Warehouse Location Problem
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[10]. In the pure exchange model the consumer side of the economy is modeled by the excess demand functions and it is assumed that production is absent and consumers exchange commodities that they initially own. Production can be introduced into this basic framework in a variety of ways by including, for example, an activity analysis model to describe the productive techniques in the economy (el. [15], [14]). The excess demand functions are aggregated demand functions over the individual consumers in the economy. They represent the difference between the market demands for the commodities and the supply of the commodities (based on the initial endowments of the consumers). Computation of economic equilibria has been, typically, based either on classical algorithms for solving nonlinear systems of equations (see, e.g., [6]), or, on simplicial approximation methods pioneered by H. Scarf [15] (see also the contributions of M.J. Todd [17], [18], J.B. Shoven [16], J. Whalley [21], G. van der Laan and A.J.J. Talman [7]). The former techniques are applicable only when the equilibrium lies in the interior of the feasible set, while the latter techniques are general-purpose
algorithms, and are capable of handling inequality constraints, such as the requirement that the prices of the commodities be nonnegative. Nevertheless, in their present state of development, they may be unable to handle large scale problems (cf. [11]). General economic equilibrium problems have been formulated as nonlinear complementarity problems (see, e.g., [8]) and a Newton-type method based on this formulation has been used by many researchers for the computation of equilibria (see, e.g., [5], [9], [13]). Although this approach has been proven to be more effective than fixed point methods, its convergence has not been proven theoretically (see, e.g., [11]). K.C. Border [1] provides a variational inequality formulation of Walrasian price equilibrium. Qualitative results using a variational inequality framework can be found in [3]. Algorithms, as well as numerical examples, as well as the relationship of the Walrasian price equilibrium problem to a network equilibrium problem can be found in [23]. Here we present the pure exchange (or pure trade) economic equilibrium model and gives its variational inequality formulation. Some fundamental theoretical results are then presented. For proofs and additional background and results, see [3], [12], [22], and [23]. Consider a pure exchange economy with l commodities, and with column price vector p taking values in Rl+ and with components Pl,...,Pz. Denote the induced aggregate excess demand function z(p), which is a row vector with components zl ( p ) , . . . , zt(p). Assume that z(p) is generally defined on a subcone C of R~_ which contains the interior Rz++ of Rl+, that is, R++ C C C R / .
Walrasian price equilibrium Hence, the possibility that the aggregate excess demand function may become unbounded when the price of a certain commodity vanishes is allowed. Assume that z(p) satisfies Walras' law, that
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