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?
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.