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Product$-$Mix Decisions With Capacity Constraints

Learning Objective $4$

Know how to choose which products to produce when there are capacity constraints

$...$ select the product with the highest contribution margin per unit of the limiting resource

We now examine how the concept of relevance applies to product$-$mix decisions, the decisions managers make about which products to sell and in what quantities. These decisions usually have only a short$-$run focus because they typically arise in the context of capacity constraints that can be relaxed in the long run. In the short run, for example, BMW$,$ the German car manufacturer, continually adapts the mix of its different models of cars $($for example, $328$i$,528$i$,$ and $750$i$)$ to fluctuations in selling prices and demand.

To determine product mix, managers maximize operating income, subject to constraints such as capacity and demand. Throughout this section, we assume that as short$-$run changes in product mix occur, the only costs that change are costs that are variable with the number of units produced $($and sold$).$ Under this assumption, the analysis of individual product contribution margins provides insight into the product mix that maximizes operating income. The same basic concepts apply in the more general cases except that we then need to consider contribution margins based on variable costs at different levels of the cost hierarchy. In many cases, a manufacturer or retailer has the challenge of trying to maximize total operating income for a variety of products, each with more than one constraining resource. Some constraints may require a manufacturer or retailer to stock minimum quantities of products even if these products are not very profitable. For example, supermarkets must stock less$-$profitable products, such as paper towels and toilet paper, because customers will only shop at supermarkets that carry a wide range of products. To determine the most profitable product mix, the manufacturer or retailer must maximize total contribution margin in the face of many constraints. Optimization techniques, such as linear programming, discussed in the appendix to this chapter, help solve these more complex problems.

Decision Point

When a resource is constrained, how should managers choose which of multiple products to produce and sell?

At the same time, managers work to relieve the bottleneck constraint to increase output and contribution margin. Can the available machine$-$hours for assembling engines be increased beyond $600,$ for example, by reducing idle time? Can the time needed to assemble each snowmobile engine $(2$ machine$-$hours$)$ or each boat engine $(5$ machine$-$hours$)$ be reduced, for example, by reducing setup time and processing time of assembly? Can some of the assembly operations be outsourced to allow more engines to be built? We address these questions in the following section$\mathrm{.}4$

Power Recreation assembles two engines, a snowmobile engine and a boat engine, at its Lexington, Kentucky, plant. The

following table shows the selling prices, costs, and contribution margins of these two engines:

Only $600$ machine$-$hours are available daily for assembling engines. Additional capacity cannot be obtained in the short run. PowerEXAMPLE

$4$

Power Recreation assembles two engines, a snowmobile engine and a boat engine, at its Lexington, Kentucky, plant. The following table shows the selling prices, costs, and contribution margins of these two engines:

$\backslash$table$[[,$Snowmobile Engine,Boat Engine$],[$Selling price,$$800,$$$1,000$

Recreation can sell as many engines as it produces. The constraining resource, then, is machine$-$hours. It takes $2$ machine$-$hours

to assemble one snowmobile engine and $5$ machine$-$hours to assemble one boat engine. What product mix should Power

Recreation's managers choose to maximize operating income?