This paper presents a fuzzy goal PROGRAMMING (FGP) methodology for solving bi-level QUADRATIC PROGRAMMING (BLQP) problems. In the FGP model formulation, firstly the objectives are transformed into fuzzy goals (membership functions) by means of assigning an aspiration level to each of them, and suitable membership function is defined for each objectives, and also the membership functions for vector of fuzzy goals of the decision variables controlled by decision maker at the first level are developed in the model formulation of the problem. To achieve the highest membership value of each of the fuzzy goals, we formulate the problem by minimizing the negative deviational variables and thereby obtaining the most satisfactory solution for all decision makers. A numerical example is given to demonstrate the proposed approach.

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.

A QUADRATIC PROGRAMMING (QP) problem is a special class of nonlinear PROGRAMMING problems with the objective function of the QUADRATIC form subject to linear constraint. The conventional QUADRATIC PROGRAMMING model requires the parameters to be known constants. However QP has been widely applied to solve real world problems. On the other hand, a wide variety of problems that we deal with in our real life, and we decide according to the solution of the mathematical models are inaccurate concepts or the sets with uncertain bounds. The parameter values used would be based on a prediction of future conditions which inevitably involves some degree of uncertainty. Consequently modeling these problems as QUADRATIC PROGRAMMING with fuzzy parameters is one of the interested fields in operations research. This paper presents a new approach to solve fuzzy QUADRATIC PROGRAMMING problems where the cost coefficients, constraint coefficients, and right-hand sides are represented by convex fuzzy numbers. This method reduces the fuzzy QUADRATIC PROGRAMMING problem to two classical QUADRATIC PROGRAMMING problems using fuzzy concepts. These conventional QUADRATIC problems can be solved using the SQP algorithm, leading to the upper bound and the lower bound of the optimal value at specific a-level respectively. Moreover we extend this method for a more general QUADRATIC PROGRAMMING problem in which all parameters in the problem are all fuzzy numbers. Finally an example is presented to describe and demonstrate the efficiency of the proposed method, of optimization problem.

Multi objective QUADRATIC fractional PROGRAMMING (MOQFP) problem involves optimization of several objective functions in the form of a ratio of numerator and denominator functions which involve both contains linear and QUADRATIC forms with the assumption that the set of feasible solutions is a convex polyhedral with a finite number of extreme points and the denominator part of each of the objective functions is non-zero in the constraint set. In this paper, we extend the procedure as suggested by Lachhwani (Proc. Nat. Acad. Sci. India, 82 (4), 317-322) based on fuzzy goal PROGRAMMING approach for the solution of multi objective QUADRATIC fractional PROGRAMMING (MOQFP) problem.The proposed technique is simple, efficient and requires less computational work. In the proposed FGP model formulation, corresponding objectives of equivalent multi objective PROGRAMMING problem are transformed into fuzzy goals (membership functions) by means of assigning an aspiration level to each of them and suitable membership function is de fined for each objectives. Then achievement of the highest membership value of each of fuzzy goals is formulated by minimizing the sum of negative deviational variables. The proposed methodology is illustrated with numerical example in order to support the proposed methodology.

Challenges such as advances in technology, demands of the global market, and limited warehouse spaces resort manufacturing industries to subcontracting. Subcontracting has been a considerable alternative in the manufacturing industries and is utilized as a strategic tool to diminish operation costs primarily to address the problem of scarcity when the firm faces a large demand on the commodity it supplies. The present study employed a mathematical model among firms engaging in subcontracting in search of an optimal schedule in the manufacture of the product and distribution of production time involved with an objective of obtaining a maximum profit. The constraints in the mathematical formulation included the total demand, processing capacity, available supply, processing rate, and time. The plausibility and the possible utility of the mathematical model has been explored employing sequential QUADRATIC PROGRAMMING algorithm in the search of the optimal solutions.

QUADRATIC PROGRAMMING has been widely applied to solve real-world problems. This paper describes a solution method for solving a special class of fuzzy QUADRATIC PROGRAMMING problems with fuzziness in relations. Then the method is generalized to a more general fuzzy QUADRATIC PROGRAMMING problem, where the cost coefficients, the matrix of the QUADRATIC form, constraints coefficients, and the right-hand sides are all fuzzy numbers. Finally, some examples are taken to the utility of our proposed method.

The linear systems are one of the most important tools for modeling real-world phenomena. Because the real-world phenomena are always associated with uncertainty, solving the fuzzy linear system have a great importance. One of the proposed methods to find the exact and approximate solutions of a fuzzy linear system is using the least squares method. In this method, by choosing an arbitrary meter and solving a QUADRATIC PROGRAMMING, they provide an approximate (or exact) solution for the fuzzy linear system. In this paper, at first, we prove that under some conditions and not depending on the selected meter the QUADRATIC PROGRAMMING is convex. Therefore, by considering three different meters and solving several examples, we compare the obtained approximate solutions.