UNCWilmington
ECN 321
Cameron School of Business
Dr. Chris Dumas
Dept. of Economics and Finance
Slacks and Shadow Values
Analyzing the Solution of a Linear Programming
Problem
In prior handouts, we considered consumer optimization problems that can be solved
using the method of linear programming. After a solution is obtained for such a problem,
the solution is often analyzed further using a variety of methods. Two of the most
common methods are the calculation of Slacks for any nonbinding constraints in the
problem and the calculation of Shadow Values for any binding constraints.
Slacks
A
Slack
is the amount of a resource that remains unused when a constraint is non
binding
. Each
nonbinding
constraint in an optimization problem has a
positive
Slack.
For example, suppose a linear programming problem features the constraint: 1L + 3B ≤
120. In this constraint, 120 is the amount of a constrained resource (money, time,
lumber, nails, etc.) that the decision maker is allocating between two goods, services or
activities, called "L" and "B." Suppose further that the solution to the problem is L =
100, B = 0. Plugging these solution values of L and B into the constraint, we find that the
constraint is nonbindingonly 100 of the available 120 units of the constrained resource
are used in the solution. The amount of the constrained resource that "remains unused" is
120  100, or 20. The value 20 is the Slack associated with the nonbinding constraint
1L + 3B ≤ 120.
Again, each
nonbinding
constraint in an optimization problem has a
positive
Slack.
In contrast, each
binding
constraint in an optimization problem has a Slack equal to
zero
.
If a constraint is binding, then all of the constrained resource is "used up" in the solution,
so the amount of the resource that "remains unused" is zero. For example, in the case of
the constraint 1L + 3B ≤ 120 considered above, if the solution to the problem had been L
= 90 and B = 10, then all 120 units of the resource would have been "used up" in the
solution to the problem, leaving a Slack of zero.
Shadow Values
A
Shadow Value
is the perunit value of a constrained resource. A Shadow Value is
equal to the change in the optimal (max or min) value of the objective that would result
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View Full Documentfrom a marginal increase in the available amount of a constrained resource
.
Each
binding
constraint in an optimization problem has a
positive
Shadow Value.
Whoa! What's that? Let's explore the definition of Shadow Value by way of an
example. Suppose a linear programming problem featured the constraint: 3L + 5B ≤ 300,
and suppose this constraint turned out to be binding when the problem was solved. If the
constraint is binding, then it will have a positive Shadow Value. For this constraint, the
"available amount of the constrained resource" is 300this is the amount of something
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 Fall '08
 Dumas
 Economics, Microeconomics, Optimization, Constraint, Dora, shadow value

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