In most of real world problems data su er from inaccuracy, thus considering an INTERVAL for data perturbation seems to be a good idea. There exist many papers in literature about INTERVAL linear PROGRAMMING problems (ILP) dealing with non negative variables. Here the general form of an ILP is considered where all the parameters and variables are considered to be INTERVALs. Moreover, in this study more general conditions for variables are considered, variables which are unrestricted in sign. Although this is the case in most of the real world problems, due to high complexity it has not been dealt with in literature yet. In this paper a new method is presented in order to obtain fully INTERVAL linear PROGRAMMING problems (F ILPP ) using a nonlinear PROGRAMMING problem (NLP). Furthermore, in order to demonstrate how the proposed method works two numerical examples are illustrated.

Among various statistical and data mining discriminant analysis proposed so far for group classification, linear PROGRAMMING discriminant analysis has recently attracted the researchers’ interest. This study evaluates multi-group discriminant linear PROGRAMMING (MDLP) for classification problems against well-known methods such as neural networks and support vector machine. MDLP is less complicated as compared to other methods and does not suffer from having local optima. This study also proposes a fuzzy Delphi method to select and gather the required data, when databases suffer from deficient data. In addition, to absorb the uncertainty infused to collecting data, INTERVAL MDLP (IMDLP) is developed. The results show that the performance of MDLP and specially IMDLP is better than conventional classification methods with respect to correct classification, at least for small and medium-size datasets.

In this paper, an INTERVAL fractional optimization problem with directionally differentiable functions is considered, and the d-invexity concept is introduced for INTERVAL-valued functions. Slater's constraint qualification and pre-invex directional derivative assumption are used to establish the necessary optimality conditions. Further, sufficient optimality conditions are derived under the d-invexity assumption, considering the LU-solution concept. As an application of INTERVAL fractional problems, a portfolio optimization problem with uncertain return and risk parameters subject to INTERVAL liquidity constraints is considered, and an optimal solution is obtained using the results developed in this paper. Also, the portfolio optimization problem is solved using the proposed global criteria method for INTERVAL optimization problems and two methods available in the literature. Moreover, to check the efficiency of the proposed method, a comparison between different methods is presented. Throughout the paper, non-trivial examples are presented at appropriate places to provide a better understanding of the results developed.

Some of the parameters in issues of the reality world are uncertainty. One of the uncertain problems with the qualitative parameters is economic problems such as bankruptcy problem. In this case, there is a probability of dealing with imprecise concepts including the INTERVALs regarding the official’s viewpoint, organizations’ managers. Accordingly, this article uses the concepts of data envelopment analysis (DEA) game theory’ applications that it is appeared in all areas of studies, and combining it with uncertainty models like INTERVALs, assess bankruptcy and specify the pessimistic and optimistic INTERVAL for bankruptcy assessment that hep us to assess uncertain concepts in economics and in the problems that we have certain, converting to INTERVAL PROGRAMMING a, is studied problems simply.

We consider INTERVAL linear PROGRAMMING (ILP) problems in the current paper. Best-worst case (BWC) is one of the methods for solving ILP models. BWC determines the values of the target function, but some of the solutions obtained through BWC may result in an infeasible space. To guarantee that solution is completely feasible (i.e. avoid constraints violation), improved twostep method (ITSM) has been proposed. Many solutions are lost in this method. By using an algorithm, we introduce closed ball method (namely, CBM) as a new method for solving ILP models. In this method, feasibility test ensures that solution space is feasible. To demonstrate the effectiveness of the proposed approach, we solve two numeric examples and we compare the results obtained through BWC, ITSM, and CBM.

In this article, multi-objective linear PROGRAMMING problems with INTERVAL coefficients in objective functions are investigated. By using the acceptance index, the concepts of different answers corresponding to such problems are presented. In order to achieve such answers, a new solution method is introduced. Characteristics of the answer/answersobtained from the new solution method are also expressed. For example, the unique optimal solution resulting from thenew solution method is a strict A-efficient solution of the multi-objective linear PROGRAMMING problem with INTERVAL coefficients in the objective functions. In order to better understand the presented concepts and solution method, numerical examples are also examined.