Henig efficiency of a multi-criterion supply-demand network equilibrium model

,T.C Edwin Cheng,Yunan Wu,,Department Of Mathematics, Chongqing Normal University, Chongqing

doi:10.3934/jimo.2006.2.269

Abstract

This paper addresses Henig efficiency of a multi-product network equilibrium model based on Wardrop's principle. We show that in both the single and multiple criteria cases, such proper efficiency can be recast as a vector variational inequality. In the multiple criteria case, we derive a sufficient and a necessary condition for Henig efficiency in terms of a vector variational inequality by using the Gerstewitz's function.

Highlights

Consider a supply-demand network that comprises some manufacturers and retailers, as well as some distributing centers or warehouses
Since we have proved that the necessary and sufficient conditions for a vector flow v ∈ D to be an equilibrium pattern flow in a scalar-valued network equilibrium problem are that it is a solution to a vector variational inequality in Theorem 2.1, by applying the Gerstewitz’s function, we suppose that the equivalence relation must hold between the vector-valued network equilibrium problem and the vector variational inequality without any convexity assumptions
A vector flow v ∈ D is a Henig equilibrium pattern flow in the generalized context of a multi-product supply-demand network equilibrium problem with a vector-valued cost function if v is a solution to a vector variational inequality: to find v ∈ D such that ξk0(v), (u − v)T ∈ R+q×q, ∀ u ∈ D

Summary

Introduction

Consider a supply-demand network that comprises some manufacturers and retailers, as well as some distributing centers or warehouses. The cost may comprise multiple criteria, which embraces tariffs, fuels, time and other relevant cost factors Such a phenomenon results when the network follows a natural law known as the user-optimizing principle or the Wardrop’s equilibrium principle (Wardrop (1952)). We establish a sufficient and a necessary condition for a Henig equilibrium pattern flow for a multi-product network equilibrium problem in terms of vector variational inequalities for the single criterion case and the multiple criteria case.

Henig efficiency of a multi-product network equilibrium model

Network equilibrium problem with multi-product products and multi-criterion

Conclusions