Interval Reliability Definition Of Interval Number And Interval Linear Programming B Stability Research

Linear programming is an important branch of operations research, Which provides a kind of effective method on how to reasonably use the limited manpower, material resources, financial and other resources to get the best economic effect in production, management, transportation and business activities. However, with the real factors of nonlinear, dynamic, open, ambiguous and incomplete information, error and so on, some parameters’metrics are not fixed values. So, how to deal with uncertain information of linear programming becomes a difficulty in the mathematics areas. So far, there has been three main linear programming models on uncertain information research:stochastic linear programming (SLP), fuzzy linear programming (FLP) and interval linear programming (ILP). This paper mainly studies interval reliability definition of interval number and the interval linear programming problem. The interval linear programming problem is the linear optimization problem that uncertain parameters is given in the form of interval numbers.The main work of this paper are as follows:As we all know, We compare the two interval numbers by the definition of creditability.We can get the first thought on solving linear programming problems---solving credible. Many scholars have put forward reliability definition of comparison between two interval numbers. In the second chapter, this paper first summaries and proposes six main reliability equivalent definitions and analyzes its excellent mathematical properties, such as complementary, symmetry, transitivity. Then, from the interval midpoint and half wide angle, this paper puts forward perfect creditability definition and establishes one to one mapping relations between this definition and the six equivalent definitions. The third chapter mainly analysis the inadequacy of credibility definitions proposed in some papers and puts forward more concise credibility definitions from the same angle. This chapter also analysis and proves this definition has good mathematical properties, such as the two interval numbers can be degradated to real number.In solving the credibility problem of interval linear programming, no strict definition easily causes that the calculation result is not accurate, therefore, this paper presents a more strict definition, and verifies the validity of conclusions proposed in the paper by numerical examples.While solving interval linear programming problems that has equation constraint, we wouldn’t be able to achieve reliability solution like the second and third chapters. Because with credibility for the equality constraint, it can cause disjoint between the equality constraints and inequality constraints, i.e. no feasible solution, however, interval linear programming problems usually have optimal solution. So Inuiguchi proposed the concept and algorithm of the necessary set and the optimal set when solving interval linear programming optimal solution set. Tong considered the interval linear programming problems which the objective function and constraint coefficient are interval numbers, and got value range--- the best optimal solution and the inferior optimal solution. but the inferior is NP-hand. This fourth chapter summarizes B stability of the interval linear programming problem and studies the three types of interval linear programming problems. First this chapter summarizes the necessary and sufficient condition, sufficient condition of B stability for the Type (A). The sufficient and necessary conditions is complicated, so the sufficient condition is importent. Analysising the unreasonable of definition proposed by Milan Hlad Golf K, this paper brings forward the modified sufficient conditions of B stability. This paper shows the sufficient and necessary conditions of optimality conditions by the formula for Type (B). B stability sufficient condition be verified by two examples of Type (A) and Type (B). Comparing the best optimum solution and the inferior optimal solution found:if interval linear programming problem is B stability, solving problem more efficient, the result more accurate. According to the principle of duality, Type (C) can be converted into Type (A) if and only if satisfied the condition of zero duality gap. At last of the paper puts forward and discusses the sufficient conditions for zero duality gap. B stability plays a key role in solving fully interval linear programming problem that has equality constraints and have no conditions for variables.