Supplier selection is one of the main discussions in the Supply Chain. The issue of assigning purchase orders to suppliers that act differently in terms of quality, cast, services, etc. criteria is one of the significant concerns of purchase managers in the supply chain. To adopt an optimal decision in this regard is related to a Multi-objective problem that the objectives are contradicting each other and have different importance and priority depending on the location. In practice, the existence of kind of ambiguity in explaining the information related to the problem constraints and complicated. In this regard, the emergence of Fuzzy set theory as a tool to describe such conditions besides presenting question Model realistically can help to solve such problems well. Despite the importance of the Model with the mentioned structure, unfortunately, few original works have been done in this field. As a result, in this paper, in addition to presenting a new Multi-objective Fuzzy Model being Modelled based on assigning purchase order to suppliers in a supply chain a solution method is introduced based on using Fuzzy Linear programming. To clarify solution process Modelling and description, a case study is included related to selecting flour supplier for providing industrial bread of Khoshkar factory. The proposed Model includes four objective functions: Aggregate costs of minimizing type, Services of maximizing type (such as packing, being faithful to promise, factory heath, discount, correct transportation, good relationships, honestly, etc.), Flour useful survival of maximizing type (regarding monthly flour buying by the factory), The purchased flour quality of maximizing type (concerning product type).  Especially in the solution process, a method is determined based on setting weight for each of the objectives concerning the major factory stockholders.

We develop an algorithm for the solution of multiobjective Linear plus fractional programming problem (MOL+FPP) when some of the constraints are homogeneous in nature. Using homogeneous constraints, first we construct a transformation matrix T which transforms the given problem into another MOL+FPP with fewer constraints. Then, a relationship between these two problems, ensuring that the solution of the original problem can be recovered from the solution of the transformed problem, is established. We repeat this process of transformation until all the homogeneous constraints are removed. Then, we discuss the multi objective programming part, for which fuzzy programming methodology is proposed which works for the minimization of perpendicular distances between two hyper planes (curves) at the optimal points of the objective functions. A suitable membership function is defined with the help of the supremum perpendicular distance. A compromised optimal solution is obtained as a result of the minimization of the The supremum perpendicular distance. The corresponding optimal solution to the original problem is obtained using the transformation matrix. Finally, an example is given to illustrate the proposed Model.

Fuzzy multiobjective Linear bilevel programming (FMOLBP) problems are studied in this paper. The existing methods replace one or some deterministic Model(s) instead of the problem and solve the Model(s). Doing this work, we lose much information about the compromise decision, and it does not make sense for the uncertain conditions. To overcome the difficulties, Zadeh’s extension principle is applied to solve the FMOLBP problems. Two crisp multiobjective Linear three-level programming problems are proposed to find the lower and upper bound of its objective values in different levels. The problems are reduced to some Linear optimization problems using one of the scalarization approaches, called the weighting method, the dual theory, and the vertex enumeration method. The lower and upper bounds are estimated by the resolution of the corresponding Linear optimization problems. Hence, the membership functions of compromise objective values are produced, which is the main contribution of this paper. This technique is applied for the problem for the first time. This method applies all information of a fuzzy number and does not estimate it by a crisp number. Hence, the compromise decision resulted from the proposed method is consistent with reality. This point can minimize the gap between theory and practice. The results are compared with the results of existing approaches. It shows the efficiency of the proposed approach.

The fuzzy primal-dual simplex method is a new and efficient method for solving Linear programming problems with fuzzy variables. This algorithm is based on duality results and, similar to the dual simplex method, begins with dual feasibility and proceeds to primal feasibility. An important difference between the dual simplex method and the primal-dual method is that in the primal-dual algorithm, it is not required that the dual feasible solution to be basic. In this paper, we develop the primal-dual simplex method for solving fuzzy multiobjective Linear programming problems. To this end, we utilize the fuzzy weighted sum scalarization method to present a fuzzy single objective optimization problem related to the fuzzy multiobjective Linear programming problem. Then, by partitioning the weights of the weighted sum problem, we generalize the single objective primal-dual algorithm to fuzzy multiobjective problems. By using the presented algorithm, we can find a set of fuzzy Pareto optimal solutions. Presenting a set of fuzzy Pareto optimal solutions to the decision maker, enables him\her to select the best solution based on his\her preferences. Finally, we apply the proposed algorithm for solving a three-objective optimization problem with fuzzy variables and compare the results with some existing methods.

Multi-objective optimization plays an important role in Modelling many problems. On the other hand, some of these problems deal with the optimization of the ratio of objectives, which are more important than the optimization of each objective alone. Since most of the real-world problems are faced with uncertainty in data and parameters, in this paper a class of Multi-objective Linear fractional programming problems with rough interval coefficients in the objective functions is considered. In order to deal with this group of problems, first the problem is taken out of the rough interval state, then the problem is solved using interval arithmetic and the weighted sum approach in Multi-objective optimization. The obtained results are investigated in a numerical example.

Optimization problems have dedicated a branch of research to themselves for a long time ago. In this field, multiobjective programming has special importance. Since in most real-world multiobjective programming problems the possibility of determining the coefficients certainly is not existed, multiobjective Linear programming problems with interval coefficients are investigated in this paper. Corresponding to such problems, four solution concepts, (, )  A b necessarily weak efficient, (, )  A b necessarily efficient, (, , )  A b C necessarily weak efficient and (, , )  A b C necessarily efficient, are introduced. Moreover, necessary and sufficient conditions for recognizing such solutions are presented. Finally, the efficiency of the results is investigated in some numerical examples.