# How should you allocate limited capacity?

Formulation of the problem

When the demand for a set of products exceeds what the company can supply (what it has in its inventory plus what it can produce), part of the demand has to be sacrificed. Managers must then know which products or services constitute the best use of available capacity.

A constrained resource is anything that is needed to operate a business and has limited capacity.

To determine which products should be prioritized in production, it is important to identify first the constrained resources, i.e. the resources which may limit the capacity of production. The most constrained resource results in a bottleneck in the production process.

A bottleneck is an operation where the work required exceeds the available capacity. It limits the total number of units which can be produced.

The appropriate procedure and tools then depend on the specifics of the situation, and more precisely whether there is a single constrained resource or multiple constrained resources.

Profit maximization with a single constraint

If there is only one constrained resource, the decision rule is to prioritize the product with the highest contribution margin per unit of that constrained resource (i.e. the unit contribution margin of the product divided by the usage of constrained resource for a unit of that product). When the demand for the first-ranked product is satisfied, production can switch to the next best ranked product. This process continues until either the demand or the resource is depleted. Prioritizing this way results in the highest short-term profit. For this exercise, you can download here data and solutions if you did not already.

Profit maximization with multiple constraints

If there is more than one constrained resource, linear programming becomes necessary.

Linear programming is an optimization technique used to maximize an outcome given multiple constraints.

Linear programming proceeds in four steps. First, build the objective function specifying the relationship between the outcome to optimize and decision variables. Here, the outcome is the contribution margin which should be maximized (fixed costs, prices and unit variable costs are not affected by the decision). The decision variables are the volume $$Q^d$$ of each product $$d$$ which should be produced. Their relationship is set by the unit contribution margin $$UCM^d$$ of each product $$d$$:

$\text{Max for CM} = \sum_{d=1}^{n} Q^d \times UCM^d$

Second, constraints must be defined. For each constrained resource, the sum of the quantities of resource used by each product should be lower or equal to its maximum capacity:

$\sum_{d=1}^{n} Q^d \times RU_j^d <= \text{Max } RU_j$

Where $$RU_j^d$$ designates the usage of resource $$j$$ consumed for every unit of the product $$d$$. Third, you should identify feasible alternatives, i.e. the combinations of quantities $$Q^d$$ which satisfy all the constraints previously identified. If there are only two products, a graphical approach can be used. However, with more than two products you have to use the solver in Excel which adopts an iterative approach to find the optimal mix. Finally, the fourth step is to identify the specific combination or mix which maximizes profit.

Qualitative factors

Prioritizing in such a way does not take into account the importance of the relationship with some customers. Customers whose demand is not satisfied may look for other suppliers. This can be an issue if they represent a large and stable portion of a company’s sales. Therefore, products with a lower contribution margin per unit of constrained resource may still get priority in production because they are destined to important customers.