Let Z*  max n

j1 cjxj

, subject to n

j1 aijxj bi, for i  1, 2, . . . , m, and xj 0, for j  1, 2, . . . , n

(where the aij, bi, and cj are fixed constants), and let (y1

*, y2

*,..., y*

m) be the corresponding optimal dual solution. Then let Z**  max n

j1 cjxj

, subject to n

j1 aijxj bi  ki, for i  1, 2, . . . , m, and xj 0, for j  1, 2, . . . , n, where k1, k2,..., km are given constants. Show that Z** Z* 

m i1 kiyi

*.