Why are shadow prices associated with nonnegativity constraints also called as reduced costs, even if they have the same interpretation as shadow prices associated with an optimal solution? Why the use of the term "reduced cost"?
In the simplex method, the term "reduced cost" is used for the marginal cost to introduce a nonbasic variable into the basis. At each iteration, reduced costs are calculated and in a minimization (maximization) problem the variable with the most negative (positive) reduced cost enters the basis via a pivot operation. One can show that the shadow price of a nonnegativity constraint is equal to the reduced cost of that variable in the final (optimal) basic feasible solution.
So "reduced cost" is not an alternate term for shadow price; it's a term related to the simplex method for solving the primal problem.
As @prubin mentioned, the "reduced cost" is not an alternative term for shadow price. Many of optimization software like AMPL, GAMS and others, have an automatic facility to calculate and show these terms.
For instance, in the classical transportation problem (using GAMS), you could try the following to show what you want easily, where the (.m) in the last line is the marginal syntax to show dual of constraints and the reduced cost of variables.
Model transport / all /; solve transport using lp minimizing z; display x.l, x.m, supply.m, demand.m;
The results are:
---- EQU supply observe supply limit at plant i LOWER LEVEL UPPER MARGINAL seattle -INF 350.000 350.000 EPS san-diego -INF 550.000 600.000 . ---- EQU demand satisfy demand at market j LOWER LEVEL UPPER MARGINAL new-york 325.000 325.000 +INF 0.225 chicago 300.000 300.000 +INF 0.153 topeka 275.000 275.000 +INF 0.126 ---- VAR x shipment quantities in cases LOWER LEVEL UPPER MARGINAL seattle .new-york . 50.000 +INF . seattle .chicago . 300.000 +INF . seattle .topeka . . +INF 0.036 san-diego.new-york . 275.000 +INF . san-diego.chicago . . +INF 0.009 san-diego.topeka . 275.000 +INF .
I hope, it would be useful.