Optimization of the product portfolio has been recognized as a critical problem in industry, manage-ment, economy and so on. It aims at the selection of an optimal mix of the products to o er in the target market. As a probability function, reliability is an essential objective of the problem which linear models often fail to evaluate it. Here, we develop a multiobjective integer nonlinear constraint model for the problem. Our model provides opportunities to consider the knowledge transferring cost and the environmental e ects, as nowadays important concerns of the world, in addition to the classical factors operational cost and reliability. Also, the model is designed in a way to simultane-ously optimize the input materials and the products. Although being to some extent complicated, the model can be e ciently solved by the metaheuristic algorithms. Finally, we make some numerical experiments on a simulated test problem.

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.

The sewer layout in flat areas significantly influences the construction and operational costs as well as reliability of the network performance. To find an optimum design of sewer networks for flat areas, this study presents a multi-objective optimization problem with the objective functions of 1- the cost and 2- the reliability. The reliability criterion is defined as the effect of a clogging in a sewer on its upstream population. To solve the problem, the NSGA-II Algorithm is coupled to two self-adaptive design algorithms for generating the layout and sizing of sewers. A case study is solved using the proposed model to demonstrate its application and advantages.

The research on supplier selection is abundant and the works usually only consider the critical success factors in the buyer–supplier relationship. However, the negative aspects of the buyer–supplier relationship must also be considered simultaneously. In this paper we propose a comprehensive model for ranking an arbitrary number of suppliers, selecting a number of them and allocating a quota of an order to them considering three objective functions: minimizing the net cost, minimizing the net rejected items and minimizing the net late deliveries. The two-stage logarithmic goal programming method for generating weights from interval comparison matrices (Wang et al. 2005) is used for ranking and selecting the suppliers. It is assumed that the suppliers give price discounts. A fuzzy multiobjective model is formulated in such a way as to consider imprecision of information. A numerical example is given to explain how the model is applied.

Compilation of optimum model for determining the extent of economic activities and the allocation of limited resource to them, regarding to reach the different goals in economic planning, is the principle research subject in this paper. In this context, it seems relevant to utilize linear programming based on Leontief's input-output system. In this model compatibility and optimality of economic activities have been taken into account simultaneously. So, compared to Leontiefs input-output model and other models like statistical model, the said optimum model, due to its more focus on planning principles would bring about more useful results. However, it would be desirable and indeed more realistic to take into account all of the goals influencing the system. Regarding these purposes, in this paper, a multiple objective optimum model based on Leontiefs input-output system has been prepared and then it has been used to economic planning in Iran for the year 1378 (1999). In the end, to evaluate the model validity, the results of the said optimum model have been compared with the results of current model being used for second economic plan for the year 1378 (1999).        

A multiobjective security game problem with fuzzy payoffs is studied in this paper. The problem is formulated as a bilevel programming problem with fuzzy coefficients. Using the idea of nearest interval approximation of fuzzy numbers, the problem is transformed into a bilevel programming problem with interval coefficients. The Karush-Kuhn-Tucker conditions is applied then to reduce the problem to an interval multiobjective single-level problem. It is shown that the solutions of this problem are obtained by solving a single-objective programming problem. Validity and applicability of the method are illustrated by a practical example.