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I have a LP with equality and inequality constraints.

When solving the LP with the excel-solver (GRG Nonlinear) the sensitivity report returns the lagrange multiplier for all constraints.

When solving the problem with the excel-solver (Simplex LP) however, the sensitivity report returns the shadow price for all constraints.

From my understanding those two should be the same but they are not. Why is this the case? For some constraints they are the same while for others they differ substantially (0 vs. 6.653 for instance).

When I use Matlab linprog I get the same results as the excel-solver (GRG Nonlinear) which confuses me since it uses the dual-simplex. My Gurobi model using the attribute reduced cost on the other hand returns the same as the excel-solver (Simplex LP).

It seems I have a misunderstanding about the equality of the lagrange multiplier and the reduced cost. Can anyone explain where this error comes from?

Model Model

Sensitivity Report comparison Sensitivity Report comparison

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If you look at the "allowable decrease" in the RHS of the highlighted constraint, it's zero. A number of the binding constraints have either allowable increase or allowable decrease zero. That means that your primal solution is degenerate, and your dual problem has multiple optima. The highlighted difference probably means that the simplex solver and the GRG solver found different dual solutions.

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