How to Calculate the Shadow Price of a Constraint

The Shadow Price of a constraint in Linear Programming indicates how much the value of an objective (optimal) function changes due to a marginal variation in the right-hand side of a constraint. It is assumed that the rest of the model’s parameters remain constant. It should be noted beforehand that the Shadow Price can be positive, zero, or negative, and these different possibilities will be discussed in the Blog.

Computational tools like Excel’s Solver can be used for obtaining the Sensibility Analysis from a Linear Programming model; however, here we will be focusing on calculating the Shadow Price of a constraint graphically, which will help us later on to understand the concepts behind the Solver results.

Calculate the Shadow Price of a Constraint Graphically

We will now calculate the Shadow Price of a constraint for the following Linear Programming model:

linear programming example

The optimal solution of this model is X=100 and Y=350 with an optimal value V(P)=3,100 according to the graphical Geogebra solution or the Solver solution from Excel. The following diagram shows the optimal solution obtained graphically at vertex C, which corresponds to the intersection of constraint 1 (R1: red color) and constraint 3 (R3: black color), this being the optimal basic feasible solution.

graphical solution linear programming

Suppose we want to know how the optimal value will change (with respect to its current value) if the right-hand side of constraint 1 is raised by one unit but without having to re-solve the problem. The shadow price allows us to answer that question and can anticipate the new optimal value resulting from a marginal variation of the right-hand side of a constraint.

A marginal variation of a right-hand side means that the new optimal solution can still be found using the current active constraints, i.e., those that are satisfied with equality (this conserves the optimal basis).

For constraint 1 if we increase its right-hand side, it will be moved upward in a parallel manner. If we want to guarantee that the new optimal solution can still be found with R1 and R3 active, we reach the vertex where R2 and R3 currently intercept corresponding to the coordinate X=166.67 and Y=350 (this is the maximum variation). In the same fashion if we decrease the right-hand side of constraint 1 and try to maintain R1 and R3 active in the new optimal, the last point where this is guaranteed is at vertex B, whose coordinates are X=0 and Y=350 (this will be the smallest variation). With this information we calculate the shadow price of constraint 1:

shadow price constraint

This shadow price is valid if the right-hand side of constraint 1 (currently b1=1,600) varies between [1,400,1,733.33]. For example, if the right-hand side of R1 increases from 1,600 to 1,700 the new optimal value would be V(P)=3,100+100*1.5=3,250. Similarly if the right-hand side of R1 decreased from 1,600 to 1,550 the new optimal value would be V(P)=3,100-50*1.5=3,025. (It is recommended you confirm these results graphically with TORA or IORTutorial).

Note that if the right-hand side variation of constraint 1 is outside of the interval [1,400,1,733.33], the shadow price cannot be used to predict the new optimal value. In an upcoming analysis we will finish calculating the shadow price of constraints 2 and 3 along with other Sensibility Analysis in solving Linear Programming models.

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