Algorithm design for a supply chain equilibrium management model

In this paper, we consider a complementary model for the equilibrium management of supply chain. In order to give an optimal decision for the equilibrium management, we propose a new algorithm based on an estimate of the error bound. This algorithm requires neither the existence of a non-degenerate solution nor the non-singularity of the Jacobian matrix at the solution. We also prove the quadratic convergence of the given algorithm. It can be viewed as extensions of previously known results. I. INTRODUCTION Supply chain management involves many aspects, including manufacturing, transportation, logistics and sales markets, etc. Manufacturers produce a commodity and then transport it to sellers. The manufacturers need to pay for the costs of production and transportation. In order to make a profit, they have to determine optimal production and transportation. Sellers need to pay for the costs in stores, and maximize their profits from the commodity price. Commodity price is the price customers are willing to pay. The customers need to determine their optimal level of consumption, which is related to commodity prices at different stages of the supply chain and the commodity handling costs. The supply chain management problem can be abstracted as a "complementary" relationship between the two sets of decision variables, expressed as a complementary model ((1,2,3)). in xy x y f x f y x y R m ££ ³- - - " I


INTRODUCTION
Supply chain management involves many aspects, including manufacturing, transportation, logistics and sales markets, etc.Manufacturers produce a commodity and then transport it to sellers.The manufacturers need to pay for the costs of production and transportation.In order to make a profit, they have to determine optimal production and transportation.Sellers need to pay for the costs in stores, and maximize their profits from the commodity price.Commodity price is the price customers are willing to pay.The customers need to determine their optimal level of consumption, which is related to commodity prices at different stages of the supply chain and the commodity handling costs.The supply chain management problem can be abstracted as a "complementary" relationship between the two sets of decision variables, expressed as a complementary model ( [1,2,3]).
Let mapping :, nn f R R  be continuously differentiable.
Find vector * , We use * X to denote the solutions of ，and assume that * X is not empty.
The model takes into consideration different behaviors and their interactions of the various participants, and provides a standard to assess the appropriate commodity prices and quantities.It is an equilibrium optimization problem.At its core is the use of mathematical methods, with computer and network as tools, to study various complex systems having this complementary relationship and their corresponding solutions.It provides scientific basis for decision makers, with the goal of ultimately achieving balanced and harmonious development of complex system operation.It is widely used nowadays.In recent years it has become a hot topic of operational research and management studies.Many researchers, especially scholars abroad, are increasingly interested, and great progress has been made in both theoretical research and practical applications.To solve (1), many algorithms have been proposed ([4, 5, 6, 7, 9]).The basic idea of these algorithms is to transform (1) into an unconstrained optimization problem, or into a simple constrained optimization problem ([4, 7, 9]), and then to use the Newton type algorithms or confidence region algorithms.Although the convergence and the corresponding rate of convergence are established, the conditions are mostly quite strong.Most of these algorithms require that the Jacobian matrix at the solution of ( 1) is non-singular or there exists a non-degenerate solution.As we know, the non-singularity of the Jacobian matrix at the solution of (1) ensures that the application of the well-known Levenberg-Marquardt (L-M) algorithm for solving (1) has the property of quadratic convergence ( [7]).Recently, Yamashita, and Fukushima have proved that under certain conditions on the local error bound, the L-M algorithm for solving nonlinear equations also has the property of quadratic convergence.These conditions on local error bound are weaker than the non-singularity of the Jacobian matrix ( [8]).This motivates us to transform (1) into a nonlinear equation, and to establish the error bound for (1).With the establishment of an error bound of a residual function from (1), we propose a new algorithm and prove its quadratic convergence, thereby overcoming the drawbacks of the above mentioned algorithms on convergence.

II. PRELIMINARIES
In this section, we mainly give a result on the error bound for (1) ( [10]), which play a key role in the establishment of the nonlinear equation and in the proof of the convergence of the proposed algorithm.
be a uniform P function, i.e., there exists a constant 0 In this section, we first transform (1) into a nonlinear equation, and then give another type of error bound for (1).Based on this, we give the new algorithm for solving (1) and establish its property of quadratic convergence, overcoming the drawbacks of requiring the non-singularity of the Jacobian matrix at the solution or the existence of a non-degenerate solution.


For (5), using Lemma 1, we have the following result.

Algorithm：
Step 1: Select an initial point 0  , In the following convergence analysis, we assume that Algorithm 1 generates an infinite sequence.From ( 7)-( 9), combined with ( 14) - (15), and the proof of Theorem 2.1 ([8]), we can obtain the quadratic convergence rate of Algorithm 1.In this paper, we present a new algorithm for solving the equilibrium management model of supply chain, and also have showed that method has the quadratic convergence, the conditions guaranteeing the quadratic convergence in this paper are weaker than those in the existing resolving methods in [7], since it does not require the existence of a nondegenerate solution of (1), nor the non-singularity of the Jacobian matrix at the solution of (1), It is uncertain whether we have that the results in this paper for the functions which contain the P functions discussed in this paper, this is a topic for further research.

Theorem 2 Function( 4 )
,(3) and Theorem 1, we can easily prove the following result.If the mapping :nn f R R  in (1) is a uniform P function, and is Lipschitz continuous ， then there exist constant 2 is not smooth.In order to give a smooth algorithm with the property of quadratic convergence, we give also the following smooth Fisher- Fx