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Linear programming (LP) is a widely used technique to model production systems. It helps managers make decisions in areas such as production scheduling, product selection, and capacity planning. Besides giving the optimal solutions, LP provides the shadow price or opportunity cost of a constrained resource. Technically, the shadow price is the change in value of the objective function per unit increase in the amount of constrained resource available (i.e., the right-hand side value associated with the constraint in an LP model). The cost behavior of the constrained resource, however, directly affects how one uses the shadow price to make optimal decisions. In other words, the correct opportunity cost of a fixed cost resource is calculated differently from that of a variable cost resource. Many management science and accounting textbooks ignore the implications of the underlying cost behavior of the resource in their explanations of how to use LP [1, 2, 3, 4, 5] . The purpose of this article is to explain how to use shadow prices in making effective managerial decisions for both fixed and variable cost resources.
THE CORRECT OPPORTUNITY COST
In LP, a resource is said to be binding if it is used to capacity. A manager will increase the capacity so long as the contribution from the unit increase in capacity is larger than the cost of acquiring that unit of capacity. Correctly using LP to determine this contribution, however, depends on whether the cost of the resource is fixed or variable. If the binding resource cost is fixed, the contribution of one additional unit of resource is the value of its shadow price; if the binding resource cost is variable, the contribution of one additional unit of the resource is the value of its shadow price plus the variable cost of that resource. Thus, managers who use only the shadow price to compute opportunity cost are likely to underpurchase (purchase too little of) the variable cost binding resources. This differential use of shadow prices relates to where fixed and variable costs are located in a contribution margin income statement. The cost of variable resources are subtracted above the contribution margin while the fixed costs are subtracted below the contribution margin. The following example explains the use of shadow prices...