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 Î¦ï¼ˆxï¼‰âˆˆR^{2ï¼ˆmn+nï¼‰}is semismooth. If FâˆˆR^{mn+n}is LC^{1}function, then Î¦ï¼ˆxï¼‰ are strongly semismooth.Lemma2.5.1The generalized gradient F Bï¼ˆa, bï¼‰ at a point ï¼ˆa, bï¼‰âˆˆR^{2}isequal to the set of all {ï¼ˆga, gbï¼‰}, such thatwhere,ï¼ˆÎ¾, Î¶ï¼‰ is any vector satisfying ï¼ˆÎ¼, Î½ï¼‰â‰¤1; the generalized gradien-tï¼ˆ?ï¼‰_{+}ï¼ˆa, bï¼‰ at a point ï¼ˆa, bï¼‰âˆˆR^{2}is equal to {ï¼ˆb_{+}a_{+}, a_{+}b_{+}ï¼‰}, whereTheorem2.5.2Let xâˆˆR^{mn+n}be given, then any matrix Hâˆˆï¼ˆ?ï¼‰_{C}Î¦ï¼ˆxï¼‰can be written aswhere, H_{1}Daï¼ˆxï¼‰+D_{b}ï¼ˆxï¼‰F ï¼ˆxï¼‰ and H_{2}D_{a}ï¼ˆxï¼‰+D_{b}ï¼ˆxï¼‰F ï¼ˆxï¼‰ with D_{a}ï¼ˆxï¼‰=diag{aiï¼ˆxï¼‰}, D_{b}ï¼ˆxï¼‰=diag{b_{i}ï¼ˆxï¼‰}, D_{a}ï¼ˆxï¼‰=diag{a_{i}ï¼ˆxï¼‰}, D_{b}ï¼ˆxï¼‰=diag{b_{i}ï¼ˆxï¼‰} be-ing diagonal matrices with entries ï¼ˆa_{i}ï¼ˆxï¼‰, b_{i}ï¼ˆxï¼‰ï¼‰âˆˆF Bï¼ˆx_{i}, F_{i}ï¼ˆxï¼‰ï¼‰ and ï¼ˆa_{i}ï¼ˆxï¼‰, b_{i}ï¼ˆxï¼‰ï¼‰âˆˆï¼ˆ?ï¼‰_{+}ï¼ˆx_{i}, F_{i}ï¼ˆxï¼‰ï¼‰.Theorem2.5.3The merit functions Î¨ï¼ˆxï¼‰ satisfy:1) Î¨ï¼ˆxï¼‰ is continuously diferentiable with Î¨ï¼ˆxï¼‰=H^{?}Î¦ï¼ˆxï¼‰, whereï¼ˆ?ï¼‰HâˆˆCÎ¦ï¼ˆxï¼‰ can be chosen arbitrarily;2) If x^{*}is a stationary point of Î¨ï¼ˆxï¼‰ and F ï¼ˆx^{*}ï¼‰ is P0matrix, then x^{*}is asolution of complementarity problem2.5.1.The algorithm are presented as the following:First Step Initialization.1) Let Î²âˆˆï¼ˆ0,1ï¼‰, Ïƒâˆˆï¼ˆ0,2^{1}ï¼‰, Îµâ‰¥0.2) For ï¼ˆ2.5.1ï¼‰, choose any x^{0}âˆˆR_{+}^{mn+n}. 3) Set k=0.Second Step Termination Check.For ï¼ˆ2.5.1ï¼‰, if Î¨ï¼ˆx^{k}ï¼‰â‰¤Îµ, STOP.Third Step Search Direction Calculation.For ï¼ˆ2.5.1ï¼‰ choose H_{k}âˆˆCÎ¦ï¼ˆx^{k}ï¼‰, Î»_{k}âˆˆï¼ˆ0,1ï¼‰ and let d_{k}âˆˆR^{mn+n}be asolution of the following system of equations:ï¼ˆHï¼ˆ_{k}^{T}ï¼‰H_{k}+Î»_{k}Iï¼‰d=Î¨ï¼ˆx^{k}ï¼‰.Fourth Step Line Search.1) For ï¼ˆ2.5.1ï¼‰, compute the smallest nonnegative integer l satisfying Î¨ï¼ˆx^{k}+Î²^{l}d^{k}ï¼‰â‰¤Î¨ï¼ˆx^{k}ï¼‰+ÏƒÎ²^{l}Î¨ï¼ˆx^{k}ï¼‰^{T} d^{k}.2) For ï¼ˆ2.5.1ï¼‰, set x^{k+1}=x^{k}+Î²^{l}d^{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 ï¼ˆ2.5.1ï¼‰. Then the sequence {x^{k}} converges to {x^{*}}if {Î»_{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. |