Introduction Definition 1.1 A function f(,x2,.·,xn)ofx为，x2,,xn is a linear function if and only if for some set of constants c1,c2,....cn. f (X1,X2,...,Xn)=c1x1 +c2x2+...+CnXn. For any linear function f(x1,x2,...,xn)and any number b,the inequalities f(xM,2,.,xn)≤b and f(灯，x2,.,xn)≥b are linear inequalities. A linear programming problem(LP)is an optimization problem for which we do the following: We attempt to maximize (or minimize)a linear function of the decision variables (objective function). The values of the decision variables must satisfy a set of constraints. Each constraint must be a linear equation or linear inequality. A sign restriction is associated with each variable.For any variable xi, the sign restriction specifies that xi must be either nonnegative (x;>0)or unrestricted in sign (urs). Xi Chen (chenxi0109@bfsu.edu.cn) Linear Programming 5/148Introduction Definition 1.1 A function f (x1, x2, . . . , xn) of x1, x2, . . . , xn is a linear function if and only if for some set of constants c1, c2, . . . , cn, f (x1, x2, . . . , xn) = c1x1 + c2x2 + . . . + cnxn. For any linear function f (x1, x2, . . . , xn) and any number b, the inequalities f (x1, x2, . . . , xn) ≤ b and f (x1, x2, . . . , xn) ≥ b are linear inequalities. A linear programming problem (LP) is an optimization problem for which we do the following: 1 We attempt to maximize (or minimize) a linear function of the decision variables (objective function). 2 The values of the decision variables must satisfy a set of constraints. Each constraint must be a linear equation or linear inequality. 3 A sign restriction is associated with each variable. For any variable xi , the sign restriction specifies that xi must be either nonnegative (xi ≥ 0) or unrestricted in sign (urs). Xi Chen (chenxi0109@bfsu.edu.cn) Linear Programming 5 / 148