In this paper, we establish a proof for a necessary condition for MULTIPLE OBJECTIVE fractional pro-gramming. In order to derive the set of necessary conditions, we employ an equivalent parametric problem. Also, we present the related semi parametric model.

MULTIPLE OBJECTIVE PROGRAMMING (MOP) problems have become famous among many researchers due to more practical and realistic implementations. There have been a lot of methods proposed especially during the past four decades. In this paper, we develop a new algorithm based on a new approach to solve MOP problems by starting from a utopian point (which is usually infeasible) and moving towards the feasible region via stepwise movements and a plain continuous interaction with Decision Maker (DM). We consider the case where all OBJECTIVE functions and constraints are linear. The implementation of the proposed algorithm is demonstrated with two numerical examples.

There are varieties of QFD combination forms available that can help management to choose the right model for his/her types of problem. The proposed MOCC-QFD-FMEA model is a right model to include variety of OBJECTIVEs as well as the risk factors into the model of the problem. Due to the fact that the model also takes into consideration the concept of Fuzzy set, it further allows management the flexibility in his/her modeling as well as decision making. The mathematical models presented in this article demonstrate the process of development of the equivalent deterministic form of chance constrained PROGRAMMING for the QFD and FMEA combined systems. The final model presented is a linear multi-OBJECTIVE goal PROGRAMMING problem that can be solved by a linear goal PROGRAMMING program.

Companies with sustainable development and competitive advantage goal, need to consider suppliers as a part of their own core competencies. Accordingly, supplier selection should be considered as an important part of the strategic decisions of these kinds of companies. Supplier selection is usually formulated as a multi criteria decision making problem. When suppliers are selected, it is essential to solve order allocation problems that means determining the amount of order for each selected supplier. So, solving supplier selection and order allocation with integrated model as a research area, has attracted many attentions. According to experience of previous researches, formulating integrated model for solving these two problems leads to a multi-OBJECTIVE mix integer program. Assuming available delivery time data as a sample of real behavior of suppliers, this paper improved a practical aspect of formulation of this problem by utilizing sampling distribution concept. A simple but relatively different solution method that guarantees at least one optimum OBJECTIVE is applied as well.

One of the main principles of the passive defense is the principle of site selection. In this paper, we propose a MULTIPLE OBJECTIVE nonlinear PROGRAMMING model that considers the principle of the site selection in terms of two qualitative and quantitative aspects. The purpose of the proposed model is selection of the place of facilities of a system in which not only it observes the dispersion principle but also reduces the system transportation costs. Moreover, the proposed model tries to select the sites that can fulfill other elements of site selection as well as dispersion in a way that it increases the trustworthiness of the selected network.

This paper focuses on the appropriate selection of suppliers and assigning order quantities to them. This problem contains MULTIPLE OBJECTIVEs and their related constraints and utilizes fuzzy decision making in its solution approach. In the proposed model, goals, constraints, variables and coefficients are all fuzzy. It is shown that with the application of the fuzzy methodology, the multiy OBJECTIVE problem is converted to a single one that can be solved easily.

In this paper, a multi- OBJECTIVE quadratic PROGRAMMING (Poss- MOQP) problem with possibilistic variables coefficients matrix in the OBJECTIVE functions is studied. Through the use of b-level sets the Poss- MOQP problem is converted into the corresponding deterministic multi- OBJECTIVE quadratic PROGRAMMING (b-MOQP) problem and hence into the single parametric quadratic PROGRAMMING problem using the weighting method. An extended b-possibly efficient solution is specified. A necessary and sufficient condition for finding such a solution is established. A relationship between the solutions of possibilistic levels is constructed. Numerical example is given to clarify the obtained results.