LP：积极减少的成本对应于积极变量？

Question

I have the next LP problem

Maximize

1000 x1 + 500 x2 - 500 x5 - 250 x6 

Subject To

 c1: x1 + x2 - x3 - x4  = 0

 c2: - x3 + x5  = 0

 c3: - x4 + x6  = 0

With these Bounds

 0 <= x1 <= 10

 0 <= x2 <= 15

 0 <= x5 <= 15

 0 <= x6 <= 5

By solving this problem with Cplex dual algorithm I get an optimal solution of 6250. But checking the reduced costs of the variables I get the next results

Variable   value    reduced cost
1          10.0          500.0 
1           0.0         -0.0 
2          5.0          -0.0 
3          5.0          -0.0
4          5.0          -0.0 
5          5.0         250.0 

Is it possible to have a positive reduced cost on a positive valued variable? Because the reduced cost value indicates how much the objective function coefficient on the corresponding variable must be improved before the value of the variable will be positive in the optimal solution, what does a positive reduced cost means on a postive valued variable?

Answer 1

Variable 1 is listed twice in the solution?

Note that you need to distinguish between nonbasic at lower bound and nonbasic at upper bound. The reduced cost indicates how much the objective can change when the corresponding bound changes by one unit.

Note also that most textbooks focus on the special case x >= 0 while practical solvers support both lower and upper bounds: L <= x <= U .