In this paper, approximate efficient (-efficient) solutions of multiobjective optimization problems are investigated. One of the most important methods for solving multiobjective optimization problems is to use Scalarization techniques. In these methods, a single objective optimization problem corresponding to the multiobjective problem is solved, and the relationship between optimal solutions of the single objective problem and (weakly, properly) efficient solutions of the multiobjective problem is investigated. In this paper, a combination of the modified constrained and elastic constrained Scalarization methods is considered, which will provide necessary and sufficient conditions for generating approximate (weakly, properly) efficient solutions. We compare the results with the necessary and sufficient conditions obtained from the modified constrained and the elastic constrained methods. The presented results can be applied for every multiobjective optimization problem without any convexity assumption for the objective functions. Unlike many of the previous methods, the obtained results are also consistent with multiobjective problems with unbounded criterion space.

In this paper, a modified Scalarization technique for finding Pareto optimal points of multiobjective optimization problems is provided. The proposed method is a combination of the normal constraint and elastic constraint method. First, we introduce the optimization problem of the modified method and then we present an algorithm for obtaining the set of Pareto points. Thereafter, the relationship between optimal solutions of this Scalarization problem and (weakly, properly) efficient solutions of the multiobjective optimization problems are analyzed. Indeed, some necessary conditions for (weak, proper) efficiency are presented. All the provided results are established without any convexity assumption. Furthermore, we propose a new algorithm for approximating the Pareto front of multiobjective optimization problems. We solve some test problems by applying the suggested algorithm and compare the results with some existing methods, including the epsilon-constraint method, the Pascolleti-Serafini method and the NBI method. The obtained results highlight the efficiency of our approach in comparison with examined methods.

In this paper, we focus on the utilization of the feasible value constraint technique to address multiobjective optimization problems (MOPs). It is attempted to overcome certain drawbacks associated with this method, such as restrictions on functions and weights, inflexibility in constraints, and challenges in assessing proper efficiency. To accomplish this, we propose an improved version of the feasible value constraint technique. Then, by incorporating approximate solutions, we establish connections between $\varepsilon$-(weakly, properly) efficient points in a general MOP and $\epsilon$-optimal solutions to the Scalarization problem.

In this paper, we study approximate proper efficient (nondominated and minimal) solutions of vector optimization problems with variable ordering structures (VOSs). In vector optimization with VOS, the partial order-ing cone depends on the elements of the image set. Approximate proper efficient/nondominated/ minimal solutions are defined in different senses (Henig, Benson, and Borwein) for problems with VOSs from new stand-points. The relationships among the introduced notions are studied, and some Scalarization approaches are developed to characterize these solutions. These Scalarization results based on new functionals defined by elements from the dual cones are given. Moreover, some existing results are ad-dressed.

Trade-off between objective functions in multi-objective optimization is one of the tools for interpreting and studying efficient solutions. Properly efficient solutions are one of the most important theoretical and practical concepts that represent the behavior of the objective functions during a process change. Actually, these solutions are those efficient solutions that filter the anomalies of objective functions at some points, and this will help the manager to decision making to choose more important solutions. One of the most important tools for obtaining solutions with bounded trade-off in multi-objective optimization field is the Sum weighted Scalarization method, which many authors have been studying it in interactive optimization field. This paper provides a method for obtaining properly efficient solutions near the ideal point with a theoretical and interactive view and using Sum weighted Scalarization method. Since being near to ideal point will be abele to a preference of decision maker; this method examines the preferences of the decision maker without sacrifice the theory. Therefore, this paper presents an approach to finding properly efficient solutions near to the ideal point.

The objective of the current study (sponsored by the Ministry of Cooperation) was to present an appropriate framework for training efficient managers in cooperation sector by relying on three aspects of education, research and extension. The research methodology was descriptive (survey) and the required statistical data were collected using questionnaires with stratified random sampling method among 200 managers of cooperative sector of Tehran Province.The distance between the current status and desirable condition of main components resulted from researches about efficient management (participative management, human resources development, empowerment, ICT, organizational learning and entrepreneurship) was determined with descriptive and inferential statistics from the aspect of the manager and through factorial analysis of their prioritization. The research findings show that human development and entrepreneurship components respectively have the most deviation from the mean compared to other components. The results of grading efficient management components shows that entrepreneurship, IT, human resources development, empowering, organizational learning and participative management components respectively rank 1st to 6th in efficiency. The results of analyzing the current status of education, research and extension in three levels of management demonstrated that the current status of education, research and extension is not desirable and requires revision in type, quality and quantity of courses in the cooperation sector. Studying the current and desirable conditions of cooperation management status of Iran implies that there is a significant difference between the current status of all efficient management components and the desirable one form the aspect of managers. With regard to the findings of the study, a conceptual framework for management of cooperatives was presented.

We study the extended scalar-tensor Gauss-Bonnet. First, we review a model in which there is a coupling between the Gauss-Bonnet scalar and the scalar field. Then we extend the model to a hybrid model with two scalar fields. We study the conditions under which the solutions are unstable and spontaneous Scalarization occurs. We show that the coupling coefficients in the hybrid state and in the presence of a scalar field are asymptotically the same but the sufficient condition of instability is different between these two models.

A problem that sometimes occurs in multi objective optimization is the existence of a large set of fairly efficient solutions. Hence, the decision making based on selecting a unique preferred solution is difficult. Considering models with fair B -efficiency relieves some of the burden from the decision maker by shrinking the solution set, since the set of fairly B-efficient solutions is contained within the set of fairly efficient solutions for the same problem. In this paper, first some theoretical and practical aspects of fairly B-efficient solutions are discussed. Then, some Scalarization techniques are developed to generate fairly B-efficient solutions.