The Properties And Algorithm Of Multilevel Decision System And Interaction System

Along with the development of science , the objects people study just change from mathematical programming descrbing simple system to mathematical programming descrbing complex system.It is the multilevel programming and interaction programming problem that mathematical programming descrbing complex system have developped recent years .In this paper, we dicuss the properties and algorithm of the multilevel programming and interaction programming problem.In first chaper , we simply introduced the content, definition and develope-ment of the multilevel programming and interaction programming problem. In second chaper, based on the research to the economic systems which invovle two equal planners,we address equilibrum solution of the Linear Interaction Programming Problem, and attain the following theorem.Theorem 2.1 Suppose u ∈ U, M(u) = 0, then M(u] is the combine of surface of the 5.Corollary 2.1 Op(y) =Σ M(u), i.e. Op(y) is the combine of surface of the S. OQ(X) = ΣM(v), i.e. OQ(X) is the combine of surface of the S.Theorem 2.2 Assume (x,y) exsits, then equilibrum solution is attainable at extreme point.As the Linear Programming is polynomial, the complexity of the LIPP is to be solved. By transforming the IPP into linear complementarity programming , we attain the following theorm.Theorem 2.3 The complexity of the LIPP is NP-hard.Theorem 2.4 Let q and M were given in lemma. If A2 and BI are non-negative matrices and the funtion f(z) = zT(q + Mz) is bounded below on the set {z ∈ Rn|z > 0, Mz > 0}, then the LCP has a solution and equilibrum solution exsists.Considering LIPP's complexity, we introduce assist LIPP to slove LIPP.Theorem 2.5 suppose (x,y) e 5, then (x,y) ∈ Op(y), (x,y)∈ OQ1(x) Corollary 2.2 suppose (LIPP) has equilibrum solution,then assist (LIPP) has equilibrum solution.Theorem 2.6 (x, y) is efficient solution of the assist (LIPP) homologous double objective , if and only if (x, y) is equilibrum solution of the assist (LIPP).Algorithm:stepl solving{ min CTX + dT}, and attaining (x1,y1), let W = (x1,y1) (x,y)∈sand T = 0.step2 let (x1,y1) go into (LIPP} and test equilibrum solution.if yes,then put out.step3 by multiobjective knowledge, attaining near efficient set of the(xi, yi) and showing Wi let T = T (xi, yi), W = W Wi Tc.step4 solving { min cTo;+dTy},and gainning(xi+1, yi+1) ,then turn step2.x,y)∈Wstep5 if none of the efficient extreme point is equilibrum solution, then (LIPP) has no equilibrum solution.As IPP is connect with BLP, BLP is used to solve IPP.Theorem 2.7 IPP has equilibrum solution, if and only if two BLP have the same optimal solution.In third chaper,we investigate three level linear programming with follower responding margrinal function to leader, when every level objective can not agree to, we attain satisfactory solution.Theorem 3.1 Suppose S is bounded, and (x*,y*, z*) is the extreme point of the S3, then (x*, y*, z*) is the extreme point of the S2, and (x*, y*,z*) is theextreme point of the S1.LemmaS.l when x is parameter , and k2 > 0,if and only ifLemma3.2 when x is parameter, and k2 < 0,if and only ifTheorems.2 Suppose k1 > 0,k2 > 0, problem(3.1) if and only ifTheorem 3.3 suppose k1 > 0, k2 < 0, problem (3.1) if and only ifTheorem 3.4 Suppose k1. < 0, k2 > 0,problem (3.1) if and only ifTheorem 3.5 Suppose k1 < 0,k2 < 0, problem (3.1) if and only ifTheorem 3.6if and only ifw is the dual vector of the z. Theorem 3.7if and only ifw1,w2 are the dual vestors of the y, z. Theorem 3.8if and only ifw is the dual vector of the z.In fourth chaper, we give a methed which is used to solve linear bilevel multiobjective programming by satisfactory index, furthmore ,we proof that optimal solution is in the efficient region.Theorem 4.1 when x is parameter,optimal solution of the programming is efficient solution of the following programming.Theorem 4.2 when x is parameter,if and only ifTheorem 4.3 solution of problem (4.5),whi...