved using standard linear programming software. Finally, an interactive decision support system based on this algorithm is developed to field test the usefulness of the algorithm. Results show that this algorithm compares favorably with an established algorithm in the field.     

无约束多目标规划的信赖域方法

本文将信赖域方法应用于多目标规划,提出了一类解多目标问题的新算法,并证明了全局收敛性。     

A modified objective function method for solving nonlinear multiobjective fractional programming problems

A new method is used for solving nonlinear multiobjective fractional programming problems having V-invex objective and constraint functions with respect to the same function η. In this approach, an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original nonlinear fractional programming problem with several ratios. Finally, a Mond-Weir type dual is associated, and weak, strong and converse duality results are established by using the introduced method with a modified function. To obtain these duality results between the original multiobjective fractional programming problem and its original Mond-Weir duals, a modified Mond-Weir vector dual problem with a modified objective function is constructed.     

Optimality conditions and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a (weakly) LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval-objective function are convex.     

Multiobjective programming problems with (G,C,ρ)-convexity

In this paper, we deal with multiobjective programming problems involving functions which are not necessarily differential. A new concept of generalized convexity, which is called (G,C,??)-convexity, is introduced. We establish not only sufficient but also necessary optimality conditions for multiobjective programming problems from a viewpoint of the new generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for general Mond-Weir type dual program.     

Multiobjective programming under d-invexity

In this paper, a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable. By considering the concept of Pareto optimal solution and substituting d-invexity for convexity, the Fritz John type and Karush–Kuhn–Tucker type necessary optimality conditions and duality in the sense of Mond–Weir and Wolfe for nondifferentiable multiobjective programming are given.     

Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity

In the paper, we consider a class of nonsmooth multiobjective programming problems in which involved functions are locally Lipschitz. A new concept of invexity for locally Lipschitz vector-valued functions is introduced, called V-r-invexity. The generalized Karush–Kuhn–Tuker necessary and sufficient optimality conditions are established and duality theorems are derived for nonsmooth multiobjective programming problems involving V-r-invex functions (with respect to the same function η).     

Hierarchical generating method for large-scale multiobjective systems

This paper investigates large-scale multiobjective systems in the context of a general hierarchical generating method which considers the problem of how to find the set of all noninferior solutions by decomposition and coordination. A new, unified framework of the hierarchical generating method is developed by integrating the envelope analysis approach and the duality theory that is used in multiobjective programming. In this scheme, the vector-valued Lagrangian and the duality theorem provide the basis of a decomposition of the overall multiobjective system into several multiobjective subsystems, and the envelope analysis gives an efficient approach to deal with the coordination at a high level. The following decomposition-coordination schemes for different problems are developed: (i) a spatial decomposition and envelope coordination algorithm for large-scale multiobjective static systems; (ii) a temporal decomposition and envelope coordination algorithm for multiobjective dynamic systems; and (iii) a three-level structure algorithm for large-scale multiobjective dynamic systems.This work was supported by NSF Grant No. CEE-82-11606.     

An Inner Approximation Method for Optimization over the Weakly Efficient Set

In this paper, we consider an optimization problem which aims to minimize a convex function over the weakly efficient set of a multiobjective programming problem. To solve such a problem, we propose an inner approximation algorithm, in which two kinds of convex subproblems are solved successively. These convex subproblems are fairly easy to solve and therefore the proposed algorithm is practically useful. The algorithm always terminates after finitely many iterations by compromising the weak efficiency to a multiobjective programming problem. Moreover, for a subproblem which is solved at each iteration of the algorithm, we suggest a procedure for eliminating redundant constraints.     

Second order duality for nondifferentiable multiobjective programming problem involving (F, α, ρ, d)-V-type I functions

In this paper, new classes of second order (F, α, ρ, d)-V-type I functions for a nondifferentiable multiobjective programming problem are introduced. Furthermore, second order Mangasarian type and general Mond-Weir type duals problems are formulated for a nondifferentiable multiobjective programming problem. Weak strong and strict converse duality theorems are studied in both cases assuming the involved functions to be second order (F, α, ρ, d)-V-type I.     

A procedure for selection of a multiobjective technique with application to water and mineral resources

The problem of selecting the appropriate multiobjective solution technique to solve an arbitrary multiobjective decision problem is considered. Various classification schemes of available techniques are discussed, leading to the development of a set of 28 model choice criteria and an algorithm for model choice. This algorithm divides the criteria into four groups, only one of which must be reevaluated for each decision problem encountered. The model choice problem is itself modeled as a multiobjective decision problem—strongly influenced, however, by the individual performing the analysis. The appropriate technique is selected for implementation by use of the compromise programming technique. Two example problems are presented to demonstrate the use of this algorithm. The first is concerned with ranking a predefined set of river basin planning alternatives with multiple noncommensurate ordinally ranked consequences. The second deals with coal blending and is modeled by dual objective linear programming. An appropriate multiobjective solution technique is selected for each of these two examples.     

Stability of multiobjective NLP problems with fuzzy parameters in the objectives and constraints functions

This paper deals with the stability of multiobjective nonlinear programming problems with fuzzy parameters in the objectives and constraints functions. These fuzzy parameters are characterized by fuzzy numbers. The existing results concerning the qualitative analysis of the notions (solvability set, stability sets of the first kind and of the second kind) in parametric nonlinear programming problems are reformulated to study the stability of multiobjective nonlinear programming problems under the concept of α-pareto optimality. An algorithm for obtaining any subset of the parametric space which has the same corresponding α-pareto optimal solution is also presented. An illustrative example is given to clarify the obtained results.     

An interactive dynamic approach based on hybrid swarm optimization for solving multiobjective programming problem with fuzzy parameters

Real engineering design problems are generally characterized by the presence of many often conflicting and incommensurable objectives. Naturally, these objectives involve many parameters whose possible values may be assigned by the experts. The aim of this paper is to introduce a hybrid approach combining three optimization techniques, dynamic programming (DP), genetic algorithms and particle swarm optimization (PSO). Our approach integrates the merits of both DP and artificial optimization techniques and it has two characteristic features. Firstly, the proposed algorithm converts fuzzy multiobjective optimization problem to a sequence of a crisp nonlinear programming problems. Secondly, the proposed algorithm uses H-SOA for solving nonlinear programming problem. In which, any complex problem under certain structure can be solved and there is no need for the existence of some properties rather than traditional methods that need some features of the problem such as differentiability and continuity. Finally, with different degree of α we get different α-Pareto optimal solution of the problem. A numerical example is given to illustrate the results developed in this paper.     

On nadir points of multiobjective integer programming problems

In this study, we consider the nadir points of multiobjective integer programming problems. We introduce new properties that restrict the possible locations of the nondominated points necessary for computing the nadir points. Based on these properties, we reduce the search space and propose an exact algorithm for finding the nadir point of multiobjective integer programming problems. We present an illustrative example on a three objective knapsack problem. We conduct computational experiments and compare the performances of two recent algorithms and the proposed algorithm.     

求群体多目标决策问题的一种可行方向交互算法

对于群体多目标决策问题,决策者可以各自的关于目标之间的权衡比表达其偏爱信息并进行决策.当个体权衡比具有加性性质时可得群体权衡比.本文以此构造一种求解群体非线性规划问题的交互算法.迭代中基于求解决非线性规划的Topkis-Veinott方法构造可行方向.在一定的条件下,算法收敛于所讨论问题的群体满意解.