Shadow price in linear programming

Posted 2020-12-08 jasperzhao

It can be understood as: the value of an addition revenue if the constraint is relaxed.

or

How much you would be willing to pay for an additional resource.

The problem:

maximize 5x1 + 4x2 + 6x3
subject to 6x1 + 5x2 + 8x3 <= 16 (c1)
10x1 + 20x2 + 10x3 <= 35 (c2)
0 <= x1, x2, x3 <= 1

Answer:

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  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.