4. SAMPLE PROBLEM NUMBER ONE


4.1 A Single Commodity, Three Region Model


 This first sample problem is fairly simple and should make clear
the process of moving from a model specification to a computer solution.
This example can be considered a quadratic programming problem as the Q
matrix is a diagonal matrix (and therefore symmetric). Although small,
the problem does demonstrate the essential features which are common among
LCP models, especially when they are used for market oriented, economic
modeling purposes.


4.1.1 Model description--In the interest of simplicity with this first
example, we will only consider a single commodity. There are three
consuming and producing regions in the model, with supply and demand
represented by linear functions. The function coefficients for regions
1, 2, and 3 are shown in Table 1. Note that these are in inverse form
since a quantity formulation is used in the model, i.e., the primal
variables are quantities, (see Takayama and Judge [5, p. 129]).





Table l.--Demand and supply functions for Regions 1, 2, and 3

 Demand Supply


Region 1 Pi = 25 .ly1 p = 4 + .2x1
 2
Region 2 P2 = 30 .5y2 p = 5 + .3x2

Region 3 P3 = 35 .7y3 p3 = 3 + .4x3





 Trade is allowed between any two regions of the model and the cost
of transporting a single unit of the commodity between two regions is
given by Table 2.