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# Slacks and Shadow Values - UNC-Wilmington Cameron School of...

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UNC-Wilmington 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 non-binding 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 non-binding 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 non-binding--only 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 non-binding constraint 1L + 3B ≤ 120. Again, each non-binding 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 per-unit 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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from 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 300--this is the amount of something
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Slacks and Shadow Values - UNC-Wilmington Cameron School of...

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