Reference suggested a method to solve nonlinear interval programming by transforming the uncertain objective functions and constraints into two deterministic objective functions and constraints. Reference defined the coefficients and variables in the linear programming in the interval form and constructed a solution by using a two-level programming approach with some additional procedures for obtaining an interval solution. Reference presented procedures to obtain optimum solution in interval form for both optimum point and optimum value. Nevertheless, the optimum point cannot be constructed in interval form. References were able to construct the interval solution at the optimum value only, which was done by combining the optimum value from the best optimum with the worst optimum problem so the interval form was obtained. Reference used two-level programming on the solving of linear programming with the interval coefficients, whereas used two-level programming on the solving of quadratic programming with interval coefficients. Two-level programming is a mathematical procedure which used to transform the interval programming model into a pair of classic programming models with special characteristics, namely, the best optimum and the worst optimum problems. Research on quadratic programming with interval coefficients has also been extended to the nonlinear interval programming which is discussed by Jiang et al. The research on quadratic programming with interval variables was inspired by linear programming with interval variables which have been discussed by Suprajitno and Mohd. ![]() All of the researches were inspired by linear programming with the interval coefficients which have been discussed earlier by Shaocheng, Chinneck and Ramadan, and Kuchta. Researches on quadratic programming with interval coefficients have been discussed by Liu and Wang, Li and Tiang, and Syaripuddin et al. If the coefficients and variables in the objective function and constraints are both of interval form, it is called quadratic programming with interval variables. Classical quadratic programming which is developed by transforming the coefficients in objective functions and constraints into interval form is called quadratic programming with interval coefficients. The special characteristic of the interval quadratic programming is the coefficients and variables of the objective functions and constraints are in interval form. It is also called interval quadratic programming. The uncertain coefficient value can be estimated using intervals based on the theory of interval analysis which is developed by Moore. But in the real world, coefficient values often cannot be certainly determined. ![]() IntroductionĬlassic quadratic programming requires the assumption that the coefficient value is certainly known. The procedure to solve the best and worst optimum problems is also constructed to obtain optimum solution in interval form. Procedure of two-level programming is transforming the quadratic programming model with interval variables into a pair of classical quadratic programming models, namely, the best optimum and worst optimum problems. ![]() In this paper, a two-level programming approach is used to solve quadratic programming with interval variables. ![]() Quadratic programming with interval variables is developed from quadratic programming with interval coefficients to obtain optimum solution in interval form, both the optimum point and optimum value.
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