一个分区线性规划问题及其算法

A PIECEWISE LINEAR PROGRAMMING PROBLEM AND ITS ALGORITHM

在生产实际中广泛存在着一类分区线性规划问题:规划Ⅰ.求一X=(x_1,x_2,…,x_n)适合下列约束条件:

In this paper the following mathematical model I and its generalization arediscussed. Model I, Find an X = (x_1,x_2,…,x_n) satisfying the following constraints: sum(a_ijx_j) j 1 from to n=a_i,i=1,2,…,m, (1) x_j≥0, j=1,2…,n,such that the objective function Z(X) = sum(|η_k(X)|) k from 1 to N achieves the minimum, whereη_k(X)=b_k+sum(b_(kj)x_j) j from 1 to n,k=1,2,…,N. Generalization. Find an X = (x_1, x_2,…,x_n) satisfying the constraints (1) suchthat the objective functionZ(X) = sum(c_jx_j) j from 1 to n+sum(d_kJ|η_k(X)| k from 1 to Nachieves the minimum (or maximum). A decision theorem for the optimal solutions to model I is given. Besides, analgorithm different from that given in [2]-[4] is presented. At last, some corresp-onding results for the generalization of model I are listed.