# Shadow prices in linear programming

I am quite confused about the meaning of shadow price from explanations on the internet.

It can be understood as the value of a change in revenue if the constraint is relaxed, or how much you would be willing to pay for an additional resource.

For example:

$$\begin{array}{ll} \text{maximize} & 5 x_1 + 4 x_2 + 6 x_3\\ \text{subject to} & 6 x_1 + 5 x_2 + 8 x_ 3 \leq 16 \quad\quad\quad\quad\,\,{(c_1)}\\ & 10 x_1 + 20 x_2 + 10 x_3 \leq 35 \quad\quad\quad{(c_2)}\\ & 0 \leq x_1, x_2, x_3 \leq 1\\\end{array}$$

Solving this problem, we get the shadow price of $$c_1 = 0.727273$$, $$c_2 = 0.018182$$.

Comparing $$c_1$$ and $$c_2$$, if one constraint can be relaxed, we should relax $$c_1$$ instead of $$c_2$$?

Here's perhaps a better way to think of the shadow price. (I don't like the word "relax" here; I think it's confusing.)

For maximization problems like this one the constraints can often be thought of as restrictions on the amount of resources available, and the objective can be thought of as profit. Then the shadow price associated with a particular constraint tells you how much the optimal value of the objective would increase per unit increase in the amount of resources available. In other words, the shadow price associated with a resource tells you how much more profit you would get by increasing the amount of that resource by one unit. (So "How much you would be willing to pay for an additional resource" is a good way of thinking about the shadow price.)

In the example you give, there are 16 units available of the first resource and 35 units available of the second resource. The fact that the shadow price of $c_1$ is 0.727273 means that if you could increase the first resource from 16 units to 17 units, you would get an additional profit of about \$0.73. Similarly, if you could increase the second resource from 35 units to 36 units then you would get an additional profit of about \$0.02.

So if you could increase just one resource by one unit, and the cost of increasing the first resource is the same as that of increasing the second resource (this assumption is not part of the model), then, yes, you should definitely increase the first resource by one unit.

• @MikeSpivey: I am looking forward to seeing your book on Operations Research, good explanations of shadow prices
– com
Commented Dec 14, 2011 at 18:44
• @com: Thanks! If I ever do write one, I'll ping your account here. :) Commented Dec 14, 2011 at 18:47
• @MikeSpivey: OK, I'll ask for a signed one :)
– com
Commented Dec 14, 2011 at 19:19
• Where can I find a more mathematical explanation of this (I am a physicist, so by mathematical I don't mean too abstract)?
– a06e
Commented Feb 13, 2015 at 19:10