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A mathematical analysis and critique of activity-based costing using mixed integer programming

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  • The acquisition and elimination of products and the resources needed to create them constitutes an important part of the business decision-making process. Activity-based costing (ABC) supports this process by providing a tool for evaluating the relative profitability of various products. It accomplishes this by allocating costs to products based on the activities, and in turn the resources demanded by those activities, required to produce them. In allocating indirect costs traditionally considered "fixed," such as equipment, administrative overhead, and support staff salaries, ABC treats all costs as variable in the long-run. However, many costs can only vary in discrete steps. For example, one usually cannot purchase a fractional piece of equipment; one chooses either to buy it or not to buy it. Also, in adding support staff, one will typically find that people demand full-time positions, so increments will come in discrete amounts. This stairstep semivariable nature of many costs runs counter to ABC's treatment of all costs as variable. In addition, different products often draw upon the same resources. This creates complex interactions, making it difficult to predict the ultimate consequences of adding or eliminating a particular product. Mixed integer programming (MIP) provides another tool for making these product/resource mix decisions. Unlike ABC, however, it can handle variables that take on integer values, and hence deal appropriately with stairstep semivariable costs. It also ensures that the decision recommended by the model will optimize profitability, given that a solution exists and the underlying assumptions hold true. In doing this, MIP automatically adjusts for all of the complex interactions that exist among the various products and resources. Using a simplified two product/two resource model, one can detail the mathematics behind ABC and MIP, and then link the two approaches through a common variable. This allows one to establish the conditions under which ABC and MIP will yield the same results, and those under which they will differ. Since MW produces an optimal solution, the fact that ABC yields a different result under specific circumstances underscores the danger of relying solely on the product margins generated by an ABC model.
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