The Research Of Nonlinear Complementarity Model And Algorithm For Supply Chain

Along with the rapid development of Internet, the global economy evolvedinto the trend of the global village, a large number of companies have begun tofocus on the global world, from raw material purchase, manufacture, product etcappear network, how to efectively utilize resources, reduce costs and make com-pany to obtain the biggest economic benefits become more and more important,this also why the study of logistics become popular in recent years. And supplychain is the key of logistics, so the research of supply chain is necessary, includingsupply chain modeling, analysis, and numerical experimentation, which will beof some guidance significance to decisions of company. This paper focuses onthe preliminary study of mathematical methods, and supply chain variational in-equality equilibrium model further reformulated into complementary model, andthen the equivalent into optimization or equations of mathematical problems,because these problems play a significant role in complementary problem of solu-tion method and existence. And NCP function plays an important role in solvingcomplementary problems, for it can equivalently transform the complementaryproblems into equations. For NCP function also plays a very important role indesigning algorithm and discussing the convergence of algorithm, so choose theright NCP function is very important.The remainder of the paper is organized as follows. The first chapter is in-troduction, briefly introduce some background knowledge, literature review andour main study of supply chain. Through the study of introduction, can helpreader understand the basic knowledge of supply chain, the research branch indomestic and foreign, and can quickly obtain general direction of this research.The second chapter is two layers of supply chain network complementarymodel, analysis the involved decision-makers in supply chain, including the o- riginal optimization problem, the variational inequality and the explanation ofeconomic significance, the definition of mapping and deduce the complementarymodel. Through the Kanzow and P etra[38] proposed nonlinear complementarityproblems of the least squares expression, discussed the corresponding propertiesof complementary model.Theorem2.5.1The mapping Φ（x）∈R^2（mn+n）is semismooth. If F∈R^mn+nis LC¹function, then Φ（x） are strongly semismooth.Lemma2.5.1The generalized gradient F B（a, b） at a point （a, b）∈R²isequal to the set of all {（ga, gb）}, such thatwhere,（ξ, ζ） is any vector satisfying （μ, ν）≤1; the generalized gradien-t（?）₊（a, b） at a point （a, b）∈R²is equal to {（b₊a₊, a₊b₊）}, whereTheorem2.5.2Let x∈R^mn+nbe given, then any matrix H∈（?）_CΦ（x）can be written aswhere, H₁Da（x）+D_b（x）F （x） and H₂D_a（x）+D_b（x）F （x） with D_a（x）=diag{ai（x）}, D_b（x）=diag{b_i（x）}, D_a（x）=diag{a_i（x）}, D_b（x）=diag{b_i（x）} be-ing diagonal matrices with entries （a_i（x）, b_i（x））∈F B（x_i, F_i（x）） and （a_i（x）, b_i（x））∈（?）₊（x_i, F_i（x））.Theorem2.5.3The merit functions Ψ（x） satisfy:1) Ψ（x） is continuously diferentiable with Ψ（x）=H^?Φ（x）, where（?）H∈CΦ（x） can be chosen arbitrarily;2) If x^*is a stationary point of Ψ（x） and F （x^*） is P0matrix, then x^*is asolution of complementarity problem2.5.1.The algorithm are presented as the following:First Step Initialization.1) Let β∈（0,1）, σ∈（0,2¹）, ε≥0.2) For （2.5.1）, choose any x⁰∈R₊^mn+n. 3) Set k=0.Second Step Termination Check.For （2.5.1）, if Ψ（x^k）≤ε, STOP.Third Step Search Direction Calculation.For （2.5.1） choose H_k∈CΦ（x^k）, λ_k∈（0,1） and let d_k∈R^mn+nbe asolution of the following system of equations:（H（_k^T）H_k+λ_kI）d=Ψ（x^k）.Fourth Step Line Search.1) For （2.5.1）, compute the smallest nonnegative integer l satisfying Ψ（x^k+β^ld^k）≤Ψ（x^k）+σβ^lΨ（x^k）^T d^k.2) For （2.5.1）, set x^k+1=x^k+β^ld^k, k=k+1,and go to Termination Check.Finally, we give the convergence properties of the given algorithm, numericalexamples also be presented to illustrate the NCP formulations and the algorithmof SCN.Theorem2.6.1Let {x^k} be a sequence generated by the above algorithm.If {x^*} is an accumulation point of {x^k} and {x^k} is an R regular solution ofthe complementarity problem （2.5.1）. Then the sequence {x^k} converges to {x^*}if {λ_k} is bounded.In the third chapter three layers of supply chain network complementarymodel, in this section we write accordance with the second chapter, and getsimilar results in the second chapter, the diference is that the dimension in thethird chapter related theorems higher than the second.